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# Clif Pickover's Extreme Challenges in Mathematics and Morals

## Exclusionary Squares and Cubes

I have a particular penchant for an unusual class of numbers called "exclusionary squares." Can you tell me what is so special about the number

639,172 ?

It turns out that this is the largest integer with distinct digits whose square is made up of digits not included in itself: 639,1722 = 408,540,845,584

Can you find the only other six-digit example? Can you find any exclusionary cubes? (I learned about exculsionary squares from my colleague Andy Edwards. So far, I am unaware of any large exclusionary cubes, and we may wish to generalize this to exclusionary numbers of the Nth order. I will report your "world records" in a forthcoming book.)

### Solutions

```
(ilan at cedara dot com)

Other 6 digit example: 203879^2 = 41566646641

Exclusionary cubes (search up to 1000000):

2^3 = 8
3^3 = 27
7^3 = 343
8^3 = 512
27^3 = 19683
43^3 = 79507
47^3 = 103823
48^3 = 110592
52^3 = 140608
53^3 = 148877
63^3 = 250047
68^3 = 314432
92^3 = 778688
157^3 = 3869893
172^3 = 5088448
187^3 = 6539203
192^3 = 7077888
263^3 = 18191447
378^3 = 54010152
408^3 = 67917312
423^3 = 75686967
458^3 = 96071912
468^3 = 102503232
478^3 = 109215352
487^3 = 115501303
527^3 = 146363183
587^3 = 202262003
608^3 = 224755712
648^3 = 272097792
692^3 = 331373888
823^3 = 557441767
843^3 = 599077107
918^3 = 773620632
1457^3 = 3092990993
1587^3 = 3996969003
1592^3 = 4034866688
4657^3 = 100999381393
4732^3 = 105958111168
5692^3 = 184414333888
6058^3 = 222324747112
6378^3 = 259449922152
7658^3 = 449103134312

A Related Problem
Jonathan Dushoff:
>>> If you drop the requirement that digits in the original be
>>> distinct you can find exclusionary squares of any length.

>> Specifically?
>
Ted Schuerzinger (and Michael Crowder in email):
> Any number of 3's.

Right.  For example, 3333333^2 = 11111108888889.  But there are no
other such series, because

*       1^2 ends in 1
*     222^2 ends in 284
*  444444^2 ends in 469136
*       5^2 ends in 5
*       6^2 ends in 6
*     777^2 ends in 603729
*  888888^2 ends in 876544
* each of 99^2, 999^2, 9999^2, etc. starts with 9.
--

> Right.  For example, 3333333^2 = 11111108888889.  But there are no
> other such series, because ...

True, but only for a narrow definition of "such series".  For example
66...7 works, as does 33...7.  Are there any with longer repeating
components, or with more than two distinct digits?

Jonathan Dushoff

Jonathan Dushoff writes:
> True, but only for a narrow definition of "such series".  For example
> 66...7 works, as does 33...7.

It's narrow to say that these don't fall under the rubric of "integers
*without* distinct digits"??
--

Martin Round:

69^2 = 4761
69^3 = 328509

The square and the cube are exclusionary.  I supose it's unique in
that all ten digits are used.
"William Rex Marshall" :

> 69^2 = 4761
^        ^
> 69^3 = 328509
^          ^
These are not exclusionary.

"Martin Round"
Sorry.  I didn't make myself clear.

The two numbers 4761 and 328509 use each of the ten digits
0,1,2,3,4,5,6,7,8,9 once and only once.

4761 is 69 squared and 328509 is 69 cubed.

I feel pretty confident that the number 69 is unique in that it
generates a square and a cube that are exclusionary to each other AND
use all ten digits just once each.

I suppose there are lots of numbers that generate squares and cubes
that are merely exclusionary to each other?  What is the first three
digit example, four digit example? ... ?

Martin.
```

German Gonzales says at http://www.devweb.cl/mathfun/:

There are 168,569 exclusionary numbers from 1 to 1,000,000 (actually to 98,654) which I Test them for Nth order.

The other 6 digits numbers is 203879

Also I test until 83nth order, which is the biggest exclusionary I found (probably there are a lot more but only with small number of the base) 2^83 = 9671406556917033397649408

This is a serie made from the number of solution versus Nth order; from 2 to 83 nth testing every exclusionary number:

142, 42, 26, 0, 10, 5, 9, 0, 3, 2, 2, 0, 3, 0, 3, 0, 0, 1, 2, 0, 2, 1, 1, 0, 2, 1, 2, 0, 0, 0, 0, 0, 2, 1, 1, 0, 0, 2, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1

See his web site for 168,569 exclusionary numbers from 1 to 987,654 and al list of all exclusonary numbers of 83th order.

## Fragile Fractions

Is it possible to construct the fraction 1/2 by summing other fractions of the form 1/x2? For example, you can choose various denominators as in: 1/32 + 1/52 + 1/102 + ... (but this is not an answer!) The solution must have a finite number of terms. 1) Fragile Fractions of the 1st Kind: No value of x may be greater than 100. No values of x may be repeated. 2) Fragile Fractions of the 2nd Kind: you may use values greater than 100. Once you have found a solution for 1/2, can you construct a solution for: 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9?

### Solutions

```
1/2 = (1/2)^2+(1/3)^2+(1/4)^2+(1/5)^2+(1/6)^2+(1/12)^2+
(1/30)^2+(1/60)^2+(1/75)^2+(1/100)^2
1/3 = (1/2)^2+(1/4)^2+(1/8)^2+(1/24)^2+(1/28)^2+(1/30)^2+
(1/40)^2+(1/56)^2+(1/84)^2
1/4 = (1/3)^2+(1/4)^2+(1/5)^2+(1/6)^2+(1/12)^2+(1/30)^2+
(1/60)^2+(1/75)^2+(1/100)^2
1/5 = (1/3)^2+(1/4)^2+(1/7)^2+(1/20)^2+(1/28)^2+(1/30)^2+
(1/35)^2+(1/60)^2
1/6 = (1/3)^2+(1/5)^2+(1/9)^2+(1/18)^2+(1/90)^2
1/7 = (1/3)^2+(1/6)^2+(1/20)^2+(1/42)^2+(1/60)^2+(1/70)^2+
(1/75)^2+(1/84)^2+(1/100)^2
1/8 = (1/3)^2+(1/9)^2+(1/36)^2+(1/45)^2+(1/60)^2
1/9 = (1/4)^2+(1/5)^2+(1/12)^2+(1/30)^2+(1/60)^2+(1/75)^2+
(1/100)^2

The are also two trivial solutions:
1/4 = (1/2)^2 and 1/9 = (1/3)^2

(ilan at cedara dot com)

David Jones  says
If were were allowed to have numbers repeat in
the denominator, the problem would be trivial

1/2 = 1/(2^2) + 1/(2^2) = 1/(4^2) + 1/(4^2) + repeated
8 times. I'm sure you can see a pattern forming.

For the first problem, the difficutly I think comes in
the fact that in order to get to a fraction like 1/2
is that it is difficult to introduce another
denominator with a prime factor that is not 2. For
example, let say that 1/(3^2) is your first term.
Well, there is no way to add more ninths terms that
get you back to 1/2. You can't cancel either of those
threes out of the demoninator (even if you let x go
over 100). So, the best you can do is only use terms
that are powers of two. But we know that 1/2 = 1/4 +
1/8 + 1/16 + 1/32 + etc and by the rules for the
problem some of these items are missing (like 1/8th)
so we know that we can never get there even we were
allowed an infinite number of terms. This is very
poorly contructed arguement, so I will not claim this
as a fact, but my supposition is that such a set of
fractions that sum to 1/2 does not exist.

Davy
```

## Five Moral, Ethical, & Scientific Questions

1. Aliens descend to Earth and plan to destroy the entire continental U. S. You plead with them not to do this. They compromise. They divide the US into four quadrants with a horizontal and vertical line going through Kansas. You get to save only one of the quadrants. Which do you choose and why? (You may consider many scenarios of your choosing, for example one in which all people and human artifacts are removed but the landscape is unharmed or another in which the destroyed quadrants are submerged under the oceans.)

2. How would civilization be different today if the Mediterranean Sea had always been landlocked. (For example, a strip of land passed between Spain and Africa.)

3. How would civilization be affected if, today, the area of Kansas suddenly disappeared and, in its place, was a large nearly-rectangular lake?

4. You are given the power to cure diseases by touching those inflicted. What is your course of action? (For example, do you keep the talent secret? Do you travel? How would you spend your next week and next year?)

5. Before you is a red button. Press it, and one person, selected at random, dies. However, as a direct, instantaneous result of your pressing the button, the stoning of suspected adulteresses in Islamic countries halts forever. Do you press the button? Would you press the button to save a loved one from certain, imminent death? Would you press the button to save a huge rain forest from imminent destruction -- say the rain forest would be guaranteed to persist for a century? Would you press the button to save the current U.S. President? Would you press the button to save yourself? Next, consider the same scenarios, except that the person, selected at random, is from the town you live in.

I'll assume I can mention your ideas in a book, with credit (for example, I can refer to you by first name and last initial), unless you tell me not to.

Cliff Pickover www.pickover.com

Davie Jones: The Grand Canyon is easily the greatest geological resource on the entire planet. If destroying this quadrant meant losing this resource, it would probably be the one I would save. As heartless as this may sound, people can be replaced; we are already overpopulating the planet as it is. Buildings can be replaced. But natural resources, when they're gone they're gone. If the landscape were not in danger of being harmed, I would probably choose to save Washington D.C.. Having the majority of our government intact would be useful in helping to reclaim and rehabitate the land.

Craig Becker: I'd likely punt and simply pick the quadrant with the most people in it -- a prosaic answer, but there you have it.

Derek Ross: How much time is given? An hour? A month? Is an evacuation possible? If the quadrants will be submerged under the ocean, how quickly will this happen? Will it be a deluge, or will there be more time for people to pack their bags into their yorkboats to get ready for the flood? Hmmm... I would probably accuse the aliens that they are in fact NOT aliens, and they are trying to pull some trick on me.

Peter Barnes: Well, naturally, I'd choose to "destroy" those quadrants where I wasn't. But OTOH if we were, as a country, to have any hope of recovering from such a disaster, I would say that we must preserve untouched that quadrant containing the largest accumulation of our resources, especially our intellectual treasures. So I would choose the Northeast quadrant to be preserved (but give me a couple of minutes to pack my bags for the trip to Boston).

"lonno7" : 1. Let's see: the Northwest has a relatively sparse population and its primary technology center is Seattle, home of Microsoft – so it goes. The Southeast has Cape Canaveral (which theoretically could be moved – or at least reproduced somewhere else) and Nashville, the home of country music – so it goes. The Southwest has Hollywood, New Age followers of every stripe, malls enough to populate a small planet, and Michael Jackson – so it goes. The Northeast has New England, the birthplace of our country, as well as agricultural, mineral and economic resources and proximity to Europe. Plus, my wife and I have always wanted to live in New England, anyway.

From Howard Rogers :
First of all, what have you got against Kansas?

Q1. I choose to save the s.w. quadrant, because that's where I live.

David Jones: I know I've seen this question somewhere before, but I can't recall where. I'm really not gifted enough of a historian to be able to make very accurate claims, but here are a few guesses.

Egypt may not have lived as long as it did. One of Egypt's most valuable resources in terms of defense is that it was very hard to get to. It was surrounded by desert on all sides but North. They were able to dissuade conquerers because they were already worn out by the harsh conditions once they got there. Still, it would have been a long journey from Spain to Egypt, but it provided the Greek and Roman Empires better access to the land.

Probably more interesting would be to see how it would have affected Greek Mythology. If I remember correctly, the River Styx was the path out of the Mediterranean which the Greeks beleived led to Hades, the land of the dead. With no River Styx, the vision of the afterlife would have been very different. Also consider that religious historians beleive that the Christian concept of hell was based on Hades. If closing the Mediterranean would have such a profound affect on Greek religion, such changes most probably would have an effect on Christianity's view of the afterlife as well.

Craig Becker We would all be wearing clown suits. Except for the Pope, who would be naked.

Derek Ross: There would also be the Mediterranean canal to go along with the Suez and Panama canals.

Pete Barnes: I'll venture that the influences of the Arabic and Middle Eastern cultures would have been vastly more widespread. For example, we may have had the use of our present system of numbering, originated by the Arabic culture, long before the middle ages. Since the MidEastern and Mediterranean civilized ares were far more advanced in culture in general and far more interested in preserving and spreading ideas than was the Western world, we _probably_ would have been much better off.

"lonno7" : It would be impossible to even guess at this, since such a seemingly small change in world geography would have enormous impacts on a number of things, such as ease of transportation between Europe and Africa (the Strait is notoriously dangerous to cross) would have had significant impacts on the local civilizations and cultures, as well as weather and climate (the warm, hypersaline water has a direct effect on the Gulf Stream, which affects Europe's weather). Considering the so-called "Butterfly Effect" of Chaos Theory, Europe might have had longer and more severe ice ages, which would have put that civilization's focus more toward mere survival than toward the advancement of learning and artistic development. We might have been only now going through the Renaissance.

From Howard Rogers : Q2. I suspect not very different - Europeans and Africans would have been able interact, Spanish and Portuguese explorers would still have been able to make their voyages. There may have been a greater Moorish influence in the Iberian Peninsula, but then again, the Romans would have been to pursue an aggressive colonization of the Maghreb.

David Jones: Umm, I don't know. The price of salmon would be cheaper because we wouldn't have to import it from Alaska? It would certainly change crop growth for the rest of the country because all that farmland would be lost.

Craig Becker: As someone noted earlier, that's a lot of prime farm land gone. And I'd speculate that the introduction of that large a body of water could cause some change to the climate. Oh, and doubtless there'd be a lot of major development along this new waterfront property.

Derek Ross: The US Gov will work hard to drain the lake while providing irrigation to the surrounding areas. They would also try to find out who was responsible, then they would go to war with them, and SMASH them!

Well, price of Cheerios would probably go way up, and the NCAA football playoffs would be somewhat less interesting :=). However, since it would be a vast freshwater resource, we would immediately have about a dozen or so political entities battling over water rights there ad infinitum (or ad naseum...). Of course, the Republicans would immediately try to grant 1000-year leases for development of the entire area to the biggest corporate donors to their party, while the Demaocrats would immediately try to impose a 10% tax on every gallon of water drawn from the lake in order to finance the administrative costs for the vast new government agency needed to control the area.

Peter Barnes: Just a footnote: that Lake Kansas would be about 82,282 sq. mi. in area, almost as big as the combined Great Lakes of North America, which at 95,000 sq. mi. of area constitute the largest body of fresh water in the world. Pete B

"lonno7" : Given the state's stand on such a basic scientific concept as evolution, the overall intelligence level of the U.S. would rise appreciably.

From Howard Rogers :

Q3. Negligible. But, we'd have to watch out for the believers in the Apocalypse trying to do something foolish.

David Jones: This is a rather scary proposition. I've commented before that overpopulation of the planet is quickly becoming one of Earth's greatest problems. I wouldn't want to see people suffer from illness, but at the same time I can see where if such a power was used optimally for saving life that I could be adding to the problem. If this were public knowledge would I ever be able to get a full night's sleep from all the people begging me to cure their little boy who was told by the doctor that he will never walk again or the little girl who has just two weeks to live because they can't find a kidney donor? What if I were kidnapped by a military organization so that soliders wounded in battle could be cured just to go back out and fight again? I'd have to keep it a secret, but how and if I used it, I'm not sure. I'd have to think about it some more.

Craig Becker: It's interesting to speculate on whether or not one could turn this into a 9-5 job. If all that's needed is a touch, then you could put the sick folks on a big conveyor belt and run 'em past you at a steady clip, you stick out your hand and touch 'em all. Sure, there might be emergency cases where you get a call at 2am, but in some ways this wouldn't be too far from how doctors operate today.

And I'll bet the insurance companies would love this -- a sick person could be flown to your office for less \$\$\$s than a

typical hospital stay. The real trick would be to engineer this in such a way that you didn't get kidnapped by some person or government and used as a private "resource".

Last: years ago, F. Paul Wilson wrote a good SF novel called _Healer_ that touches on this a bit.

Derek Ross: I'd grow a beard, let my hair grow out, wear a bath robe, buy a pair of sandals, and tell the world, "I'm Baaaaack!"

Pete Barnes: I would spend the year touching as many of the ill as possible, but I fear one with such power would either be kidnapped or killed, long before the year was up, by those wanting to gain control of the gift.

"lonno7" : Obviously, selectivity is key here; no single individual could cure all the world's sick people in only one lifetime. Of course, some selfishness would likely play a part – I certainly would cure my wife's Interstitial Cystitis in a flat second if I could. Family, friends and those who would have the greatest beneficial impact on society would be my guiding criteria.

From Howard Rogers :

Q4. Maybe I would cash in on it faith healer style. I suspect that I would keep it a secret and go about my days as normal but I would probably visit my remaining family for a "check-up".

David Jones: No. The bottom line here is that by pressing the button I am killing somebody. For that person, the responsibility of that death lies on me. And, not to sound selfish, but I haven't really made any better change in the world in exchange for accepting that responsibility. For the woman being stoned, while I do have sympathy for her, what will this really accomplish other than to trade a life? The people stoning her won't change their beleifs; they will be more than happy to stone the next woman who the feel is wrong. I've accomplished nothing. Besides, if her life is really that horrid that she has to live in such a society, I would think she would be better put out of her misery. I hate to put a value on other people's lives, but consider that the person I kill by pressing the button life a happy life in a liberated country. If I had to choose one person to kill, should I kill the happy free woman or the repressed one? Finally, if you believe that there is some sort of universal justice (God, kharma, whatever), and I do, I've just let these @\$\$&0!#\$ off the hook and agreed to add their negative karma to my life balance. They were doing something bad, and then they stopped. It's like I've handed them a get out of jail free card and landed on the "Go to Jail" space at the same time.
```> Would you press the button to save a loved
> one from certain, imminent death?
```
Much harder question. I should say no for a variety of ethical reasons, but it's a lot easier to see what you gain personally out of such a bargain. I can't honestly say that I would have the fortitude not to push it, espically if by "imminent" you mean that I don't have more than a few minutes to ponder the situation. > Would you press the button to save a huge rain > forest from imminent destruction -- say the rain > forest would be guaranteed to persist for a > century? No. Going back to the karma arguement, I'm letting somebody off the hook for destroying the forest and accepting the blame for a death. Much like the stoning, there is no guarantee that they just won't go do it to a different rain forest or other valuable natural resource. And what after 100 years? If we still haven't learned how to respect our ecosystem it will be gone then. See, the deal has involve some sort lost lasting educational benefit to all of humanity before it becomes worthwhile. In situations like this, its not enough to just stop people; they have to understand why its important to stop. Besides, if humanity is so stupid as destroy a major source of our oxygen supply, maybe the planet would be better off if we all became extinct as a result of it. Nature manages in its own way.
```> Would you press the button to save the current U.S.
> President?
```
I could give quite a few reasons why I would be tempted to NOT push the button, but I appreciate the fact that this group has never become overly political and I'd like to keep it that way, so I'll keep this one under my hat. But even if it were a President that I valued, still probably not.
```> Would you press the button to save yourself?
```
See "loved one"
```> Next, consider the same scenarios, except that the
> person, selected at random, is from the town you
> live in.
```
Wouldn't change anything.
```> I'll assume I can mention your ideas in a book, with
> credit (for example, I can refer to you by first
> name and last initial), unless you tell me not to.
```
Heck, I'd actually prefer David C. Jones so I could tell all my friends "Look everybody, I'm in a book!"

From: Quinn Tyler Jackson
```

If I this Button before me now press,
Someone unbeknownst to me perish,
Yet someone saved whom I dearly cherish,
For Greater Good this Act -- oh what distress
To make such choice! And yet, is this choice less
Ethical than others make when they wish
To rid the world of evil with their harsh
Wars and soldiers, who to their duties pass
To service of a greater call? To halt
My hand in uneasy ethic to save
Some unknown at the cost of one I love,
At this my conscience would later find fault,
And thus, I'd press and pray forgiveness brave,
Rather than some notion of ethics prove.

II

To end the madness that is stoning death,
To save a multitude from fate rock hit,
Again, I'd play at being God, unfit
Though I am, even for a single breath,
To pluck that random price from all the wealth
Of those who would thus be from my choice hit,
And though some random martyr pay for it,
I'd ask forgiveness for my act uncouth.
O fickle Button! Why do you now place
Such power over life and death on me?
I am not worthy to yield such Decide,
But since no other has fall'n to my place,
I'll press you, damn it, though respectfully,
And pray my decision I may yet abide.

-- Quinn Tyler Jackson (if you use it, credit me as you will)
```

Craig Becker: Just a general comment: whenever I see questions like this, I find myself making the distinction between "What *should* I answer?" and "What would I *really* do?" Of the above, the only question I can answer with any certainty is re pressing the button to save someone I love: when it's a matter of my son or daughter vs some Random Stranger, well -- I'm sorry, Random Stranger . . . Oh, also, consider the psychological consequences of being put in the situation where one has to _make_ this kind of decision. I'm reminded of the movie _Sophie's Choice_, wherein Meryl Streep is forced to choose one of her children to be sent to the ovens, so that the other child would be spared. As I recall, she had a few 'issues' afterwards.

Derek Ross:
```> 5. Before you is a red button. Press it, and one person, selected
> at random, dies. However, as a direct, instantaneous result of your
> pressing the button, the stoning of suspected adulteresses in Islamic
> countries halts forever. Do you press the button?
```
This variation of this question has been asked already: "Before you is a red button. Press it, and ten-million prople, selected at random, die. However, as a direct, instantaneous result of your pressing the button, the form of government known as communism halts forever.
```> Would you press
> the button to save a loved one from certain, imminent death? Would
> you press the button to save a huge rain forest from imminent
> destruction -- say the rain forest would be guaranteed to persist for
> a century? Would you press the button to save the current U.S.
> President? Would you press the button to save yourself? Next,
> consider the same scenarios, except that the person, selected at
> random, is from the town you live in.
```
Since I don't believe in magic, I always wonder how exactly the button would work. I think there would be much profit to be made if one could invent a button that killed people at a distance.

Pete Barnes:
```5. Before you is a red button. Press it, and one person,
selected at random, dies. However, as a direct, instantaneous result of
Islamic countries halts forever. Do you press the button?
```
No, let nature take its' course
```> Would you press
> the button to save a loved one from certain, imminent death?
```
Probably, after all who wouldn't?
```>Would you press the button to save a huge rain forest from imminent
> destruction -- say the rain forest would be guaranteed to persist
for a century?
```
No, not worth the human life.
```Would you press the button to save the current U.S. President?
```
Not if it meant someone else arbitrarily and undeservedly dies. What that amounts to is trading one life for another. No.
```> Would you press the button to save yourself?
```
Same as above re loved one, yes, human nature prevails.
```> Next, consider the same scenarios, except that the person, selected at
> random, is from the town you live in.
```
No change

"lonno7" :
```·	Would you press the button to save a loved one from certain,
imminent death?
Absolutely.

·	Would you press the button to save a huge rain forest from
imminent destruction – say the rain forest would be guaranteed to
persist for a century?
Survival of the rain forest would have an impact on more than just
one person, so yes.

·	Would you press the button to save the current U.S.
President?
I would be gambling on the possibility that the randomly chosen
person might have more long-term positive influence than a barely
elected good-ole-boy who can barely utter a coherent sentence
(besides, he has enough brainpower surrounding him that his removal
would barely be registered).  No.

·	Would you press the button to save yourself?
Yes. (Is an explanation really necessary?)

·	Next, consider the same scenarios, except that the person,
selected at random, is from the town you live in.
The initial "one person, selected at random" to me implied anyone on
Earth – meaning that the odds of a friend or family member's being
selected were exceedingly low.  However, limiting the choice to a
population of a few hundred thousand greatly increases those odds …
so No.
```

From Howard Rogers :

Q5. No, I would not press the button to stop stonings. I don't think that I would press the button to save a loved one. I don't think that I would save the rainforest - but I could be swayed. To save the president - of course not! To save myself - of course! The point of origin of the random person is immaterial. A button that could people at a distance may be useful - see Star Trek episode "Mirror Mirror" for more information on the Tantalus Device. Wars could be nipped in the bud - of course, I have sole, permanent control the button.

## The Problem of the Rich Jeweler

Your friend, a jeweler in New York city, walks you to a large room filled with three kinds of valuable objects: gems, cubical chunks of special alloys, and bottles of rare spices.

Before you are four kinds of gems: one pink, another beige, a third yellow, and the last green. Among the gems, the number of pink gems is equal to one-fourth plus one-third the number of beige gems plus the number of green gems. The number of beige gems is one-seventh plus one-third the green gems plus half the number of yellow gems. The number of green gems is one-fifth plus one-eleventh the (pink plus the beige) gems.

We also are gazing at four colors of precious alloys. Among the alloys, the number of pink alloys is one-half plus one-fifth of pink objects in the room plus twice the number of yellow alloys. The number of beige alloys is one-third plus one-half the total of the green alloys. The number of green alloys is one-fifth the number of pink alloys plus one-sixth the total number of the yellow alloys, and the number of yellow alloys is one-eighth plus one-third the total of the green objects in the room.

There are four colors of spice bottles. The number of pink spice bottles is one-half the number of beige objects in the room. The total number of green and yellow spice bottles is equal to the total number of pink and beige spice bottles.

What is the least number of precious objects that are in the room?

### Solution

David Jones gives his answer here:

http://www.geocities.com/davypi/jeweler.html

David writes: In the problem, there are three types of objects: Alloys, Gems, and Spices. Each type has four colors: Beige, Green, Pink, and Yellow. Each item will be referred to with two letters, the first by color and then second by type. For example, Pink Spices will be annoted by PS, Beige Alloys are annotated by BA, and so on.

The second paragraph of the problem talks exclusively about gems. The problem will be started and dealing with alloys and spices will come later. An elementary interpretation of the paragraph yields the following equations, labeled A, B, and C.

• A: PG = (1/4 + 1/3)BG + GG
• B: BG = (1/7 + 1/3)GG + YG/2
• C: GG = (1/5 + 1/11)(PG + BG)

Algebraic manipulation allows us to rewrite these equations as:

• A: (7/12)BG + GG - PG = 0
• B: -BG + (10/21)GG + (1/2)YG = 0
• C: (16/55)BG - GG + (16/55)PG = 0
The table below represents the minimum solution, according to David.
```Color	Type	Variable	Amount
Blue	Gems	BG		98280
Green	Gems	GG		63840
Pink	Gems	PG		121170
Yellow  Gems	YG		135760

Blue	Alloys	BA		871035
Green	Alloys	GA		1045242
Pink	Alloys	PA		4802600
Yellow  Alloys	YA		508332

Blue	Spices	BS		105
Green	Spices	GS		6
Pink	Spices	PS		484710
Yellow  Spices	YS		484809

Total				8615889
```

As corollary, I wondered how much room would be required to store all of this. David says: The total number of gems is 419050, alloys is 7227209, and spices is 969630. When you go to geologic tourist trap stores near Yosemite or The Grand Canyon, you can usually by sample gems and rocks that are no more than 1/2 inch in each direction, so assume that a gem or alloy sample is 1/8 in^3 in volume. In geology class, I have seen powder samples stored in test tubes 3/8th inch across and 2 inches tall, so assume 9/32 in^3 for spices. This totals up to a net volume of just under 711 cubic feet. Assuming that the walls are 8 feet high as in most houses, you would need a room with 89 square feet of floor space just to store all of the object. Keep in mind that this does not count whatever boxes or materials you use to store the objects or the space you would need for a human being to walk through to retreive items. I would think that a room the size of a normal high school chemistry lab would be sufficent.

## New York Gambit

For 40 million dollars, tax free, would you spend the rest of your life confined to New York City, unable to leave the city for any reason?

### Solutions

Marcus Rauchfuss : No way! I could imagine doing that if Rhode Island or San Francisco would be offered. But NOT New York City. April P says: I'd answer "Yes" to that in a New York minute!! The city has everything I could possibly want. After all, I'm most likely going to be staying here in Reno for the rest of my life, with only a few excursions to Calif, (which I could live without for 40 million bucks), so if I'm content here for free, I'd jump at the chance to stay in NYC for all that money! April

Peter Barnes: Nope, not a chance. What if I got the 40 mil, but came down with a life-threatening disease the week after, the only cure for which was offered by a doctor in a foreign country who absolutely refused to travel to New York because his hostile government would not allow him to go unless he used his children as ransom for his return? Or my child was held hostage in, say, Arkansas by a demented lunatic demanding I personally come there to negotiate with him or my child woud be killed? A no-brainer choice, IMHO Pete B

April responds: But Pete, most likely those scenarios would not happen. Using that logic, you'd better not get out of bed tomorrow morning, because on the way to work a cattle truck might overturn and cause a traffic pileup, leading you to be a half-hour late, but right on time to be the victim of a drive-by shooting.

[Cliff says, "New York City has the best doctors in the world. If I had to get sick, it would be in New York. On another topic, has anyone wondered why April is the only person of the female persuasion who routinely posts to this discussion group? What do you think this means?"]

Pete Barnes: April, we are not talking about traffic jams here. We are talking about NEVER being able to leave New York City. That stipulation is just as unlikely as the one I posed. I claim equal time to improbable events!!

The doctor scenario is just one of many, many others of equal or greater peril that could arise. Here is just one more such scenario: scientists find a comet headed straight for Earth, it is going to hit the Atlantic Ocean off the coast of New Jersey. Fortunately, it is not big enough to cause any of the doomsday events predicted now and then, but it will cause a massive tsunami to strike the East Coast. So starting tomorrow, we will be evacuating New York and all other coastal cities until it passes a week from now.

Darn, and just when I was getting ready to go shopping with the \$40 Mil I got last month, now I am going to drown instead....
```> [Cliff says, "New York City has the best doctors in the world.
> If I had to get sick, it would be in New York.
```
Probably true, but what if the only treatment and diagnostic equipment for your illness was offered at the Mayo Clinic in Minnesota? Myself, I would hate to be restricted.

Lon says: Unable to leave NYC for *any* reason? Given that I've never really wanted to live in NYC, and I have lived many other places (England, Germany, Colorado, So. Calif.) and have visited even more, what good would \$40 MM do me if I had to be impriso- I mean, confined - in NYC. The world has much more to offer me than does NYC. Also, I fully agree with peteb re: illnesses and other emergencies. lon

Derek says: It would be sad if I was trapped in NY and all my friends were going to Mars for holidays. The rest of your life is a long time... you don't know what kind of things will be possible in 5 decades.

April says: * True, but anything that's worth doing is bound to show up in NYC. And even if vacations to the moon or to Mars became available, I could live without them. I'd be too old anyway to enjoy such excursions by the time that kind of stuff came along.

Craig Becker : April asks: (and how would escape from New York be prevented BTW?) How about a bomb implanted in a very difficult to reach part of one's body that will detonate if it goes outside of a certain set of GPS coordinates? I'm sure there are other ways, too. Craig

David Jones :
```--- Pedersen  wrote:
> * Ha. But freakish events can strike anywhere at any
> time. Might as well have fun spending the 40 million
> carpe diem style!
```
Ok, but what about less freakish events? What happens when my best friend gets married? Can't go to the wedding unless I'm willing to pay to fly the whole party to NYC (that is, if the bride and groom are willing). When my dad is on his death bed, am I going to be there? When both mom and dad die how will I take care of the estate? Ok, I guess for 40 million I can move the family out to live with me before any of that happens. I've always wanted to go scuba diving off some tropical reef where I can swim with the fish. That one's out the window for sure. (I suppose not if they have an aquarium, but what kind of grant would I have to give for a dip in the tank?) I'm definitely never going to get to see the Pyramids of Egypt which I've wanted to do for most of my life. For me personally, I have asthma. I don't know that the higher pollution levels would good for me and no amount of money is going to cure that. I think the main thing is that being just under 30, I have too much life left to life. 40 million dollars probably comes out just enough to comp for all the things I would miss doing, miss out on never being able to do, and the expenses of transporting the valuable things of my life with me. Make the offer again in, say, 20 or 30 years and I would probably take it.

From: Howard Rogers
Of course I'd take the 40 million. I've never been to New York, and would view this as an extended vacation. Also, I'm a junior college geography instructor, and come into contact on a daily basis, with people who have never been more than 30 miles from downtown LA. Sad isn't it?

## The Search for Isoprimes

11 is an isoprime, a prime number with all digits the same. Do any other isoprimes exist? 101 is an oscillating bit prime. Do any others exist? For example, 10101 is not prime. Neither is 1010101. Cliff www.pickover.com

### Solutions

David Jones: If any more exist, they would also have to be all ones. If you take any other number, say repeating 3's, the number is automatically divisible by 3. Furthermore, the number of 1s in the number would have to be prime itself. Say I string together 15 1s. Since 3 and 5 divide 15, I can quickly conclude that this number is divisible by 111 or 11111. [Cliff says, "Do you think humanity will ever find a 1111... isoprime?"]
```> 101 is an oscillating bit prime. Do any others
> exist? For example, 10101 is not prime. Neither is
> 1010101.
```
Much like the example above, we know that such primes will have to end with the same number they start with. For example, 737373737373 can't be prime since 73 divides it. We need to add a seven to the end to make it a viable candidate. I don't have any other brilliant leads beyond this.

I seem to recall a few months ago that you proposed other people come up with special classes of numbers. A few suggestions were made about "heavenly primes" and the like. If memory serves, some of them were oscillating primes and a handful of them were found. Davy

chuck2051" says: According to The Penguin Dictionary of Curious and Interesting Numbers, the next isoprime is 1,111,111,111,111,111,111. The third one is 11,111,111,111,111,111,111,111.

David Jones: Well, the first couple aren't that hard to find. Define a function ISO(n)= n number of ones. For example, ISO(2)=11, ISO(3)=111, ISO(4)=1111 and so on. By toying with Maple I found that ISO(19) and ISO(23) are both prime which I verified with my 49G. I think I tested it up to about ISO(83) before I put it down. I could have easily made an error as I was entering numbers by hand and may have miscounted digits. I don't know Maple well enough yet that I could write a script for it to create and test numbers for me and PrimeForm won't let me define functions like that. Maybe tommorrow at work I'll play with it a bit more.

Michael Benedetti says: In the world of factoring and primality proving 11 is considered a repunit prime. All repunit primes in base 10 can only be composed of 1's. the others known have 19 digits, 23 digits, 317 digits, and 1031 digits. The next 2 which are believed to be prime but are not proven such yet are 49081 digits and 86453 digits. For any info regarding prime numbers there is a great web site run by Chris Caldwell at http://primes.utm.edu/

Daniel Dockery: Unless I err, the fourth isoprime would be:
```
11,111,111,111,111,111,111,111,111,111,
111,111,111,111,111,111,111,111,111,111,
111,111,111,111,111,111,111,111,111,111,
111,111,111,111,111,111,111,111,111,111,
111,111,111,111,111,111,111,111,111,111,
111,111,111,111,111,111,111,111,111,111,
111,111,111,111,111,111,111,111,111,111,
111,111,111,111,111,111,111,111,111,111,
111,111,111,111,111,111,111,111,111,111,
111,111,111,111,111,111,111,111,111,111,
111,111,111,111,111,111
```
(317 digits) Both Primo and Maple certify the number as prime. At 20020127T0850, Cliff wrote:
```> 101 is an oscillating bit prime. Do any others exist?
> For example, 10101 is not prime. Neither is 1010101.

```
I've found no others less than 10**700 (the highest I've searched so far). Daniel

David Jones: Just to give you an idea of how hard it is to find these things (isoprimes), lets go back to our problem of Specially Augmented (aka "Jim") Primes from a few months ago. Jim Primes and Repunit primes grow at about the same rate for N. On November 4th I complete proving that n=3,011 was prime. At the same time, I had a separate computer looking for more and it was at n=23,000. Today, 85 days later, I am testing n=47,461. My N value currently goes up by about a 1,000 every week.

Using a log(log(n)) fit to predict where the next SAP should be, it should have been at about 39,000 but it didn't appear. The next one should be at around n=245,000. I haven't quite figured out a reliable timetable yet, but an optimistic prediction shows that I won't get there until about April of next year unless I either get a faster computer, a faster testing program, or more help.

If/When I do find one, it will only have passed a preliminary test for "probable" primality. Doing a full primality proof on n=3011 took a couple of weeks if I remember correctly. I would imagine that proving a prime on n=245,000 would take at least three months. The individual proving ISO(86,453) has my best wishes as I would hate to lose months of computing time just to find out my candidate is NOT a prime number.

The moral of the story here though is that, now days, beating any sort of prime record requires a ton of patience and even more persistence. Davy

Daniel Dockery:At 200201281158, David Jones wrote:
```> The individual proving ISO(86,453) has my best wishes
> as I would hate to lose months of computing time just
> to find out my candidate is NOT a prime number.
```
On a first run, PFGW yielded a 3-prp result for the 49081 digit item after about 10 minutes (with thread priority; all the other runs were set for only idle cycles), and a "composite" result for the 86,453 digit item after about 45 minutes. A second run repeated 3-prp for 49081, but yielded 3-prp for 86453 this time, both with about the same speed. Running a primality test (pfgw -t -f -l) cycled through, I think, base 17 before declaring 49081 composite; time was close to 30 minutes. I haven't yet attempted the full test on 86453, and may not at this point. Daniel

" Cliff wrote:
```>
> 11 is an isoprime, a prime number with all digits the same. Do any other
> isoprimes exist?
```
It's conventionally called a "repunit", standing for "repeated unit", i.e. string of 1s. A list of the largest known repunits can be found on Professor Caldwell's Prime Pages http://primepages.org/
```In particular:
http://primes.utm.edu/glossary/page.php/Repunit.html

> 101 is an oscillating bit prime (OBP). Do any others exist? For example,
> 10101 is not prime. Neither is 1010101.  Humans have not disovered any
> other OBPs, as far as I know.
```
What base is that in? You mention 'bit', so should I assume it's base 2? In which case, these are simply the repunits in base 4. If it's base 10, then these are simply the replunits in base 100. Repunits in bases other than ten are typically called Generalised Repunits. In base 10, primes with the generalisation of this pattern are typically called "undulating". There have been some very famous undulating primes, such as Landon Curt Noll's '37' one. Rudolf Ondrejka's Top Ten lists may well contain some of your '101...01' type , it contains all sorts of things. http://www.utm.edu/research/primes/lists/top_ten/
```
> After you read through the discussion on this subject at:
> http://sprott.physics.wisc.edu/pickover/extremec.html
>
> you'll discover that the largest isoprime published in a book is:
>
> 11,111,111,111,111,111,111,111.
```
Book? What kind of 16th century medium is that? The Cunningham Project has been finding factors of numbers of such form (and thus finding primes too) seemingly for ever, well over a decade. http://www.cerias.purdue.edu/homes/ssw/cun/
```> And, we think we have discovered a new one:
>
> 11,111,111,111,111,111,111,111,111,111,
> 111,111,111,111,111,111,111,111,111,111,
> 111,111,111,111,111,111,111,111,111,111,
> 111,111,111,111,111,111,111,111,111,111,
> 111,111,111,111,111,111,111,111,111,111,
> 111,111,111,111,111,111,111,111,111,111,
> 111,111,111,111,111,111,111,111,111,111,
> 111,111,111,111,111,111,111,111,111,111,
> 111,111,111,111,111,111,111,111,111,111,
> 111,111,111,111,111,111,111,111,111,111,
> 111,111,111,111,111,111
>
> My question: do you think humanity will ever discover a larger
> isoprime?   Upon what do you base your reasoning?
```
You've obviously never heard of Harvey Dubner! (Having said that it wasn't Harvey but Lew Baxter who found a probable repunit prime with 86453 digits only 15 months ago.) The largest known _proved_ repunit is the 1031-digit one found to be PRP by Harvey Dubner some time in the mists of time, but proved to be actually prime only more recently. Phil

Jan Kristian Haugland :

```George Weinberg wrote:
> On Wed, 30 Jan 2002 08:40:04 -0500, Cliff
> wrote:
> >11 is an isoprime, a prime number with all digits the same. Do any other
> >isoprimes exist?
>
> Funny thing, when I first read this I somehow got the
> impression you were talking in binary,  maybe it's
> the oscillatong but thing.  11 seems to be prime in quitre a lot of
> bases.

Yes, it "seems" to be prime whenever the base is a
prime minus one. ;-)

(...)
> >My question: do you think humanity will ever discover a larger
> >isoprime?   Upon what do you base your reasoning?
> >
>
> No.  Didn't read the discussion, so I may be wrong, but I think
> that the frequency of isonumbers falls off fast enough and the
> frequency of primes falls off fast enough that even going out to
> infinity there's likely only a finite number.

The frequency of primes falls off, but not very fast, so
it is not unreasonable that there may be infinitely many.
That however doesn't necessarily mean humanity will find
too many others. Compare with Mersenne primes.
```
>Phil Carmody wrote:
```>
>>> My question: do you think humanity will ever discover a larger
>>> isoprime?   Upon what do you base your reasoning?
>
>> Facts and research.
>
>In 1993 Dr. Ron Rivest did a calculation on the number of MIPS-years
>it takes to factor a number (it's in the appendix of my O'Reilly PGP
>book, dunno if the calculation has been updated). You could probably take
```
the iso-prime research and run it through his calculator to determine >when humanity could find the next isoprime. You'll need to take into account improvements in factoring -- I believe there have been several since 1993 and I see no reason to believe there won't be more. -- Matthew T. Russotto mrussotto@speakeasy.net --

## Galactic Set A or Set B

Astronomers estimate that there are about 125 billion galaxies in the universe. (All the stars visible to our naked eyes are part of Earth's galaxy, the Milky Way.) Aliens come to you and give you a horrifying choice. Set A contains the Milky Way galaxy. Set B contains every galaxy in the universe, except for the Milky Way galaxy. Due to some strange physical phenomena all the stars in either set A or set B will experience an accelerated death (give off no light or heat within about a decade). You have a choice to save set A or B. Which do you choose? Cliff www.pickover.com

### Solutions

Derek Ross says:I'd find out where the aliens are from, and choose the set that results in their solar system being wiped out. That'll learn 'em for posing these conundrums! Derek.

April responds to Derek: But what if their solar system was in set A, the set that contains our galaxy? Is revenge worth our galaxy's destruction? Then again, if left alive, the aliens could pose this question again until someone chose set B to be saved. Hmm, now I'm thinking, does it really matter which set the aliens live in? Regardless, I'd choose to save set A on the anthro-centric biased assumption that the Milky Way is the only galaxy in the universe that has intelligent life until proven otherwise.

Marcus Rauchfuss says: The logical choice would be choice B. Save the many and sacrifice the few. However, I would also kill the aliens who want me to choose. On second thought: If the aliens are not from the Milky Way I should choose the Milky Way out of sheer spitefulness. After all, it's the aliens fault to choose me to choose.

"lonno7" :"Save the many and sacrifice the few?" The many what? Stars? Gasbag planets? Comets? Asteroids? Are you assuming there are, in fact, other advanced civilizations "out there" on the basis of no evidence whatsoever (except for a couple of aliens) and that we should snuff ourselves for them? for remotely possible civilizations that might not even exist at all? Everybody, keep this guy AWAY from the button! My opinion: Let's see, all the stars in the universe that have no effect on the Earth, or all the stars in the Milky Way -- including our Sun. I'm not sure I see the dilemma here, but for the record, I'm all for saving our home galaxy.

```--- Pedersen  wrote:
> * But what if their solar system was in  set A, the
> set that contains our galaxy? Is revenge worth our
> galaxy's destruction? Then again, if left alive, the
> aliens could pose this question again until someone
> chose set B to be saved. Hmm, now I'm thinking, does
> it really matter which set the aliens live in?
```
Davy says: I'm thinking it does, but only in the sense that I'm voting to kill the aliens regardless of where they live.

Suppose the aliens live in Set A. If we choose set B then that means that we are condemned to live out the rest of our lives with this alien race. Now, if they have the power to destroy the rest of the universe, then surely they have the ability to dominate us. If they are actually willing to destroy the rest of the universe, then I surely don't want to condemn the human race to be subject to whatever other kinds of cruelty they have in mind. I'm a give me liberty or give me death kinda a guy, so I say we take them out even if we have to go with them. (Furthermore, if it does turn out that there are at least two sentient races in are galaxy, this increases the likelyhood that there are other lifeforms in other galaxies that we are saving.)

Suppose the aliens live in Set B. If we sacrifice ourselves for the sake of the rest of the universe, then the alien race just moves on to the next life form and makes them the same deal. I can't imagine the EVERYBODY would sacrifice themselves, so eventually somebody would have chosen to take us out anyway. Might as well beat them to the punch and take away the evil alien race at the same time.

I still haven't decided what to do if we don't know where the aliens live, but I would probably choose B.

Davy

VIDYANANDA says:

Unless Milky Way Galaxy has some super-cosmic importance it is logocal that the answer on the first question would have to be B: "The need of the many outweigh the need of the few." -- Mr Spok

MRBW545 says: My instant answer is A, although I appreciate Vidyananda's reasoning for choosing B. If given more time and information, I would like to know where the aliens come from and what their take on the situation is. Would their galaxy be destroyed by my choice? It would seem to me more complex than what you have presented here. What an awesome amount of death I would be responsible for. I would have to question, "Why was I chosen for this decision? . As I continue to think about this, what would the aliens do if I am unable, or refuse, to make a choice? This would be an interesting choice for a human like myself to have to make.

Trozman says: "The need of the many outweigh the need of the few." -- Mr Spok
The last time I checked, the other 124,999,999,999 galaxies aren't sending out distress signals. Obviously I'd choose A... survival of the human race? Nah, more like personal need for survival. Hey, I'm a mammal, what can I do? :D

## Gordon Question of Big Numbers

What is the largest number (integer) that a computer could in principle work with or maybe yield as a result of a computation? The computer that I am envisioning is one that is not an idealized or theroretical model, but one limited by known physical laws such as the speed of light, electrical resistance, RC delay, atomic dimensions, world supply of silicon, finiteness of programs and data, and other known phenomena. Can such a machine work with numbers so huge they would require Donald Knuth's arrow notation to express them? (Dennis Gordon posed this question)

John Rivaz: If the computer can be programmed to work in text mode (as a very young child doing arithmetic with numbers above 9) the limit is storage capacity rather than length of registers in the processor.

Therefore to get your answer one needs to consider the largest number that can be stored using all the matter in the universe. If a quantum computer is to be considered, then all the matter in the multiverse is the figure you need.

Carl Speare: These answers pose further questions: is the universe infinite? Does the multiverse really exist? If yes, then even if the universe is finite, are there an infinite number of them? Well, if we consider John de Rivaz's idea that an integer is represented a string in memory, the limit would be the memory. And quite a large number it would be for even small computers - 64K of RAM (TRS-80, C64) would yield about 60,000 digits (if you assume a small chunk of memory necessary for a resident microkernel to handle the computation). There is a fairly simple implementation of a large- decimal class that uses a string-based integer representation in many languages (even BASIC).

However, a string based representation is rather ineffecient, because you get only one digit per "byte" of memory. But each byte (actually just 8 bits) can store up to 256 values. So, more efficient (both faster and more space efficient) methods go about their method by representing the number in base 16. (Even more efficient methods go right to base 256, but I'll stick with base 16 for now.) So, the number 141 takes 3 bytes in the string method, but only 2 bytes in the base-16 method: 8D. Now, that does not mean all numbers save only a byte of space. Consider,

6498212189713214987894231234654987984651 in decimal is now: 13

18B624E4B9E074CE3F0777DAF07A6F0B in base 16 (hex) saving 6 bytes of memory. (As one would assume, base 256 saves even more space for large numbers.) So, 40 digits became 34; in fact, you save

truncate ( NCHAR / { ln(16)/ln(10) } ) + 1 characters. So, ( 40 / 1.2 ) + 1 = 33.333 + 1 = 33 + 1 = 34. In general the formula works only for larger strings (it fails in the case of 3 digits, like 141 -> 8D). As you can see, base 256 saves yet more:

truncate ( NCHAR / { ln(256)/ln(10) } ) + 1

would be: 40/2.4 + 1, or about 18 characters.

So, where does this leave us? Going to base 256, on a common computer (64MB of memory, 50MB free) we have:

50 * 1048576 bytes, or 52,428,800 bytes of memory. This would allow up to 52,428,000 digits in the old method, or about 125,829,120 digits using a base-256 representation. So, on a common desktop, 125 million digits is the largest you could go to (reasonably).

Here are some other memory values, based upon having 78% of the memory available for use, which is a reasonable assumption based upon modern operating systems. The list shows the old method [string], then the new method [base 256, or hexhex] storage spaces. As a caveat, all values are approximate.

Mem : [string] : [hexhex]
```016MB: 13,107,200 : 31,457,280
032MB: 26,214,400 : 62,914,560
064MB: 52,428,800 : 125,829,120
128MB: 104,857,600 : 251,658,240
256MB: 209,715,200 : 503,316,480
384MB: 315,572,800 : 754,974,720
512MB: 419,430,400 : 1,006,632,960
```
So, in less than half a gig of free memory, we can store over 1B digits. So, you can go ahead and figure that a 64GB machine (IBM 390, Sun E10000) we could go ahead and go for broke (same 78% free):

64GB: 53,687,091,200: 128,849,018,880

So, there you are: a man-made computer, available today, can put together 128B digits.

## Zygotic personhood

"Zygotic personhood" (the idea that a fertilized egg is a person) is a recent concept. For example, before 1869, the Catholic church believed that the embryo was not a person until it was 40 days old. (Aristotle agreed with this 40-day threshold.) Thus, the church did not believe a human had a soul until day 40. Pope Innocent III in 1211 determined that the time of ensoulment was anywhere from 12 to 16 weeks.

Given this background, if your child was acerebrate (had no brain) would you consider terminating the pregnancy? If born and on life-support for 20 years, would you ever consider terminating the life and donating the organs so they could save others?

Are there any religions today that the set the time of personhood later than the zygote stage of development? In 200 years from now, do any of you feel that the Catholic church will set a different standard for personhood?

"John de Rivaz" : On the basis that soul= program and data, in any instance where some programming and/or some data unique to that individual exist, then there is a person. If some parts of personality are provided by the genes, then it is logical to regard a fertilised egg as "a person" as parts of the unique program are already there. Exterminating it is therefore murder. But it should be noted that in nature many fertilised eggs are exterminated without human intervention, possibly because they would produce unsatisfactory results. In religious terms, gods often require mankind to behave in a similar manner - "do as I do" - "do likewise" and so on. So this doesn't really solve the abortion dilemma. Similarly at the other end if a person who has died is cryopreserved, then the cryopreserved body contains some of the program and data that made the person. If future technology can extract this, then the cryopreserved body is still "a person" and deliberately thawing it is murder. In addition, it is still murder to prevent a dead person from being cryopreserved, because you are destroying the program and data.

## The Problem of the Tanks

You are a tank traveling (up, down, left, right and diagonally) through electric fields. If you travel through two - signs in a row, your battery is drained and you are stuck. If you travel through two + signs in a row, your battery overcharges and explodes. How can you travel from Start to End through every cell once and survive?
```- + - + -
+ + - - +
- + + + S
- + E - -
- + + - +
```
I am curious to see how many people can solve this and how long it took. If you are able to solve this, rather than post the answer, why not just mention you were able to solve it and then tell as about how long it took to solve it. Is there any correlation with the ability to solve this puzzle and age, gender, or profession? How many different solutions does this have?

Teachers: why not give this to students and see how well they do? Have them create their only "tank" problems.

Programmers: why not write a computer program to solve problems of this kind?

Graphics experts: why not write a computer program that displays this more attractively and show the computer trying to solve this or shows the player moving the tank around the field.

Visionaries: why not extend this puzzle to the third dimension?

If you wish, you can redraw the figure on graph paper, or format it as a table, for greater clarity.

Do you think anyone on this planet could solve this problem within 30 seconds? What kind of correlation would exist between the ability to solve this problem and IQ?

Cynthia Sue Larson: What a great puzzle this is! I redrew it on a piece of paper, and worked on solving it while doing some other computer work. I re-drew the puzzle using a pen, and then used a pencil to trace possible solution paths. I solved the puzzle in about ten minutes, using a technique of working from both the "Start" and "End" points simultaneously, and finding where they could best meet in the middle. This is how I noticed when I had a problem (there were two plus signs at the middle), so I erased the line coming from the shorter of the two paths and started over. This puzzle will be great to show kids -- it's a perfect summertime activity! love, Cynthia Sue Larson

Quinn Tyler Jackson" : Pickover: "Do you think anyone on this planet could solve this problem within 30 seconds? What kind of correlation would exist between the ability to solve this problem and IQ?" Yes, I do believe someone on the planet could solve it in 30 seconds. It would be interesting to see a curve of solution times. Pickover: "Is there any correlation with the ability to solve this puzzle and age, gender, or profession?" Please compile that information and let us know. :-) Pickover: "Programmers: why not write a computer program to solve problems of this kind?" I considered writing a G.A. that attempts this and comparing fitness algorithms and results. The alleles would have 8 states -- once for each direction. The chromosome would have to have at least 25 alleles. "Fitness" would be fairly simple -- the number of moves before a chromosome gets stuck. In fact, Dr. Pickover, may I have permission to use that particular maze in my work? I've been using a 64-bit-blackbox key for my experimentation with G.A. -- and I feel that this maze actually represents a more "difficult" (yet solvable) problem for a G.A. experiment I am currently working on. With citation as regards the source of the maze, of course. I will print it out and let my three children have a go at it tonight. -- Quinn Tyler Jackson http://QuinnTylerJackson.n3.net/ Quinn Tyler Jackson" : Using genetic algorithms to solve problems.... [Cliff responds, That's great you are interested in this puzzle. Let me know what you discover.] I managed to get a GA going that was able to solve the puzzle as you posted it in 600 ms, which is pretty fast. Due to the non-deterministic nature of GA, however, and due to the nature of this particular problem, a recursive maze solver would probably solve it more quickly than a genetic algorithm. The problem that I am currently using for the GA experiments is fairly straightforward compared to the maze: A random 64-bit number is generated. The GA attempts to "guess" in as few tries as possible what that number is. ("Pick a number between zero and 15 trillion," or something like that.) The fitness score for guesses is the number of BITS in the guess that are correct, regardless of their position. ("Getting warmer...") I managed to get a 64-bit key solver down to about 45 generations with n^4 fitness scaling and a few other tweaks. The maze problem, however, is more interesting for a GA because it is possible to have paths that are relatively fit, but that do not reach the goal. This has proven to be a real challenge, since the GA often gets to 21/24 fitness and then plateaus. The 600ms result was only obtainable by introducing allele chain reversals during chromosome mating. This allowed for the introduction of completely new path attempts in populations that had grown fit on a 'near miss' path. I will keep at this until the thing is tweaked to always solve it because I need a solvable problem for some other things I'm doing -- and I find the 64-bit-key thing far too "computer-oriented" -- the maze fits the experiment better. Cheers, -- Quinn Tyler Jackson

genie@megasociety.com : > > Do you think anyone on this planet could solve this problem within 30 seconds? No. It's an easy problem, but a lot of people are not that smart. Many would not even be able to decipher the requirements. They use a maze subtest in the WISC IQ battery (the tank puzzle is really a type of maze puzzle with constrained pathways). These types of puzzles may be useful for IQ evaluation because they tend to be "culture- fair" and not too complex. I wrote some puzzles that list members may find amusing: http://www.ultrahiq.net/UltraPuzzle.html Spatial problems like the tank problem are typically easier for men to solve because of the positive (but ultimately curvilinear) relationship between testosterone level and spatial skills. There was a discussion on gender diffs on another list a few months back that inspired this brief FAQ on gender differences and IQ for one of our ezines: http://www.ultrahiq.net/Ubiquity/SprUbiq01/Gender.htm

David Jones : About 10 seconds. My first run through failed, second run through worked. For me, I think the contributing factor was having been exposed to topology and some graph theory. There are certain parts of the "maze" that can only be traversed a certain way. Once you see where certain lines must be drawn, you peice the remaining bits together like a jigsaw puzzle. > Visionaries: why not extend this puzzle to the third > dimension? Your question reminds me of an acquaintence I knew in Phoenix. There was something in his brain that allowed him to have a phenomal ability to visualize three dimensional patterns and figures, but he couldn't understand plane geometery to save his life. While he was of average intillegence, this is the sort of thing he probably wouldn't have been able to do in the time requested. I tend have a backwards problem in that I can't understand anything three dimensional unless I can actually see it. (Its a wonder how I did do well in my Vector Analysis class.) Something odd about myself though is that I cannot play chess on a computer. Even though chess is a 2 dimensional game, I only seem to get a minimal understanding of the dynamics of position when looking at it on a flat surface. If I play computer chess, I have to have a board sitting next to me so I can see the game in 3D if I want to play well. Based on these two examples, I might suggest that the ability to solve this type of problem might be based on how one learns to use spatialization in problem solving. My friend would probably solve the same exact tank problem faster than I if we were actually sitting in battlefield and looking out over a set of markers denoting the charge fields. Davy

"Carlos Paula Simoes" : Hi Cliff, Quite easy :-) I have placed this question (slightly changed, same rules) in the puzzle contest I run at www.cpsimoes.net. The complete address for that specific question is: http://planeta.clix.pt/cpsimoes/puzzle1a5.html

German Gonzales says:

Software and solutions can be found here:

http://www.devweb.cl/mathfun/

How many possible paths are there (disregarding + and - signs), so we can get a hint at what percentage of random trials are likely to lead to a solution?

Answer: there are 912,072 path disregarding signs, therefore the percentage is: 28 / 912072 = 3.06E-5

## Search Packs for Transcendence

At http://www.pickover.com, I speak of "seach packs" that consist of 2 to 4 words that, when typed into the Google search engine, yield fascinating results. Do you have any search packs to suggest that encourage the mind to transcend its Earthly bounds (or that yield very interesting web sites)?

Here are some of my favorite search packs from www.pickover.com. You just type the words into Google input field, without quotation marks:
```Search 1: "god synchronicity transcendence afterlife"
Search 2: "mind lsd dimensions"
Search 3: "third temple jerusalem"
Search 4: "parallel universes beings god"
Search 5: "pickover hawking jesus"
Search 6: "genius iq savant"
Search 7: "quiet mind tome"
Search 8: "dreams transcendence geometry"
Search 9: "apocalyptic dreams time travel"
Search 10: "jesus never died"
Search 11: "problem miracles"
Search 12: "alien inside brain"
Search 13: "rephaim gilgal"
Search 14: "god mathematics knowledge"
```
What search pack do you particularly like?

Comment Jason Edwards: Here are three search packs I thought were interesting:
```1. "mystical sequence numbers"
2. "automated reasoning Turing Babbage"
3. "eschatology Abraxas"
```
Using the first "search pack" I came across a page that referred to this sequence: 3,7,13,39,69,... as a "mystical sequence." It wasn't exactly clear from the web page why the author called it mystical however. Has anyone else seen this sequence before or know why the author calls it mystical? Can anyone come up with the next term? Can anyone come up with a generating formula?

Lately I have been intrigued with "automated reasoning" and the concept of computer-assisted proofs. One of the main things that is fascinating to me is the hint of controversy surrounding the subject. In 1976, when Wolfgang Haken and Kenneth Appel came up with a computer-assisted proof for the long standing Four-Color problem, many mathematicians refused to accept it because they felt Haken and Appel's "brute-force" approach to exhausting all of the possibilities wasn't as rigorous as a traditional deductive argument. A couple of years ago Tom Hales and Sam Ferguson offered a proof of the Kepler sphere-packing conjecture. Their proof consisted of almost 300 pages and was also computer-assisted, which makes it extremely difficult to verify. I remain rather skeptical about computer-assisted proofs, nevertheless I find the subject especially fascinating. Does anyone else have an opinion on automated reasoning? Jason

Cliff, "Carlos Paula Simoes" : Search packs (using Google)
```mind increase power
reach superior intelligence
alien inside brain
god mathematic knowledge
time light brain"
```

"Quinn Tyler Jackson" : "transcend consumerism meaning"

## Synchronicity

In my book Dreaming the Future, I discuss synchronicity.

Psychologist Carl Jung and biologist Paul Kammerer believed in "synchronicity" (or meaningful coincidences -- or the theory that there is a mysterious connectedness of objects and events in space and time). Paul Kammerer (1880-1926) wrote, "We thus arrive at the image of a world-mosaic or cosmic kaleidoscope, which, in spite of constant shufflings and rearrangements, also takes care of bringing like and like together." He compared events in our world to the tops of waves in an ocean. We notice the tops of the isolated waves, but beneath the surface there may be some kind of synchronistic mechanism that connects them. He also believe that because of this, we often also see "streaks" of luck or misfortune in situations ranging from sports to gambling to family catastrophes. Modern examples of this might include clusters of airplane accidents, many momentous events happening in the same year, the sudden simultaneous arrival of musical geniuses and music, the "curse" of the Kennedy family, and the skiing accidents of celebrities, which, while not causally connected in any formal sense, are connected through meaning. Skeptics would suggest this connection arises from our tendency to affix patterns to ANY sequences of events, regardless of their significance.

In the following new experiment with members of this group, I will now mention two words that one would not normally hear in daily conversation, or read in the newspapers, or hear on TV:

or
2) Scarab

If any of you do hear or read these words within 48 hours of reading about this present experiment, let us know. (Don't go out of your way to find these words -- just let your life flow naturally and see what happens.) Feel free to invite your friends to participate in the "Madagascar Experiment" and tell us their findings in this group. -- Cliff

Cynthia Sue Larson" says: Hello, everyone! With positive expectations that the Synchronicity Experiment would yield interesting results, I read and agreed to participate. For the first 24 hours, I didn't notice any reference to the two key words. As the end of the 48 hour period since I'd read about "Madagascar" and "scarab" neared, I was very pleased to find two separate emails that came to me containing those two rather unusual words. The Madagascar reference pertains to the June 21 solar eclipse, and the scarab reference was part of a forwarded article on the subject of (what else?) -- Synchroncity! lots of love, Cynthia

martyn27015@yahoo.com: There's this statistics, or probability, professor that i read about who would give an assignemnt the first class of the term. He would tell his students to flip a coin 100 times and record the results. The twist was that half the class were to just make up the results without him knowing who made it up. The next class, the professor asked some students, one at a time, to let him take a glance at the results, and he'd tell them who really did it, and who faked the results. And he did do that, with nearly 100% accuracy. Most classes he made no mistake. His technique was this: according to some probabalistic principle, the odds are VERY high that when flipping a coin a hundred times, at least once there will appear a streak of either six heads or six tails. The ones who faked it, trying to be "random," had at most three or four consecutively. My point is, that people really don't have a natural feeling for probability at all, and if we see some obscure pattern we instinctively think there is a cause attached to it.

martyn27015 talks about how difficult it is to make up a list of random heads and tails. Similarly, I have done experiments in which humans find it very difficult just to type a "random" sequence of 1s and 0s, as in 1010111011100010101010010100101 -- Cliff

rachi11er@yahoo.co.uk: Hello Cliff, Does it count that two days ago I read about the Jung/scarab beetle incident in a completely unconnected book? I will now look out for Madagascar and it's peculiar lemurs at every turn. There are theories that synchronicity works because we have a brain which actively looks for certain elements in the world and even goes so far as to "buffer" its perceptions in such a way that, if you sit beside two people talking in a cafe, you will probably not hear their conversation at all - unless they mention your name. At that point you will realise you had in fact heard not just your name, but quite likely the entire sentence it was contained in. Of course, that theory rather disintegrates in the face of some synchronicities - surprise visits from real scarab beetles among them! Regards, Chiller (and thanks for the invite)

"Brandon J. Van Every" : r skill or gift that few have. Freud helped to open the doors for me ) I have a completely different armchair opinion on the subject of dream interpretation. I believe all people can understand the meaning of their dreams if they are merely willing to follow their instincts. The dreams are occurring, after all, so that the subconscious can work out and make sense of what's going on in the person's life. Why should it be hard to interpret a dream when the subconscious is trying to make it easier for you? All you have to do is match the elements of the dream against what is occurring in your life, and use your instinct as the "aha" that links the two. Problems with dream interpretation are created by our societies. Instead of realizing that each person owns their own dream, and is the person best qualified to interpret it, society tells us that dreams are something terribly mysterious that only experts can understand. It's rather similar to white anthropologists telling blacks in Africa what their culture really is. We should be suspicious of people who argue from authority about phenomena occurring within us. Much as it might have helped Freud or Jung or any of those other guys' academic credentials, there are no universal dream symbols. There is no possible way for a psychiatrist to understand a person's dreams without the complete context of that person's life. More information on dreaming and synchronicity here: http://www.fortunecity.com/westwood/vivienne/598/asd-17/abstracts/Rita_Dwyer_17.html

j snow : re: finding the scarab I had some "midnight" uneasiness last night. Before i went to bed, i realized belatedly, that as i was busy writing about "the synchronicity experiment" a lady bug had been tapping on my computer screen. It came back several times, but i was too dense to realize that the lady bug is a flying beetle as is the scarab. If one complains that a ladybug is not a scarab, well, neither was the one that flew into Jung's room when he was with the patient... Synchronicity deals with symbols. My main reference is a SUNY press book that is temporarily misplaced. sorry. It is hard to be scientifically objective since many a dream image will find its counterpart in the consensual reality world, and then bring the dream back into consciousness. For example, one morning i saw a woman on the NYC subway carrying a book with a mostly white cover and a splash of color that reminded me of the cover of "Iron John" by Robert Bly. And that reminded me of my dream that included that book, which provided the key to the meaning of the dream. But that does not provide convincing evidence of synchroniciy since many objects may have reminded me of the dream, or the dream may have repeated the theme many times.... But i know other stories ........ enuff said, The following article may be of interest http://www.gis.net/~fpnma/sermons/scarab.htm THE SCARAB AT JUNG'S WINDOW by R.M. FEWKES One of this year's best selling books is called SMALL MIRACLES: EXTRAORDINARY COINCIDENCES FROM EVERYDAY LIFE by Yitta Halberstam and Judith Leventhal. The book is a collection of stories of extraordinary coincidences in the lives of ordinary people. "Coincidences," say the authors, quoting the writer Doris Lessing, "are God's way of remaining anonymous", or the universe's way of telling us that there is more to reality than meets the eye. Coincidences are small miracles that can awaken us "to the rich promise of a bounteous universe and the splendor lying dormant within your soul. Coincidences are everywhere and can happen any time. When your soul is ready, they will come. All that is required is that you open your heart." (p.xiii)

j snow : In an earlier post, Cliff gave us two secret words, Madagascar and scarab, as part of a syncronicity experiment. We were to report if we had come across the words within a short period of time after Cliff gave the words to us. summary of my results: i read the experiment on tuesday april 17 while i was on the computer, lady bug kept tapping the screen. (beetle) On friday july 27, NPR radio mentionned madagascar. ( 3 feet from my computer is a CD that i picked up for a buck. Key song was madagascar. This may have interfered with experiment :-- ) peace. J

## Simultaneity in Science

Several famous, simultaneous discoveries exist in the history of humankind. For example, Newton and Leibniz "discovered" calculus independently, but at the exact same moment in history. The same is true for Darwin and Wallace, who both separately developed the theory of evolution at the same time. What other examples can you give, and what significance is there to such simultaneity?

### Conments

Dennis W. Gordon" <74644.1103@compuserve.com> : Cliff, It appears that the mechanical computer was simultaneously invented by both Charles Babbage and George Biddell Airy (Astronomer Royal in the United Kingdom) and another un-named man working with Airy. I am not sure of the significane is, but maybe human pettiness and jealousy figured in. Dennis

petebarnes@home.com : How about the discovery of the telephone, which was patented by Alexander Graham Bell and his business partners. Bell beat his competitor, Elisha Gray, in filing the patent by a matter of hours, although Gray was filing for a device that he had not quite perfected, but which would have given him the credit for the invention. Western Electric, cofounded by Gray, became one of the Bell System's major competitors. Pete B

## Would you pay \$2000 for a Turbing?

In 1950, Alan Turing proposed that if a computer could successfully mimic a human during an informal exchange of text messages, then, for most practical purposes, the computer might be considered intelligent. This soon became known as the "Turing test," and it since led to endless academic debate.

Opponents of Turing's behavioral criterion of intelligence argue that it is not very relevant. This camp suggests that it is primarily important that the computer demonstrates cognitive ability regardless of behavior. Some say that computers can never have real thoughts or mental states of their own. The computers can merely simulate thought and intelligence. If such a machine passes the Turing Test, this only proves that it is good at simulating a thinking entity. Holders of this position also sometimes suggest that only organic things can be conscious. If you believe that only flesh and blood can support consciousness, then it would be very difficult to create conscious machines. But to my way of thinking, there's no reason to exclude the possibility of non-organic sentient beings.

I call these "humanlike" entities Turing-beings or "Turbings." If our thoughts and consciousness do not depend on the actual substances in our brains but rather on the structures, patterns, and relationships between parts, then Turbings could think. If you could make a copy of your brain with the same structure but using different materials, the copy would think it was you.

Would you pay \$2000 for a Turbing -- a Rubik's-cube sized device that would converse with you in a way that was indistinguishable from a human? Why?

"Rhonda Eudaly" : Though where I could see where it might come in handy in rehearsing all the possible outcomes of those conversations you know you have to have but don't want to, I don't think I could justify \$2K on it. Other than that, no, not me. I don't even like telephones, can't imagine conversing with a calculator (for example) - and it's always been my contention that talking to yourself is eccentric, it only becomes a problem when you start answering yourself. And in a way, couldn't this be that last step from eccentricity? Rhonda

"John de Rivaz" : If a "Turbing", the device the size of a Rubic Cube that will hold a human like conversation, costs \$2K now, it will be \$200 in a few years time. Assuming you have plenty of other things to do then it would seem sensible to wait. Possibly every sensible person thinks this way now and that is an explanation as to why the technology industry is in the doldrums. The idea of making early adapters pay development costs could dying. Instead shareholders have to pay them, and maybe it is this re-organisation that has caused the present re-valuation of technology shares. However once the change has worked its way through the system, technology should again be a highly profitable investment. For links to technology shares corporate web pages, please see http://www.geocities.com/longevityrpt/shares.htm

John W Burgeson : Cliff wrote: "Would you pay \$2000 for a Turbing -- a Rubik's-cube sized device that would converse with you in a way that was indistinguishable from a human? Why?" Cliff -- my PC is a little bit larger (not much) than a Rubik's cube, but it also costs about 1/2 as much as your example. Now I might assert that I have already paid for that function, for this device allows me to converse with you (and many others) on an as needed basis. Often the conversations are quite interesting; sometimes, for instance when a "fundie" shows up on Compuserve's RELIGION forum, they are pathologically sad. My point is, of course, that I am quite unable, except on a very occasional basis, to determine whether or not all these people, including you, are "Turing test enabled" computers or real live persons. You, for instance, I have never met in person, even though we once collaborated (you did most of the work) on an invention disclosure. So the answer is "yes," and I assert I have already done so. This sounds like your neat thought-problem of 10 years ago in which you asked us to reflect on what a scientist of 1910 would have thought if an IBM PC were to be given him. John Burgeson (Burgy)

Will Dye : Mr. Pickover wrote: > I call these "humanlike" entities Turing-beings or "Turbings." I don't care for the "Turbing" naming convention. For one thing, "Turing being" may be easily confused with the well-established term "Turing machine". I'm also concerned that the tie-in to the Turing test may inadvertently contribute to the perception that machine intelligence is properly measured against the sole yardstick of engaging in human-style conversation. All this fussing about terminology may seem obsessive (see .sig), but please keep in mind that one of the core problems in this field is parsing out more precise meanings of words. Take it from an old-timer: in this hyar swamp, ta move faster, ya gotta move slower. > Would you pay \$2000 for a Turbing -- a Rubik's-cube sized device > that would converse with you in a way that was indistinguishable > from a human? Would you pay \$2,000 for a human slave, occasionally brought out so that you could have a conversation with someone who has no choice but to listen? I doubt if it will be easy to instill as complex a human-like behavior as good conversation without instilling other human-like traits such as ambition, loneliness, desire, or even agony. Perhaps I'm wrong, and the technology, once developed, will be of the Star Trek variety -- all 'logic' and no 'emotion'. Perhaps regulatory and/or humanitarian concerns will encourage engineers to create entities that are "OK" with being a conversation-in-a-box. Still, at \$2,000 a pop, the engineers will be under a lot of pressure to create an entity that only... just... barely parrots back the right answers, at the right times, to be classified as unworthy of true liberty. Still others simply don't care if the box is genuinely happy or not, or at least don't care enough to bother tackling the difficult issue of providing dangerously-intrusive legal protections for manufactured entities. I can't be sure if we can or should attempt to instill regulations in such matters, but hopefully we'll at least be able to control our own actions. I, for one, pray that I never develop a mindset that would allow me to financially encourage such a trade. --Will

cliffpickover@hotmail.com : I would pay \$2000 for a Turbing device for several reasons, including: 1) You (or whomever you give the device to) would never be lonely. After all, we posited the fact that the device responds in a way that is indistinguishable from a human. 2) The technology would be quite advanced. If we could somehow reverse engineer this, we would be millionaires! Note that we are currently far from being able make a machine that passes an unconstrained Turing Test in which a human may converse with the machine for an extended period of time on any subject. (Judges can always tell they are talking to a machine if they ask enough questions and have no constraints.) I wonder if we will be able to create such a Turbing by 2010?

Will Dye : Mr. Pickover writes: > I would pay \$2000 for a Turbing device for several > reasons, including: > > 1) You (or whomever you give the device to) would never be > lonely. After all, we posited the fact that the device > responds in a way that is indistinguishable from a human. Ewww. If the only way you can sustain a conversation is by enslaving the other participants, then no wonder this list is so quiet. Seriously, if someone in an age of advanced communications and medical technology can't maintain a relationship with anyone who has a choice to leave, creating quasi-human slaves is decidedly not the optimal response. > 2) The technology would be quite advanced. If we could > somehow reverse engineer this, we would be millionaires! Oh, brother. There are so many dubious assumptions in that scenario that I don't even know where to begin. I regret that I sound rather grumpy here, but I'm not sure how else I can express my level of disagreement. --Will

"John de Rivaz" : If you read some of the newsnet postings, particularly on groups discussing religious matters, it does appear that some people have tried to write computer programs that mimic human response, albeit somewhat badly, and connected them to newsnet. For example, in a debate on whether citizens should have the freedom to chose whether their bodies are autopsied or not after they have died, one such entity merely posts messages to the effect that people who want such freedom should be made into dog meat. The quality of the messages seldom change, although the content is the same. It is interesting to note that those who have taken the trouble to vote on the web site http://www.autopsychoice.com are more or less in favour of people being allowed an informed choice, so perhaps the entity is a hungry dog!

John de Rivaz" : In previous posts to this discussion group, we've talked about the idea of living in accurate, virtual reality simulations. Here is related information: From CryoNet: >>>>>>. Message #16948 From: Azt28@aol.com Date: Tue, 10 Jul 2001 17:50:42 EDT Subject: Re: CryoNet #16924 Simulation From Lee Corbin: > >I think we don't live in a simulation, not because it is not > >possible but because we can't produce it now. > > But the most common conjecture is that the simulation wasn't > constructed "now", but in, say 20,000 A.D., and that we just > don't know the right date. I have advocated the idea of Black Magic: A quantum computer simulate a system and then project that simulation onto a material quantum system on a vacuum fluctuation. We may see such phenomena in the laboratory 10 to 20 years from now. Well before AD 20 000 that technology would be able to project the full observable Universe as a simulation. We are too few a part of that simulation to be its main purpose. You don't create 10^20 stars to watch GWB in the White House or anything else on Earth. In God I don't trust, and I don't think the Universe we see has been built by anyone, less by anyone with the objective to watch us. We are nothing and we have to learn this fact. May be the Universe is a simulation, I like the idea of a self simulation. Degenerate quantum state, as found inside a neutron star may form spontaneously a simple very repetitive quantum computer. It seems indeed that the quantum domain may be reduced to two quantities: one finite number: zero and one undefined one, you can as well call it one or infinite or undefined. If there was any thinking behind the simulation, the parameter choice would have been somewhat smarter. It seems we are in the dumbest possible world, a nearly proof that nobody is at the command level. Yvan Bozzonetti. Brandon J. Van Every" : Regarding the question about whether we could be living in a simulation... > We are too few a part of that simulation to be its main purpose. You don't create 10^20 stars to watch GWB in the White House or anything else on Earth. The universe doesn't have to be simulated, only Earth. Any time we request data farther than the boundaries of Earth, they can feed us any data they like. Could be a combo of data from real telescopes hooked up to the simulation, and blot-outs of anything they don't want us to see. One question of the simulation is whether we're clever enough to pierce the veil of illusion.

math@antiquark.com : I'd buy 100 Turbings and startup my own telephone soliciting call center, with turbings doing the calls. It was be amazingly profitable... no need to pay unreliable humans \$5 an hour. For one-time fee of only \$2000 per turbing, you'll have a reliable, around-the-clock telephone solicitor. Of course, this assumes that turbings are utterly devoid of souls and ethics. Derek.

Chuck: I'd have mine screen my phone calls and hang up on telephone solicitors. Yes, I'd probably buy one unless they got awful reviews from other users.

"peteb_hwp" : Well, certainly not if I were the very first one to know about it. Imagine what your friends would say about you being caught conversing with a Rubik's Cube at the Christmas party! :=) Seriously, I would not pay 2K for a copy of myself (I already talk to myself, and I don't even like the answers most of the time), but I would gladly pay 2K or more for such a device that was a copy of, say, Albert Einstein's intellect. Or any of a number of other highly interesting individuals that come to mind, say Carl Sagan, Isaac Asimov or Mark Twain. Hey, how about, how much would you pay for a Turbing of your mother- in-law? :=) Myself, I think the true test would be to have such a Turing machine show that it was self-aware and capable of independent thought. The mimic test you mention is just evidence of super good computer programming IMHO. In fact, as I am sure you know, there are quite a few of these Elisa programs floating around, some are astounding. Pete B [Cliff comments, "One wonders if people might actually fall in love with their Turbings."]

"peteb_hwp" : I think love involves a lot more than just intellectual interactions, there are emotions, physical responses, etc., so I doubt that would happen. Although, if as you proposed, somebody has a Turbing of their own mind, and that person is extremely narcissistic (sp?), then perhaps they would become enamored of their own thoughts being fed back to them. Thinking about it further, though: would a self-centered personality be any good in a Turbing machine, or would it just be absorbed in reflecting on its' own new state of existence? Imagine how bad that would be, to have a Turbing of yourself, and have the machine refuse to talk to you! Or worse yet, it gets mad at you for ignoring it. "Yeah, I got rid of the darned thing, everything was always all about it, never about me, you know what I mean?" :=) Wonder if a Turbing could get a headache? Get mad or jealous? And to turn your question on its' head, could a Turbing machine fall in love with its' owner, do you suppose? Pete B

"Quinn Tyler Jackson" : Cliff: "Would you pay \$2000 for a Turbing -- a Rubik's-cube sized device that would converse with you in a way that was indistinguishable from a human? Why?" Quinn: No, because I already have a desktop version of that (the Internet) that costs me about \$25.00/month.

-- "Quinn Tyler Jackson" : Bobomutin: Applications would include, home psychologist, philosopher, medical advisor, pastoral counselor, baby-siter, entertainer, educator, lawyer, and so forth. Certainly, the depressed, the bereaved, the elderly, the emotionally disturbed, would be a central market for such a comodity. But also conventional younger people would need a bit of chat now and then. Reply: Pocket Poet(R) 1.0 For Pickover Turbing(R) Models 5.2+ O Owner, in whose hand I gently whir, I know all the poems that ever were, And if you solve the puzzle of my face, I shall some cinquain gently, humbly pur. But don't forget my Double AA's refresh, And period interface with mesh, For if my charge becomes uncharged, you know, I'll rhyme a word like orange with Gilgamesh. And though I can recite all forms of verse, For just two thousand taken from your purse, There is one thing like no other poet: I shall not, if scorned, versify some curse. My guarantee completely covers that, And so, you're safe to loudly curse and spit If some epiphany of mine is off: Just take me back to be quickly refit. Indeed, though poets voice of their age Are supposed to be, if I should enrage You with a metered observation's foot, You're free to flip and switch and turn my page. But if you seek some succor in my mete, Just listen 'til my stanza is complete -- Do not return me if you find me curt, If I should kick you with my neuram feet. But since you paid your two thou, what's to do? My software wasn't beta'd 'pletely through, So I may, like a poet, shatter dreams .... Perhaps just download debugged version 2. -- Quinn Tyler Jackson

John de Rivaz" : > No, because I already have a desktop version of that (the Internet) that costs me about \$25.00/month. Good Grief! in the UK you can get it effectively for nothing. Companies like NTL offer you free unmetered connection provided you spend UKP15/month with them for voice telephone calls. On the basis that you (or your wife :-) ) spends that much on phone calls anyway, then the Internet is free.

## What will society be like in 1000 years?

Would you care to speculate about what civilization and society will be like 1000 years from now? -- Cliff

"John de Rivaz" : If technology advance continues, such speculation is futile, unfortunately. The major change is the end of ageing and relative rarity of death, but as this will come so much sooner than 1000 years that one would be looking at nearly 1000 years of humans living as ageless beings. (Never immortal, as you have to have lived for eternity in order to be sure you are immortal, and if you then die you are not immortal) Some speculation about this has been made in George Bernard Shaw's play cycle "Back the Methuselah." It would be easy to speculate, but so hard to be anywhere near reality!

Derek Ross" : In Msg 141, you ask "What will we be doing 1000 years from now?" In my opinion -- even though this does not sound futuristic -- we will probably be doing the same activities we were doing 1000 years ago, and still continue doing today. Off the top of my head, I can think of a few obvious ones: - Riding around on four-legged animals. - hitting/kicking/batting/swatting a variety of balls for fun (or watching others do the same). - Reading books and writing letters. - Pitching cloth tents in the wilderness, crapping in the bush, and staring at flaming logs. (AKA camping). - Growing tomatoes in your garden. - Going to church. - Painting pictures using pigments dissolved in oil. - Going to restaurants. - Capturing fish with a small metal hook attached to a string, using a worm as bait. I'm sure you get my point -- we will continue engaging in activities that appear to be useless and archic, that could easily be eliminated by a quick technological fix. But if people find some activity enjoyable, then it will continue to be practiced. Just my \$0.02. Derek.

## What is the most forgettable license plate?

pmosk@watson.ibm.com: I read an article about someone who claimed to have devised the most forgettable license plate, but did not divulge it in the article. What is the most forgettable license plate? Is it a random sequence of eight (the maximum allowed in New York) letters and numbers: for example 6AZL4QO9, or perhaps a set of visually confusing numbers or letters, for example MWNNMWWM? What do you think?

-Paul

David Jones : I remember once watching a news show about some yearly competition to find out who has the best memorization skills. They have tests like each person has to memorize the most digits of a random number within 30 minutes and so forth. They talked with some of the competitors and the main trick they all use is mneumonics (sp?). For example, with MWNNMWWM I just have to remember four words: MoWN NaMe WoW Me (remove the vowels.) So, the most forgettable license plate would have to be one that somehow minimizes mneumonic compression. Consecutive numbers are easier to compress than consecutive letters. Your brain stores 53 as a single number, but it stores 5 and 3 and two separate numbers. On the other hand, there is no easy way of compressing something like QK. Yes, you can use the word QuicK but you still have to know what two letters in the word you care about. My point though is that the order of number and letter placement can complicate things. Im guessing that something like AA#AAA#A would do the job as it seems like with well chosen letters you could require the memorizer to use a minimum of six mneumonics. After that, I don't have too many good ideas. GJ for the first two letters might be a good start as I cannot quickly think of a word that uses those two letters in that order and they are phonetically similar enough that they can be easily confused. Davy Cliff asks, What about a binary number sequence like 01011011...? This would be hard to remember.

## "Omega Browser" Survey

Imagine you have a strange web browser called Omega. This is your only browser for the rest of your life. If your Omega browser was locked in such a way that you could view five web sites, what would they be? (You may consider that the web site you choose has access to all its "subsites" at that domain name; e.g. www.britannica.com would have access to all its articles.) Cliff

Derek Ross : Google. They have a cache of every page they reference, so in essence, google contains the entire internet. Derek.

David Jones : Well, I could pick a site like safeweb or version99 which allows me to browse other websites anonymously, but in turn would allow me to bypass the Omega limitation. Similarly, Google.com allows you to access the cached pages they use for their search engine so, although much more limiting, I still achieve the effect of viewing other websites while staying on the same domain. I will assume, however, that answers like these are not in the "spirit" of the question. I guess that first and foremost would be nightswimming.com. I own the domain and also must have access to it in order to change its contents. Second would be yahoo. I mean, what CAN'T you do on Yahoo? I get mail, games, maps, shopping, news, calendars, umm, I know theres more, but you get the point. Third would be pogo.com. They have much better bridge players there than yahoo games does. Fourth would be ZDnet so I can download shareware and game demos and still get access to tech news. Fifth would be classicgaming.org - gotta have my emulation/nostalgia fix. On second though, I might scratch the fifth one in favor of soa.org - I would need to have access to information for my actuary certifications. Davy

## Special Augmented Primes

One of my favorite bizarre questions involving prime numbers relates to the wonderful "special augmented primes." You can augment a prime simply by placing a 1 before and after the number. The augmented prime is "special" if it yields an integer when divided by the original prime number. For example, 137 is such a number because 137 is prime and because 11371 / 137 yields an integer, namely 83. Are there other such numbers? How rare are special augmented primes? It turns out that there is a general set of numbers of the form 90909...01 that are special augmented primes. For example, Jim * discovered that 909090909090909091 -- when augmented by ones to form 19090909090909090911 -- has a factor of 10**19+1 and, similarly, 909090909090909090909090909091 has a factor of 10**31+1. For those of you with access to computer programs that can manipulate large integers, how many more of these Jim * augmented primes can you find? Amazingly, Jim found this incredible extension to the 9090 trend:
```909090909090909090909090909090909090909090909090909090909090909090\
909090909090909090909090909090909090909090909090909090909090909090\
909090909090909090909090909090909090909090909090909090909090909090\
909090909090909090909090909090909090909090909090909090909090909090\
909090909090909090909090909090909090909090909090909090909090909090\
9090909090909090909090909091
```
which, when supplemented, has a factor of 10**293+1. What smaller special augmented primes are lurking in our number system waiting to be discovered? Cliff (Note: I am waiting to get Jim's last name.)

"Daniel Dockery" : For any using the software, this is a Maple routine that will produce augmented primes for the nth prime number: auP := proc(n) local p; p := ithprime(n): (10^length(p)+p)*10+1; end: E.g., auP(1) = 121, auP(2) = 131, auP(33) = 11371, etc. > The augmented prime is "special" > if it yields an integer when divided > by the original prime number. For > example, 137 is such a number because > 137 is prime and because 11371 / 137 > yields an integer, namely 83. Are > there other such numbers? Searching the range of the first 50,000 primes, I found only four. p( 5)= 11, auP( 5)= 1111, auP/P: 101 (itself a prime) p( 6)= 13, auP( 6)= 1131, auP/P: 87 p( 33)= 137, auP( 33)= 11371, auP/P: 83 (prime) p(1128)=9091, auP(1128)=190911, auP/P: 21 Unless I missed one or more in my search, the above would suggest they have, at least in the above cited number range, a density of around 0.008%, I think. > How rare are special augmented primes? > > It turns out that there is a general > set of numbers of the form 90909...01 > that are special augmented primes. > For example, Jim * discovered that > 909090909090909091 -- when augmented > by ones to form 19090909090909090911 > -- has a factor of 10**19+1 and, > similarly, 909090909090909090909090909091 > has a factor of 10**31+1. In both of the cases given above, by my calculations, the prime 90...91 (i.e., the prime itself, not the augmented form) equals (10**n+1)/11. That is, 909090909090909091 * 11 = 10**19+1, 909090909090909090909090909091 * 11 = 10**31+1. (10**n+1)/11 will, I think, always produce a number of the form 90...91 with (n-1) digits when n is an odd number greater than or equal to 5. Checking the range of {5<=n<1000}, it appears to create prime numbers when n=5, 7, 19, 31, 53, 67, 293 and 641. I find it interesting to note that in all cases, here, for the 90..91 numbers, the integer auP/P is always 21. > For those of you with access to computer > programs that can manipulate large integers, > how many more of these Jim * augmented > primes can you find? If, as above, you define a JauP (Jim augmented Prime, naturally) to be (10**n+1)/11*21 for some odd n (will n always be prime?), then a casual search for n<1000, reveals there are JauPs for n in {5, 7, 19, 31, 53, 67, 293, 641}. JauP(641) = 190909090909090909090909090909090909090 \ 90909090909090909090909090909090909090 \ 90909090909090909090909090909090909090 \ 90909090909090909090909090909090909090 \ 90909090909090909090909090909090909090 \ 90909090909090909090909090909090909090 \ 90909090909090909090909090909090909090 \ 90909090909090909090909090909090909090 \ 90909090909090909090909090909090909090 \ 90909090909090909090909090909090909090 \ 90909090909090909090909090909090909090 \ 90909090909090909090909090909090909090 \ 90909090909090909090909090909090909090 \ 90909090909090909090909090909090909090 \ 90909090909090909090909090909090909090 \ 90909090909090909090909090909090909090 \ 909090909090909090909090909090911 > Amazingly, Jim found this incredible extension > to the 9090 trend: > > 909090909090909090909090909090909090909090909090909090909090909090\ > 909090909090909090909090909090909090909090909090909090909090909090\ > 909090909090909090909090909090909090909090909090909090909090909090\ > 909090909090909090909090909090909090909090909090909090909090909090\ > 909090909090909090909090909090909090909090909090909090909090909090\ > 9090909090909090909090909091 Hmm. Maple and Pari both return prime(n)=false for the above large number; as best I can tell, it's of the form 6*x+1 for x = 15151515151515151515151515 \ 15151515151515151515151515 \ 15151515151515151515151515 \ 15151515151515151515151515 \ 15151515151515151515151515 \ 15151515151515151515151515 \ 15151515151515151515151515 \ 15151515151515151515151515 \ 15151515151515151515151515 \ 15151515151515151515151515 \ 15151515151515151515151515 \ 15151515151515151515151515 \ 15151515151515151515151515 \ 151515151515151515 > which, when supplemented, has a factor of 10**293+1. Using n=293, (10**n+1)/11 yields 909090909090909090909090909090909090909090909090909090909090909090 \ 909090909090909090909090909090909090909090909090909090909090909090 \ 909090909090909090909090909090909090909090909090909090909090909090 \ 909090909090909090909090909090909090909090909090909090909090909090 \ 9090909090909090909090909091 which is probably what was intended with the earlier number, I imagine, which simply had an extra row of 9090..9090 attached. :) > What smaller special augmented primes are > lurking in our number system waiting to > be discovered? I wonder if there are any other integer multiples auP/P besides the four found and cited earlier (i.e., 21, 83, 87 and 101). In my searches so far, I've found none. Since it seems that all JauP produce the same integer multiple, 21, I wonder if there are similarly other primes which produce their auP when multiplied by the four earlier integer multiples? (I've searched through the first 1000 primes in each of the four cases and found none.) It should be possible to speed up searches for such numbers by noting that since our augmented number must end in the digit 1, and must be the product of an integer i and the prime itself, that for primes ending in the digit 1, i mod 10 must equal 1, primes ending in 3 can only have an i mod 10 equal to 7, primes ending in 7 can only have an i mod 10 equal to 3, and primes ending in 9 must have i mod 10 equal to 9. Theoretically, it seems that it should be possible for a prime ending in 9 to produce its auP when multiplied by an integer ending in 9, but so far I've found no actual example of this happening. Any suggestions from anyone as to why? Am I simply missing something? I also wondered if there were interesting properties to be discovered by generalizing the augmented prime idea to allow for the prepending and subtending of any n to a prime, rather than 1 only? (E.g., 21372, 31373, etc.) Interestingly, it looks like any generalized augmented prime formed by prefixing and suffixing a digit x, from 1 to 9, to the nth prime, gauP(n)/P(n) will always be an integer for n in {5, 6, 33, 1128}. E.g., n: 5, x:1, p: 11, auP: 1111, auP/p:101 n: 6, x:1, p: 13, auP: 1131, auP/p: 87 n: 33, x:1, p: 137, auP: 11371, auP/p: 83 n:1128, x:1, p:9091, auP:190911, auP/p: 21 n: 5, x:2, p: 11, auP: 2112, auP/p:192 n: 6, x:2, p: 13, auP: 2132, auP/p:164 n: 33, x:2, p: 137, auP: 21372, auP/p:156 n:1128, x:2, p:9091, auP:290912, auP/p: 32 n: 5, x:3, p: 11, auP: 3113, auP/p:283 n: 6, x:3, p: 13, auP: 3133, auP/p:241 n: 33, x:3, p: 137, auP: 31373, auP/p:229 n:1128, x:3, p:9091, auP:390913, auP/p: 43 n: 5, x:4, p: 11, auP: 4114, auP/p:374 n: 6, x:4, p: 13, auP: 4134, auP/p:318 n: 33, x:4, p: 137, auP: 41374, auP/p:302 n:1128, x:4, p:9091, auP:490914, auP/p: 54 n: 5, x:5, p: 11, auP: 5115, auP/p:465 n: 6, x:5, p: 13, auP: 5135, auP/p:395 n: 33, x:5, p: 137, auP: 51375, auP/p:375 n:1128, x:5, p:9091, auP:590915, auP/p: 65 n: 5, x:6, p: 11, auP: 6116, auP/p:556 n: 6, x:6, p: 13, auP: 6136, auP/p:472 n: 33, x:6, p: 137, auP: 61376, auP/p:448 n:1128, x:6, p:9091, auP:690916, auP/p: 76 n: 5, x:7, p: 11, auP: 7117, auP/p:647 n: 6, x:7, p: 13, auP: 7137, auP/p:549 n: 33, x:7, p: 137, auP: 71377, auP/p:521 n:1128, x:7, p:9091, auP:790917, auP/p: 87 n: 5, x:8, p: 11, auP: 8118, auP/p:738 n: 6, x:8, p: 13, auP: 8138, auP/p:626 n: 33, x:8, p: 137, auP: 81378, auP/p:594 n:1128, x:8, p:9091, auP:890918, auP/p: 98 n: 5, x:9, p: 11, auP: 9119, auP/p:829 n: 6, x:9, p: 13, auP: 9139, auP/p:703 n: 33, x:9, p: 137, auP: 91379, auP/p:667 n:1128, x:9, p:9091, auP:990919, auP/p:109 Note that auP/p appears to be a function of x, and a number apparently connected in some way to the individual primes involved: e.g., for n=5, or the prime 11, auP/p equals 91*x + 10; for n=6, p:13, auP/p is 77*x+10; for n=33, p:137, auP/p is 73*x+10; and for n=1128, p:9091, auP/p is 11*x+10. E.g., n=5,x=1: auP/p = 91*1 + 10 = 101 n=5,x=7: auP/p = 91*7 + 10 = 647 etc. Can anyone offer any explanation or insight as to why these particular values (i.e., {91, 77, 73, 11}) correspond to these particular values of n or p? It would also seem that n=1 is true when x is even. n: 1, x:2, p: 2, auP: 222, auP/p:111 n: 1, x:4, p: 2, auP: 424, auP/p:212 n: 1, x:6, p: 2, auP: 626, auP/p:313 n: 1, x:8, p: 2, auP: 828, auP/p:414 auP/p = 10 + x/2*101 n=2 holds when x is 3 or a multiple: n: 2, x:3, p: 3, auP: 333, auP/p:111 n: 2, x:6, p: 3, auP: 636, auP/p:212 n: 2, x:9, p: 3, auP: 939, auP/p:313 auP/p = 10 + x/3*101 n=3 when x=5: n: 3, x:5, p: 5, auP: 555, auP/p:111 auP/p = 10 + x/5*101 and n=4 for x=7: n: 4, x:7, p: 7, auP: 777, auP/p:111 auP/p = 10 + x/7*101 Thanks for the information, btw; fascinating topic. ---Daniel

"Jason Edwards" : Special augmented primes of order 2 (instead of placing a 1 at the beginning and end of the prime, you place a 2): 2, 11,13, 137,... Example: 21372./137 = 156.0000000000000000000000000 Order 3: 3, 11, 13, 137,... Order 4: 2, 11, 13, 137,.. Do I see a pattern here?

Interesting analysis Daniel. I am conducting a little search myself. Here is my Pari code in case anyone is interested. I am sure it isn't efficient by any means: aup(n)=((10^l(prime(n))+prime(n))*10+1) l(n) = ceil(log(n)/log(10)) ? for(n=1,10^6,x=aup(n)/prime(n);if(frac(x)==0,print(prime(n),"\t",x))) 11 101 13 87 137 83 9091 21 909091 21 5882353 27 The program is still running. --Jason

"Daniel Dockery" : > 5882353 27 Excellent. 5882353 also holds for all nine generalized cases, with gaugP/P being 17x+10. I wonder if the next gaugP/P will be higher or lower? For the first four cases, gaugP/P continued to decrease, yet the value here of 17x+10 is higher than the 11x+10 of the 90..91 series of numbers. > The program is still running. Please let us know what else you discover. Thanks. ---Daniel

"Daniel Dockery" : Greetings, Davy wrote: > We've been talking about a special class of > prime numbers. Look out below! This is a > LONG post. Excellent, & many thanks. I was wondering if my posts on the subject had simply been boring everyone else; glad to see that Jason & I aren't the only ones interested in the subject. :) > The first thing that is interesting is to > just simply look at "augmented" numbers by > themselves. For now, the prime aspect is > going to be ignored. Yes, I've been considering the general case of positive integers, myself, augmented by any digit x {1<=x<=9} [for, indeed, as you mention in your post, the rules change altogether for x>9]. > First, lets define some notation: > > a | b means that a divides b evenly or > equivantly a is a factor of b. Ok. > 1n1 will refer to the augment of n. > For example, 131 is the augment of 3. Similarly, I've used the generic xNx to designate N augmented by x, with au(N,x) -> (10^k*(x%10)+N)*10+(x%10) > So starting with these two notations/defintion, > we can say that an augmented number is any > integer n such that n | 1n1. Actually, that would be an special augmented number, would it not? At least, as I understood it, an augmented number is simply the xNx number, and is considered a special augmented number when N | xNx. E.g., from Dr. Pickover's original post on the subject: > The augmented prime is "special" > if it yields an integer when divided > by the original prime number. For > example, 137 is such a number because > 137 is prime and because 11371 / 137 > yields an integer, namely 83. etc. > Well, it stands to reason that since we can > find the number of digits n has, call it > k, that 1n1 = 10^(k+2) + 10*n + 1, Hmm. That method, actually, always seems to insert a 0 digit between the first digit 1 and the number n, so that the result is 10n1. 1n1 = 10^(k+1) + 10*n + 1 seems to work as intended, though. Or 1n1 = (10^k+n)*10+1. > therefore n | 10^(k+2) + 10*n + 1. Um, I may be missing something here, but are you saying that 10^(k+2)+10*n+1 is always going to be divisible by n? Testing for {0 n 1n1 1n1/n 1 1011 1011 7 1071 153 73 10731 147 Even rectifying the equation to remove the leading 0-digit, there are only six cases {0 A nice property of a | b is that if a | b > and a | c it follows from the distributive > property that a | (b+c) > > From this, we can derive that > > If n | 10^(k+2) + 10*n + 1 > Then n | 10^(k+2) + 10*n + 1 - 10*n (since n | -10*n) > Thus n | 10^(k+2) + 1 Excellent. I'd foolishly missed this completely, but it clears up something for me which I'd observed, but hadn't yet worked out. Namely, the significance of the numbers 11, 13, 77 and 91. In an earlier post, I had noted that the integer xNx/N appeared to be a function of x and a number somehow related to N, by noting that for N xNx/N 11 91*x+10 13 77*x+10 137 73*x+10 9091 11*x+10 Or, to phrase it alternatively, for specific values of N, we can create a number xNx with a pre- and post-fixed digit x by multiplying N by the formula a*x+10. E.g., if N=11, and x=2, then xNx=11*(91*2+10)=11*(192)=2112. If N=9091 and x=5, then xNx=9091*(11*5+10)=9091*65=590915, etc. I had wondered how or why the values 91, 77, 73 and 11 are connected to the primes 11, 13, 137 or 9091. When searching through the positive integers, however, and not only primes, for numbers which matched the pattern xNx=N*(a*x+10) for all values of {0 All of our augmented numbers are going to start > following patterns that match reciprocals of two > digit primes. Again, excellent observation. > Lets go to our next prime, seventeen. 11 minus 1 time 10 plus 1 > modulo 17 yields 16. Repeat. The process becomes 11, 16, 15, 5, 7, > 10, 6, 0, 8, 3, 4, 14, 12, 9, 13, 2, 11. So starting at k=8, every > 17th k gives us an augmented number when divided by 17. I suspect > that, yet again, we will find a pattern here in the augmented numbers > that matches 1/17, but the integer library on my HP49G won't let me > perform the division to verify this. For prime P=17, 10^(8*w)+1, odd w: 8*w-1 = l(10^(8*w)+1) w N 1 5882353 3 58823529411764705882353 5 588235294117647058823529411764705882353 7 5882352941176470588235294117647058823529411764705882353 9 58823529411764705882352941176470588235294117647058823529411764705882353 xNx/N = P*x+10. The pattern being 1/17 = 0.05882352941176470588235294117 For prime P=19, 10^(9*w)+1, odd w: 9*w-1 = l(10^(9*w)+1) w N 1 52631579 3 52631578947368421052631579 5 52631578947368421052631578947368421052631579 7 52631578947368421052631578947368421052631578947368421052631579 9 5263157894736842105263157894736842105263157894736842105263157894736842105263 1579 xNx/N = P*x+10. 1/19 = 0.05263157894736842105263157894. For prime P=23, 10^(11*w)+1, odd w: 11*w-1 = l(10^(11*w)+1) w N 1 4347826087 3 43478260869565217391304347826087 5 434782608695652173913043478260869565217391304347826087 7 4347826086956521739130434782608695652173913043478260869565217391304347826087 9 4347826086956521739130434782608695652173913043478260869565217391304347826086 9565217391304347826087 xNx/N = P*x+10. 1/23 = 0.04347826086956521739130434782. And so on. :) Interestingly, there appear to be no solutions for the primes 31, 37, 41, 43, 53, 67, 71, 79, and 83, in terms of primes<100. For the curious, the other primes < 100, for which I found solutions along the lines of the above model: P=47, 10^(23*w)+1, odd w: 1/47 = 0.02127659574468085106382978723 P=59, 10^(29*w)+1, odd w: 1/59 = 0.01694915254237288135593220338 P=61, 10^(30*w)+1, odd w: 1/61 = 0.01639344262295081967213114754 P=73, 10^( 4*w)+1, odd w: 1/73 = 0.01369863013698630136986301369 P=89, 10^(22*w)+1, odd w: 1/89 = 0.01123595505617977528089887640 P=97, 10^(48*w)+1, odd w: 1/97 = 0.01030927835051546391752577319 [...] ---Daniel

Message 567 of 1707 | Previous | Next [ Up Thread ] Message Index Msg # From: David Jones :
```
> Lets go to our next prime, seventeen. 11 minus 1 time 10 plus 1
> modulo 17 yields 16. Repeat. The process becomes 11, 16, 15, 5,
> 7, 10, 6, 0, 8, 3, 4, 14, 12, 9, 13, 2, 11. So starting at k=8,
> every 17th k gives us an augmented number when divided by 17. I
> suspect that, yet again, we will find a pattern here in the
> augmented numbers that matches 1/17, but the integer library on my
> HP49G won't let me perform the division to verify this.
```
Small mistake here. Every 16th k gives us an augmented number, not every 17th. I couldn't verify the pattern here because I was dividing the wrong numbers. A pattern does exist here just like it does for 11 and 13. > Essentially if we continue this process for all 23 of our chosen > numbers, we have a formula for finding every augmented number that > exists. I wrote a small program to find these formulae for us. For the number 11, we get that every 10^(2k+1) + 1 / 11 yields an augmented number. For 13, the pattern is every 10^(6k+3)/13 yields an augmented number. The table below continues:
```11 - 2k + 1
13 - 6k + 3
17 - 16k + 8
19 - 18k + 9
23 - 22k + 11
29 - 28k + 14
47 - 46k + 23
49 - 42k + 21
59 - 58k + 29
61 - 60k + 30
73 - 8k + 4
77 - 6k + 3
89 - 44k + 22
97 - 96k + 48
```
A few quick notes: You might notice that 31, 37, 41, 43, 67, 71, 79, and 83 are missing from the above list. These number never divide 10^k + 1 so this significantly decreases our search. Also of note is that 13 and 77 have the same pattern of 6k + 3. The means that except for k=0, all of the augmented numbers produced by 13 and 77 are automatically non-prime. So in the search for augmented prime numbers, we have already narrowed down our candidates to 12 lists of numbers generated by the process mentioned in the previous message. Davy

## Kelley Primes

Kelly" : i think i remember from reading one of cliff's books about how certain numbers can conjure up strong emotions in people. 666 has always been one of these since primes can only end in 1,3,7,& 9 i thought of looking for what i call "diablo primes." i don't know if anyone has done this befroe me. maybe they have, hell, i don't know. but anyway, they would look like this
```6666...6661
6666...66663
6666..66667
6666....66669

diablo primes exist for 666...6661
-------------------------------------------------------
61
661
6661
6666666661
666666666666666661
666666666666666666661
6666666666666666666661
6666666666666666666666666661
6666666666666666666666666666666666666666661
666666666666666666666666666666666666666666666666666666666666666661
[..many more found but their too large to include.]

66666...6663
---------------------------------------
I found one diablo prime that ended with 3. 3 itself but that it isn't
really a diablo prime because it is only 3 without any 6s.
ha ha....

6666...6667
----------------------------------
7
67
666667
66666667
666666667
66666666667
66666666666666666667
66666666666666666666667
66666666666666666666666666666666666666667
666666666666666666666666666666666666666666666666666666666666667
666666666666666666666666666666666666666666666666666666666666666667
[.there are many more to. ]

but notice that 7 isn't really one because of our previous stipulation of no 6s.

6666...6669
------------------------------------------------------------------
i didn't find a single stinkin' one. might there be one lurking out there in
the infinite?

gaze in wonder at the mystery of the digits.
```
Cliff says, Don't forget that it would be great to submit discoveries like this in the following form, so I can catalog and promote:

## Heavenly Primes

Jason Edwards" : This represents the first submission in Cliff Pickover's catalog of interesting number classes. I have used Cliff's official template for the submission. Cliff may maintain a web page of these and someday include them in a book, with credit to the discoverer.
```Entry 1: Heavenly Primes

1. Name of Number Class:

"Heavenly Primes"

2. Reason you Chose Name:

Because I recall reading somewhere that 7 is God's number and something
about 666 is the number of the beast and so 777 is the number of God or some
such hogwash.

3. Inventor(s) of Number Class (e.g. first proposer or someone
to make a discovery):

Jason Edwards, based on Kelly's idea.

4. Definition of Class (clear, for a general audience):

Primes of the form 77777....7777 with the last digit of course ending in
either 1,3,7 or 9.

5. Examples (of numbers in class):

type 7777...7777:
---------------------
7 is the only one I can find. (There's probably an obvious reason why.)

777...7773
--------------------

3
73
773
77773
777777773
777777777773
777777777777773
777777777777777777773
Jackpot!
There were more as well.

777...7779
-----------------
79
777777777777777777777777777777777777777777777777777777777777777779
7777777777777777777777777777777777777777777777777777777777777777777777777777
7777
777779
7777777777777777777777777777777777777777777777777777777777777777777777777777
7777
7777777779
7777777777777777777777777777777777777777777777777777777777777777777777777777
7777
7777777777777777777779
[more found.]

Those babies are looong.

777 ...7771
---------------
71
7777777777771
77777777777777777771
77777777777777777777771
7777777777777777777777777777771
7777777777777777777777777777777777777777777777777777777777777777777777777777
7777
77777777777777777771
7777777777777777777777777777777777777777777777777777777777777777777777777777
7777
7777777777777777777777777777777777777777777777777777777777777777777777777777
7777
7777777777777777777777777777777777777777777777777777777777777777777777777777
7777
1
[more here too..]

6. Largest Known Number in Class:

If you decide to use these I will do a search for each and see the
largest value of n I can find. Example:
Running for a short while for 777...7771 I get values of n:

he(n) = ((7*10^n-61)/9)
for(n=1,10^6,if(isprime(he(n)),print(n)))

2,13,20,23,31,100,241,275,

7. Special Software Used in the Quest for Examples

PARI version 2.0.17 beta.

But if someone else wanted to use one of the more sophisticated primality
proving programs they are more than welcome.

Are there infinitely many of these heavenly primes? Or will they
eventually terminate? What is the largest someone can find. Is it easier to
find more of one of the types (e.g. 1,3,9) than another? why?

9. Credits (e.g. names of people contributing to the quest):
?

10. References (if this closely relates to other classes of
numbers -- or you got the idea from a related problem.
Be as complete as possible with the references):

Kelly.

"This number class was first discussed in
http://groups.yahoo.com/group/CliffordPickover/"

11. Details (in this section add whatever you like):

I knew that 7./9 produced .7777777...

So I just messed around until I came up with some formulas that seemed to
work. (If anyone can write these more efficiently have at it.)

777...7777 : (7*10^n-7)/9
777...7779: (7*10^n+11)/9
777...7773: (7*10^n-43)/9))
777..77771: ((7*10^n-61)/9)

Enjoy the divine splendor and phantasmagoria of the digits...

--Jason
```

Daniel Dockery" : Greetings, Kelly wrote: [...] > 66666...6663 > I found one diablo prime that ended with 3. > 3 itself but that it isn't really a diablo > prime because it is only 3 without any 6s. > ha ha.... [...] > 6666...6669 > i didn't find a single stinkin' one. > might there be one lurking out there > in the infinite? Just a quick note re the two forms of your sequence which you mentioned above: since all the digits of both forms listed here (66...63, and 66...69) are each divisible by 3 itself, no numbers of this form can be prime --- they'll all be 22...21 x 3 or 22..23 x 3. As far as numbers which could get a response from people, or numbers related to the "apocalyptic" 666, etc., your 66...67 series is likely to be the best. How so? Well, besides the obvious 666 number from the biblical book the "Apocalypse of John", some proponents of isopsephia* note that while 666 is the number of "the beast" of the "Apocalypse", 667 is the name of the woman mentioned riding the same in chapter 17. I note with some amusement that the specific 666667 number (i.e., both values side by side) is actually prime, too. :) *Note: isopsephia (or gematria, in Hebrew) is the technique of assigning a numerical value to a word or phrase based on adding together the numerical values held by its letters when the latter are used as numerals. Of course, this only applies to languages whose alphabet *did*/*does* do the double duty of having both alphabetic and numeric values (Hebrew, Greek, etc.) [As an aside, isopsephia *shouldn't* be confused with modern day "numerology".] For the curious: how do isopsephia supporters purport that 666 and 667 relate to the Beast and the Woman of "Revelations"? According to Rev. 13.18, the number of "the beast" is 666; the Greek phrase "to me/ga the/rion" [where e represents the letter Eta, and the / indicates that the preceding vowel takes the accent/stress mark], translating generally as "the great beast" enumerates to 666; viz., tau + omikron = 370 mu + epsilon + gamma + alpha = 49 theta + eta + rho + iota + omikron + nu = 247 370 + 49 + 247 = 666 In Rev. 17.3 & 4, we have the reference: "Kai\ ei)=don gunai=ka katheme/nen e)pi\ theri/on" ["And I saw a woman sitting on a beast"] ... "kai\ e( gune\ e(=n peribebleme/ne porphurou=n kai\ ko/kkinon" ["And the woman was clothed in purple and scarlet"]. The Greek phrase "e(/ kokki/ne gune/", meaning "the scarlet woman", enumerates to 667: eta = 8 3*kappa + omikron + iota + nu + eta = 198 gamma + upsilon + nu + eta = 461 8 + 198 + 461 = 667 [...] Jason wrote: [...] > Because I recall reading somewhere that > 7 is God's number and something about > 666 is the number of the beast and so > 777 is the number of God or some such > hogwash. As to 7 alone, it depends on the particular numero-philosophical system consulted as to what it "means". (E.g., in many, 7 is simply a reference to the planet Venus in its mythological context, etc.) 777, though, is sometimes used with "divine" significant by Christian commentators through (aberrant) enumeration of things like the Greek word "stauros", etc. (I say "aberrant" because in order to make "stauros", or "cross" [its translation], evaluate to 777, you have to use the *very* unusual concept that the letters 'st' taken together have a value of 6, instead of the sum of the individual values of sigma and tau.) E.g., st + a + u + r + o + s = 6 + 1 + 400 + 100 + 70 + 200 = 777 unlike s + t + auros = 200 + 300 + 771 = 1271 etc. As an aside, 888 seems to hold more Christological significance in that the Greek spelling of Jesus is I)esous, iota + eta + 2*sigma + omikron + upsilon = 10 + 8 + 2* 200 + 70 + 400 = 888 [...] > 5. Examples (of numbers in class): > > type 7777...7777: > 7 is the only one I can find. (There's > probably an obvious reason why.) This is similar to the 66...63 and 66...69 examples from Kelly's sequence in that in this all the digits are divisible by 7 itself, so no matter how many are prefixed, the number will always be a composite number of the pattern 1...1 x 7. > 11. Details (in this section add whatever you like): > > I knew that 7./9 produced .7777777... > > So I just messed around until I came up with some > formulas that seemed to work. (If anyone can write > these more efficiently have at it.) > > 777...7777 : (7*10^n-7)/9 In Maple: ones := proc(n) option remember: if n < 1 then 1 else ones(n-1)*10+1 fi; end: ones(0) := 1: for n from 0 to 100 do ones(n)*7 od; 7 77 777 7777 77777 ... This formulation makes it easy to see why there are no primes for this pattern. > 777...7779: (7*10^n+11)/9 for n from 0 to 100 do ones(n)*7*10+9 od; 79 779 7779 77779 ... For primes only: for n from 0 to 100 do t := ones(n)*7*10+9: if isprime(t) = true then print(t) fi: od: 79 7777777777777777777777777777777777777777 \ 77777777777777777777777779 7777777777777777777777777777777777777777 \ 7777777777777777777777777777777777777777 \ 777779 7777777777777777777777777777777777777777 \ 7777777777777777777777777777777777777777 \ 7777777779 7777777777777777777777777777777777777777 \ 7777777777777777777777777777777777777777 \ 7777777777777777777779 The same formulation could be used for the others fairly simply by defining: j := proc(n,a) ones(n)*7*10+a end: Then, for n from 0 to 100 do t := j(n,1): if isprime(t) = true then print(t) fi: od: or the same but with t := j(n,3): [or] t := j(n,7): [or] t := j(n,9): etc. To pursue other patterns, one can render an even more abstract formula k := proc(n,m,a) ones(n)*m*10+a end: So that something like for n from 0 to 100 do t := k(n,6,7): if isprime(t) then print(t) fi: od: would produce one of Kelly's diablo sequences: 67 666667 66666667 666666667 66666666667 66666666666666666667 66666666666666666666667 66666666666666666666666666666666666666667 ... Something like av := [1,3,7,9]: for m from 1 to 9 do for n from 0 to 100 do for i from 1 to 4 do avi := av[i]: if gcd(m,avi) = 1 then t := k(n,m,avi): if isprime(t) then print(t) fi: fi: od: od: od: will search for all primes of the form m...m1, m...m3, m...m7 and m...m9 when m is coprime to 1, 3, 7 or 9, for n ranging from 0 to 100 (n is two less than the number of digits in the number output). Hope that helps. ---Daniel

## The Leaning Tower of Pisa

Sam Loyd (19th century American puzzlemaster) proposed "The Leaning Tower of Pisa" problem: If an elastic ball is dropped from the Leaning Tower of Pisa at a height of 179 feet from the ground, and on each rebound the ball rises exactly one tenth of its previous height, what distance will it travel before it comes to rest? How would you solve this? How would you extend the problem to something even more challenging?

Derek Ross : Here's my non-standard way of calculating this particular infinite series: 179 times 10% repeatedly, accumulated is: sum = 179 + 17.9 + 1.79 + 0.179 + 0.0179 ... Separate out the 100's, tens and ones into separate infinite series: = 100 + 10 + 1 + 0.1 + 0.01 ... + 70 + 7 + 0.7 + 0.07 + 0.007 ... + 9 + 0.9 + 0.09 + 0.009 + 0.0009 ... Get the sum of each series: = 111.111111... + 77.777777... + 9.999999... Add 111.11 and 77.77, replace 9.99 with 10.0 = 188.888888 + 10 = 198.888888 And to answer the second question, I don't think this answer would help to solver other problems... it's just too specific. Derek. Cliff says, Loyd gives a different answer... I think we can find out why after more thought. I think you could extend this problem to something even more challenging by saying that the ball is strange and its successive heights go as some sequence such as the Fibonacci sequence or something more interesting.

"Daniel Dockery" : First, I'll generalize to n, instead of 179. We know the ball falls n the first time. Then it goes back up n/10 --- but it must also fall n/10 again, so for each successive sequence, rising and falling together, we must add 2*(n/(10^b)) with b being the number of times the ball has fallen. E.g., When the ball is dropped, distance = n, when it has fallen 1 time, distance = n + (n/10^1) [up] + (n/10^1) [down], when 2 times, distance = n + 2*(n/10^1) + 2*(n/10^2), when 3 times, distance = n + 2*(n/10^1) + 2*(n/10^2) + 2*(n/10^3), etc. I think the sum of 2*(n/10^b) converges to n*(2/9), so n+n*(2/9), when n=179, is something like 218+7/9 feet?
```> How would you extend the problem to
> something even more challenging?
```
There are probably any number of other ways which I'm not realizing, but it comes to mind that you could perhaps involve horizontal movement as well as vertical (i.e., in the above we had only enough data to suggest strictly up and down motion --- but by how much would the distance travelled change if we had also to consider how far it moved horizontally with each bounce?), or you could have it bouncing down a hill/slope or flight of stairs so that even though it may go up on each bounce exactly one-tenth of the distance it previously fell, the distance it falls next will depend not only on the previous distance risen, but also on by how much lower the next landing spot is than the previous, etc. Just a few thoughts.

David Jones : I have been trying as well to think of ways of making it more interesting. I don't know that this more "interesting" so much as more difficult, but I have something at home called a "crazy ball". The ball, instead of being round, is composed of several different flat surface of various shapes, sizes, and areas. (Imagine taking a knife to sphere in such a manner that you cut of thin slices of the surface and random angles.) The ball now bounces off the floor at different angles in a "crazy" fashion. The result is that sucessive bounces off the floor go at different heights. Also, because the bounces can convert vertical motion into horizontial motion and vice versa, the ball can actually end up bouncing higher on one bounce than it did on its previous bounce (but obviously less farther.) Finding the expected value of the horizontal distance traveled would certainly be a challenge, but I think its more like a nightmarish excersize in probability than it is a "good" puzzle. I've also been toying with the idea of giving the ball a horizontal motion and making it bounce off of a wall. This too would cause the ball to lose energy. However, figuring out exactly when the ball hits the wall and applying the loss of energy to its horizontal height requires some knowledge of physics, after which it becomes more of a heavy duty calculus problem. Perhaps even simpler yet not overly difficult: Suppose the ball had to fall down a flight of stairs such that for, say, the first ten bounces it ends up falling an extra foot after each bounce. The up distance would not be the same as the down distance for these bounces. Is there an easy way to find the distances for the first ten falls without resorting to a step-by-step "brute force" method?

"Steve Brazzell" : How does one answer the question, in laymen's terms, of the old paradox about the ball having to travel an infinite number of half-distances before it even hits the ground in the first place, thus never reaching the ground? I understand the math, but how would you explain this in understandable terms to a non-math-person? Steve

David Jones : The simplest way I have heard it explained is that the paradox relies on your beleif that an infinite series also has an infinite sum. Once you realize that an infinite series can have a finite sum, things start to make more sense.

"Derek Ross" : That's right, the answer is wrong. I forgot that a bounce of X inches actually travels X*2 inches (as they say, what goes up must come down). My revised answer is: 198.88 * 2 - 179 = 218.7777 You have to multiply my original answer by two to get the full travel of a bounce, but subtract 179, because the initial drop wasn't the result of a bounce. I am now confident that this answer is correct, because it's the same as Dan's answer. (unless we're both wrong!) Derek.

Marcus Rauchfuss" : I say the ball travels a combined 219(179 + 17.9 + 17.9 + 1.79 + 1.79 + 0.179 + 0.179 ...) and something feet after which the fraction it travels becomes less than the natural vibration of molecules and I simply presume that at that point the ball still continues to bounce but the distance bounced is impossible to tell from the natural vibration ("Braunsche Molekularbewegung" sorry, I'm not familliar with the proper English term). I know this is really picky of me but the answer is so much simpler this way...

## Angel Number Guessing Game

An angel materializes on your doorstep.

"I am thinking of two whole numbers," the angel says. "Their product is 1000 times larger than their sum. What are the two numbers?"

"Wait," you say. "That is not enough information."

The angel shakes its head. "It is enough information, and you have 24 hours to give me an answer. If you are correct, you will live the rest of your life in bliss."

"Pedersen" : * Setting myself up as a fool here, are the two numbers "1,000" and "2"? 1,000+2= 1,002 (sum) 1,000x2= 2,000.(product) but hmm, 2,000 isn't 1,000 times larger than 1,002, is it, it's only twice as large...... or maybe "1,000" and "10"? Someone, help!!! April

"Marcus Rauchfuss" : This refers to previous posts on the Angel number guessing game. Angel, the numbers you think of are 1250 and 5000! Am I correct?

"Daniel Dockery" : There are, I think, twenty-five possible solutions (fifty if you consider also the cases where b holds the values of the a column, below, and a holds the following b values), for the natural numbers>0:
```
a b
1001000 1001 a = 1000 * b
501000 1002 a = 500 * b
251000 1004 a = 250 * b
201000 1005 a = 200 * b
126000 1008 a = 125 * b
101000 1010 a = 100 * b
63500 1016 a = 125/2 * b
51000 1020 a = 50 * b
41000 1025 a = 40 * b
32250 1032 a = 125/4 * b
26000 1040 a = 25 * b
21000 1050 a = 20 * b
16625 1064 a = 125/8 * b
13500 1080 a = 25/2 * b
11000 1100 a = 10 * b
9000 1125 a = 8 * b
7250 1160 a = 25/4 * b
6000 1200 a = 5 * b
5000 1250 a = 4 * b
4125 1320 a = 25/8 * b
3500 1400 a = 5/2 * b
3000 1500 a = 2 * b
2600 1625 a = 8/5 * b
2250 1800 a = 5/4 * b
2000 2000 a = b

such that (a+b)*1000 = a*b.

If we let c be a/b, then you'll notice that c*1000
is either a power of 5, or a power of 5 multiplied
by a power of 2.

c*1000
1000000 = 2**6 * 5**6
500000 = 2**5 * 5**6
250000 = 2**4 * 5**6
200000 = 2**6 * 5**5
125000 = 2**3 * 5**6
100000 = 2**5 * 5**5
62500 = 2**2 * 5**6
50000 = 2**4 * 5**5
40000 = 2**6 * 5**4
31250 = 2**1 * 5**6
25000 = 2**3 * 5**5
20000 = 2**5 * 5**4
15625 = 2**0 * 5**6
12500 = 2**2 * 5**5
10000 = 2**4 * 5**4
8000 = 2**6 * 5**3
6250 = 2**1 * 5**5
5000 = 2**3 * 5**4
4000 = 2**5 * 5**3
3125 = 2**0 * 5**5
2500 = 2**2 * 5**4
2000 = 2**4 * 5**3
1600 = 2**6 * 5**2
1250 = 2**1 * 5**4
1000 = 2**3 * 5**3
```
---Daniel

"German Gonzalez" : I have an answer to the problem Cliff posted previously. The numbers are: x:1001000 y:1001 x+y = 1002001 x*y = 1002001000 Is this the correct answer?

"Marcus Rauchfuss" : Hey April, relax, you do not set yourself up as a fool. The only fools are they who declare others to be fools. Here's how I got there: since X*Y has got to be exactly 1000 times x+y I had to go for sums and products with "round" results (please excuse the fact that I do not use correct English maths terminology). In order to get round results you can basically only operate with multiples of 2 and 5 (1000 is not dividable by 3,5,6,7 and of course 9) I figured the X had to be a "straight" multiple of 5 (5, 50, 500 etc) the Y I tackled in steps of 25. The product of course had to be >1000. Try to find it out now! Good luck!

Let's start this collection of paradoxes with a simple one. Perhaps you heard this sort of fallacy when you were a child. If you know children, why not try this on them? Let me know their response.

For now, I want to try it one of my regular customers.

I go up to Big Tony who is currently wolfing down a sausage and pepper calzone. "Big Tony," I say. "Pardon me for the interruption." I present two pepperoni pizzas to Big Tony. Just by looking at the two pizzas, you can see that one pizza has three pepperonis and the other has three pepperonis making a total of six pepperonis. "Big Tony," I say, "how many pepperonis do you see on these two pizzas?"

Pepperonis symbolized by * symbols

* * * * * *
Pizza 1 Pizza 2

Big Tony flexes his biceps and looks suspicious. "Luigi," he says, "are you trying to pull something over on me?"

"Not at all. I'm trying to teach you about paradoxes and fallacies."

"What's the difference?"

"A fallacy produces a wrong answer using explanations that sometimes appear to be very logical. A fallacy is a statement or an argument based on a false or an invalid inference. A paradox is a seemingly contradictory statement that may nonetheless be true."

Big Tony takes a bite of his calzone and then looks at me and then back at the pizzas. "Luigi, I see six pepperonis."

I shake my head. "Let me prove to you that you are wrong." I point at the pepperonis, in turn, on the first pizza and start counting, "1, 2, 3," and then I pause and point to the second pizza and continue counting, "4, 5, 6." I pause again. "Hold on. I think I made an error." I start counting backwards while pointing to the second pizza, "6, 5, 4." I am now pointing at the last pepperoni on the second pizza and have said the word "four." Next, I stop and say, "Four, and three more on the other pizza makes seven."

Big Tony looks at me, very confused. "Hold on."

"Let me try again." I take out a magic marker and start drawing on his table cloth.

Pepperonis on Pizza 2 Pepperonis on Pizza 1
* * * * * *
6, 5, 4 Plus three more make seven total

I start counting backwards while pointing at the second pizza, "6, 5, 4." When I say the word "four," I then point to the first pizza and say, "and three more on the first pizza makes 7. This means the pizza has seven pepperonis."

Big Tony grumbles, gets up, and starts running after me.

What is wrong with this argument? How would children of various ages respond? (If you have access to a child, please tell us how they responded.) Are there similar problems that are slightly more complex? What's wrong with the argeument?

Cliff Pickover, http://www.pickover.com

chuck1863@qwest.net : By continuing the backward count to the first pizza you'd count 6, 5, 4, 3, 2, 1. There's only one piece of pepperoni. Who makes these pizza anyway? If I ordered a pepperoni pizza and there were three pieces of pepperoni on it I wouldn't be going back there again.

David Jones : The fallacy is that when you count seven pepperoni's that you are mixing up addition and subtraction in the middle of a counting process. This is the same fallacy that occurs in the classic, yet more complex, problem of three gentlemen who rent a hotel room for \$30 but then get a \$10 discount and one of the dollars gets lost in the process. Davy

## Would you increase your IQ by 40 points?

Would you be willing to live one year less if it meant that today your IQ were boosted by 40 points? Example: if you were going to die at age 75 with your current IQ of 130, you would now live to age 74 with an IQ of 170.

Would you give up two years for an 80-point increase?

Would you pay \$20,000 for an 80-point increase?

Regarding the question of downloading yourself to a virtual-reality simulation of a mall, I would certainly do it I were dying. If the mall were sufficiently realistic, you could do the same things you would enjoy doing today from meditation to rock climbing (e.g. climbing areas of the mall). If the people in the mall were sufficiently kind, you could establish loving, meaningful relationships. The key, for me, would center on the question of who would be in the mall with me. For the moment, let's assume that the inhabitants are people you would enjoy or tolerate being with.

After all, what's the current alternative except for the blackness and blankness of death and the rotting of your corpse?

: While I would definitely download myself into the mall,I would not pay for an IQ boost.I would take one for free, though (after all, it's free). I'm sufficently satisfied with my IQ and giving one year of your life is far to high a price to waste it on increased intelligence. Besides, being vastly intelligent is not necessarily a blessing. Many not so intelligent people often live happier lives because there are some problems they don't understand and ignore or don't bother to think about. I would consider it if I could have a condition attached to the IQ boost. If I could specify what exact benefit which the IQ boost would give me I might consider spending \$20.000.

"Quinn Tyler Jackson" : >Would you be willing to live one year less if it meant that today >your IQ were boosted by 40 points? Example: if you were going to >die at age 75 with your current IQ of 130, you would now live to age >74 with an IQ of 170. If my current IQ were 130 -- yes I would. >Would you give up two years for an 80-point increase? Again, if my current IQ were 130, yes indeed I would. >Would you pay \$20,000 for an 80-point increase? Again, if my current IQ were 130, yes indeed, I would -- iff I had the money. But then again, maybe I would not. Assuming I had an IQ of 130, I would not know what delights 170 had in store for me. However, if my IQ were between 105 and 115, I would not opt for the 40 point increase, since that would put me in the 145-155 zone, which is known to a few in UHIQ circles as the "danger zone." The D.Z. is quite a phenomenon -- it is where, if hypotheses are correct, one is quite smart enough to be extremely smart, but not *quite* smart enough to realize that there are others who are smarter. Smart enough to read and understand extremely tricky theory, but not quite smart enough, when a stumbling block is reached, to make that "leap of faith" that Gallois alluded to. So, for no amount of money, if my starting IQ started at some number, could you pay me to land into that zone. Of course, it's easy to say: "I would not want to be stuck in the D.Z. rut." If my IQ were 105, would I believe that such a rut even existed? Once I arrived there, having sacrificed that year to do so -- would I even recognize I was there? If an IQ rises in the forest, and there is only a dogmatic thinker there to hear it -- did it rise? Here is a twist on your question, Dr. Pickover -- a variation on a line from Speed, starring Hopper, Reeves, and Bullock: "You're traveling in a bus full of 50 people. You've been told by aliens that they'll all be allowed to live if you allow them to reduce your IQ from 180 to 90. Whatya do?"

"Quinn Tyler Jackson" : > After all, what's the current alternative except for the blackness > and blankness of death and the rotting of your corpse? Well, I will step outside my belief system for a moment, and take the position of someone who believes the above. Without regard for an afterlife -- we die, and in doing so, make room for the next generation to progress. Let us imagine a world where the wealthy and powerful's lifespans were extended by 300 years. (They'd be the ones who could afford to rot latest.) The world would *still* be full of those who committed the atrocities of WW I and II. Pol Pot would still be a young buck. J.E. Hoover would still be dancing the two step on everyone's civil liberties. There would former slave owners sitting on their plantations, pining for the days when .... But these people were allowed by time and space to pass, and with them, many of their mistakes, misconceptions, and biases went as new generations took their places. The fabric of society is allowed, by means of a simple enough thing as a short, finite lifespan, to correct itself. The societal alternative to death and replacement is stagnation. Cold introspect can be Onan's lover, Spilling seed to waste in times of feasting, Whose offspring never come. Plenty's fleeting, Though, and the famine comes, but there's no cover From the gatherer of souls, no keeper Of the parents who never were. Counting On their own strength, week now from age, they sing To their Reason for a helping hand. Where Is the young hand to lift them from the grave? Their barren house has no heir to protest When hunger hits and waits at their white door. Onan's children do not speak up for him When a reckoning is made. No young breast To offer comfort when the eyes grow dim.

chuck1863@qwest.net : I'd gladly pay \$20,000 for 80 points of IQ. I'd expect to make it back almost immediately in the form of a signing bonus from one of the rich corporations that would then want me. I'm not sure about trading years. I might find myself already dead. If I knew I'd live to 74 for sure then I'd do it just to explore the consequences of this knowledge. I could have all sorts of fun if I knew I couldn't die. Well, maybe not. If I did something stupid I might spend those years in coma. But it's still sort of like time travel even if it's just information. I'm 50 now. I think I'd take the 80 points for 2 years deal. The 23 remaining years would be much more insteresting with an extra 80 points. Maybe I could have myself frozen at the end, or is such cheating of death not allowed? If I didn't know how many years I had left then the deal doesn't sound nearly as good.

genie@megasociety.com : Contrary to popular belief, that is not at all the case. The truth is that *who you know* means a heckuva lot more than most other factors in virtually every field and every hiring situation. And hiring a real genius is more of a "package" deal than most corporations want. Make no mistake about it ;) Corporations are not looking for the very best and brightest, they're typically looking for competent yes-men. Many human resource departments look for the moderately-gifted and shun the ultra- bright. I can't say that I blame them all that much. A larger percentage of people with 4-sigma+ IQs (over 164) are natural born leaders, more apt to be stubborn, headstrong and visionary, dissatisfied with the rote and mundane, preferring to follow their own goals rather than the corporation's. They are not looking for that. An interesting question is: What would happen if *everyone's* IQ was raised by 50 points, say over the course of a year? That would change the structure of society very suddenly and very profoundly. Johns Hopkins announced last year that within 10 years "smart drugs" would be available that would be able to raise human IQ by 50 points on average (I imagine the increase would be greatest on the low end). Who will get these drugs? Should they be generally available or will economics be a factor? And who should pay? Can they be made generally available without dire consequences to social structure? If everyone gets them, who will flip the burgers and sweep the floors? On the other hand, can't humanity use all the brainpower it can muster to solve the scientific and ethical dilemmas on the horizon? ~~~ Dr. Gina Lynne LoSasso Clinical Neuropsychologist

How naive. Of COURSE economics would not only be the determining factor, it would really be the ONLY factor. The rich would get richer, and the poor would get the dregs. When has it been otherwise? That is assuming that an increase in IQ is found to be worth anything at all.... Pete B

Quinn Tyler Jackson" : Pete Barnes: "Such people might be capable of amazing things with respect to solving complex problems in swift order or undertaking daunting tasks with success, but most of them are sorely lacking in the social graces and motivational skills necessary to be a great leader, primarily because they usually lack any sense of empathy." Considering how rudely you worked your reply, perhaps that lack of empathy encountered was simply the result of projection? Pete Barns: "Yes-men do not get very far in companies, nor do they find great success in life." I agree 100%! Here here! Barnes: "politicians (who have, perhaps, a _collective_ IQ of 160, if you get enough of them together in one place e.g. Congress)" No elitism *here*, eh Pete? I'm finding less to win as more is won, I'm finding less to praise as praise is sung, And, since I have said what I came to say, I'm finding less to say as time goes on. When I was younger, I was full of say, And at each insult quickly joined the fray, But now I'm more inclined to hold my tongue, For I have found that more is said that way. My name: but one syllable to defend; My honor: just a concept to offend; My pride: it has never fed me or mine; That these concepts get hurt, I won't pretend. Surely, on my name some will howl a claim, Some dirt to my honor they will proclaim, And my pride will hurt if I don't respond: If I let them suffer, mine is the blame. But before charging lance-held, I must think, Is the greater damage worth all the stink? For if the greater damage is not done, My name, honor, and pride shall not then shrink. But if I charge into a useless fight, Pretending I defend a Greater Right, And all I defend is petty interest, The words are wasted in their empty might.

"peteb_hwp" : I found the statement claiming that "A larger percentage of people with 4-sigma+ IQs (over 164) are natural born leaders", to be untrue, derogatory, and even mildly insulting, although I am sure it was not really intended that way. Indeed, the statement exemplified the lack of sensitivity and empathy to which I referred. So the tone of my reply was quite justified. OTOH, I admit that perhaps I may have been overreacting. You are correct that my language was unnecessarily harsh.

Hardly, since feeling elite when in the company of politicians is a natural reaction, just like the reflexive impulse to hold on tightly to your pocketbook or wallet in such company. :=) If you doubt, go to a political convention sometime as a delegate..... Pete B

"dg123478" <74644.1103@compuserve.com: Pete, factor, it would really be the ONLY factor. The rich would get > richer, and the poor would get the dregs. When has it been otherwise? > It has often been otherwise. The rich do not always get richer; they often get exterminated. Look at the butchery, expropriation, or expulsion waged against the jews in Germany, the Chinese in Indonesia, the wealthy in the former USSR, the Ibo in Nigeria, the prosperous and productive in communist countries such as Cuba and China (until very recently) among others. Maybe this is one reason why we in the US maintain a large defense establishment. And, then there is the constant whining of US liberals about the 'evil rich' and tax policies such as the alternate minimum (federal) tax designed to punish high income earners. There is an entire chapter entitled 'The War Against Wealth' (Chapter 9) in 'Wealth and Poverty' by George Gilder. Dennis

Quinn Tyler Jackson" : Chuck1863 said: "I'd gladly pay \$20,000 for 80 points of IQ. I'd expect to make it back almost immediately in the form of a signing bonus from one of the rich corporations that would then want me." I apologize for replying to this with a URL.... but the saying goes: "Be careful what you wish for..." http://QuinnTylerJackson.n3.net/writing_editing/shorts/TheVitruvian.html The "danger zone" I referred to in an earlier post has the benefit that it tends to be where the rich companies retain enough control over the employee for them to be worth the effort. Bright enough to be able to tackle most any problem, but not quite enough to be overflowing with dangerous new ideas. Would you believe that many in the IQ range you're referring to above actually have learned to HIDE their IQs?

genie@megasociety.com : > Would you believe that many in the IQ range you're referring to above actually have learned to HIDE their IQs? > I know Quinn has read "The Outsiders", but some other list members may be interested in this well-written and insightful piece: http://www.megafoundation.org/Ubiquity/Outsiders.html The author, Grady Towers, was a case in point. With an IQ of about 160 (rarity approx. 1/25,000), Grady worked as a security guard. (Tragically, Grady was murdered last year while investigating a break- in at the park facility he guarded.)

"Quinn Tyler Jackson" : Quinn: "Would you believe that many in the IQ range you're referring to above actually have learned to HIDE their IQs?" Gina: "I know Quinn has read 'The Outsiders,' but some other list members may be interested in this well-written and insightful piece...." Yes, it is Grady's best piece, IMO. A very important part of our cultural literature. Further to the notion that someone with an 80 point boost would be gobbled up by the Big Corporation is the following piece: http://qtj.n3.net/writing_editing/articles/ShameOfIntelligence.html But we're not gonna take it anymore. ;-)

: I think we need a sliding scale for this offer since one year will mean more for an older individual like me at 59 than it would for someone of, say, 20 years of age. We might need an exponential function relating age to IQ points offered. That is, to make it more attractive for me maybe the offer should include more IQ points for that year of sacrifice than would be offered for a youth. And, what if someone of age 75 were to accept and find out that the current year was the one being subtracted? > > Would you pay \$20,000 for an 80-point increase? > 20,000.00 for 80 IQ points. Wow. What a deal. I'll take it. Any preference - credit card or personal check for payment? I would expect that the additional IQ points would make me smart enough to yield a handsome return on that 20 grand. And, besides this would be a nice down payment on a partial reincarnation to replace some of the lost brain cells destroyed during my mis-spent youth. Dennis

math@antiquark.com I always got the impression that the power of the IQ paled in comparison to the power of the MQ -- the Motivation Quotient. I would definitely prefer to have my motivation increased. I think that motivation leads to far more success than a high IQ. For example, how many people know of some super-smart person who hasn't done much with his/her life because they're just not very motivated? And, how many people know of someone who was of average intelligence, but went vary far simply because they worked their butts off? I'm making up the idea of the MQ, but I imagine it would be a formula like: 100*(Things you have actually done)/(Things you say you will do). Thus, if you said that you were going to write 10 novels, but actually wrote 5, then your MQ would be 100*5/10 or 50. Notice that the inverse of the MQ is the BSQ, or Bullshit quotient. Derek. More info on MQ here: http://groups.yahoo.com/group/CliffordPickover/message/1213

## Ant Problem

Aliens capture you and seat you in front of a large terrarium containing colonies of red ants (R), black ants (B), fire ants (F), and army ants (A), each species at the corners of a square. Your captors ask you to create a tunnel out of plastic tubing linking all four species of ants together. You must use as little tubing as possible and still allow access from any colony to any other. Your captors provide you with a little glue to help stick pieces of tubes together, if necessary. What is the best solution?

Here is one solution with intersections at 120 degrees. Is it the best?

### Solutions

From "Mark Ganson" :

I believe the tube configuration on your website using 120 degree intercepting angles is the optimal one for this puzzle. Optimal, at least, in the sense that it uses the least amount of tubing. Depending upon the size of the tubing in relation to the size of the ants, there might be some traffic congestion in the vertical tube, leading to possible hostilities, especially among the fire and army ant colonies. (Tube rage?)

Using Mathematica, I wrote a program that calculates the tube lengths used with varying intersecting angles. The program test angles from 90 degrees to 179 degrees. (It can't directly test 180 degrees as written because it gives a divide by zero error, but that's okay, because we can readily figure 180 degrees in our heads.)

The configuration consists of 4 diagonal sections of tubing (2 on top and 2 on bottom) and one vertical tube, which connects to the 4 diagonal sections. The 2 top diagonal tubes connect to each other at an intersecting angle of 120 degrees. The 2 bottom tubes connect to each other at the same angle. The vertical tube connects these intersections together. The total length of tubing being used will vary with the intersecting angles. My program has revealed that the 120 degree angle is the one that results in the least amount of tubing needed to create the pathways.

With an intersecting angle of 90 degrees, we have a vertical tube length of 0. In fact, we have no vertical tube at all because the diagonal tubes will all intersect in the middle of the ant farm. The configuration resembles an "X". At first, I thought this would be the way to do it with the least amount of tubing being used since the hortest distance between 2 points is a straight line. Using the Pythagorean Theorem, we can calculate that the total tube length for this criss cross configuration would be 2 X the square root of 10^2 + 10^2 (totalTubeLength = 2 x Sqrt[10^2 + 10^2] = 28.284).

```
tubeLengths = {};
Do[
farmSide = 10;
baseAngles = (180 - intersectingAngles)/2;
intersectingAngles *= Degree;
baseAngles *= Degree;
diagonalTubeLengths = (farmSide * Sin[baseAngles])/
Sin[intersectingAngles];
verticalTubeLength =
farmSide - (2*(farmSide/2 *
Sin[baseAngles]/Sin[intersectingAngles/2]));
totalTubeLength = 4 * diagonalTubeLengths + verticalTubeLength;
AppendTo[
tubeLengths, {intersectingAngles / Degree,  N[totalTubeLength]}];
Print[ "intersectingAngles = ", intersectingAngles,
" totalTubeLength = ", N[totalTubeLength], "  vertical tube = ",
N[verticalTubeLength]];
, {intersectingAngles, 90, 179}
];
ListPlot[tubeLengths, PlotJoined -> True];

Here is the output:

intersectingAngles = 90  totalTubeLength = 28.284271247461902  vertical
tube
= 0.
intersectingAngles = 91  totalTubeLength = 28.213668666324637  vertical
tube
= 0.173027368
intersectingAngles = 92  totalTubeLength = 28.14638407226283  vertical tube

= 0.3431122519
intersectingAngles = 93  totalTubeLength = 28.082323786485514  vertical
tube
= 0.510354332
intersectingAngles = 94  totalTubeLength = 28.021398360595292  vertical
tube
= 0.674849138
intersectingAngles = 95  totalTubeLength = 27.963522358010902  vertical
tube
= 0.836688259
intersectingAngles = 96  totalTubeLength = 27.908614149149123  vertical
tube
= 0.995959557
intersectingAngles = 97  totalTubeLength = 27.85659571937542  vertical tube

= 1.1527473544
intersectingAngles = 98  totalTubeLength = 27.80739248881396  vertical tube

= 1.3071326218
intersectingAngles = 99  totalTubeLength = 27.760933143181028  vertical
tube
= 1.459193145
intersectingAngles = 100  totalTubeLength = 27.717149474872773  vertical
tube = 1.609003688
intersectingAngles = 101  totalTubeLength = 27.67597623359938  vertical
tube
= 1.7566361418
intersectingAngles = 102  totalTubeLength = 27.637350985913272  vertical
tube = 1.902159668
intersectingAngles = 103  totalTubeLength = 27.601213983029698  vertical
tube = 2.045640833
intersectingAngles = 104  totalTubeLength = 27.56750803638441  vertical
tube
= 2.1871437349
intersectingAngles = 105  totalTubeLength = 27.53617840041569  vertical
tube
= 2.3267301202
intersectingAngles = 106  totalTubeLength = 27.50717266209657  vertical
tube
= 2.4644594989
intersectingAngles = 107  totalTubeLength = 27.480440636778738  vertical
tube = 2.600389249
intersectingAngles = 108  totalTubeLength = 27.455934269942183  vertical
tube = 2.734574719
intersectingAngles = 109  totalTubeLength = 27.433607544474633  vertical
tube = 2.867069321
intersectingAngles = 110  totalTubeLength = 27.41341639313202  vertical
tube
= 2.9979246179
intersectingAngles = 111  totalTubeLength = 27.395318615856812  vertical
tube = 3.127190413
intersectingAngles = 112  totalTubeLength = 27.379273801653838  vertical
tube = 3.254914831
intersectingAngles = 113  totalTubeLength = 27.365243254745046  vertical
tube = 3.381144388
intersectingAngles = 114  totalTubeLength = 27.353189924743845  vertical
tube = 3.505924068
intersectingAngles = 115  totalTubeLength = 27.343078340608283  vertical
tube = 3.629297391
intersectingAngles = 116  totalTubeLength = 27.334874548148655  vertical
tube = 3.751306480
intersectingAngles = 117  totalTubeLength = 27.328546050880856  vertical
tube = 3.871992118
intersectingAngles = 118  totalTubeLength = 27.324061754031003  vertical
tube = 3.991393809
intersectingAngles = 119  totalTubeLength = 27.32139191150995  vertical
tube
= 4.1095498357
intersectingAngles = 120  totalTubeLength = 27.32050807568877  vertical
tube
= 4.2264973081
intersectingAngles = 121  totalTubeLength = 27.321383049817317  vertical
tube = 4.342272218
intersectingAngles = 122  totalTubeLength = 27.323990842938706  vertical
tube = 4.456909485
intersectingAngles = 123  totalTubeLength = 27.328306627162235  vertical
tube = 4.570443003
intersectingAngles = 124  totalTubeLength = 27.334306697165992  vertical
tube = 4.682905683
intersectingAngles = 125  totalTubeLength = 27.341968431809384  vertical
tube = 4.794329494
intersectingAngles = 126  totalTubeLength = 27.35127025774292  vertical
tube
= 4.9047455050
intersectingAngles = 127  totalTubeLength = 27.36219161491045  vertical
tube
= 5.0141839194
intersectingAngles = 128  totalTubeLength = 27.374712923845166  vertical
tube = 5.122674114
intersectingAngles = 129  totalTubeLength = 27.388815554667723  vertical
tube = 5.230244673
intersectingAngles = 130  totalTubeLength = 27.404481797699848  vertical
tube = 5.336923418
intersectingAngles = 131  totalTubeLength = 27.421694835612882  vertical
tube = 5.442737444
intersectingAngles = 132  totalTubeLength = 27.440438717035576  vertical
tube = 5.547713146
intersectingAngles = 133  totalTubeLength = 27.460698331550088  vertical
tube = 5.651876250
intersectingAngles = 134  totalTubeLength = 27.48245938600988  vertical
tube
= 5.7552518379
intersectingAngles = 135  totalTubeLength = 27.50570838211693  vertical
tube
= 5.8578643762
intersectingAngles = 136  totalTubeLength = 27.530432595200104  vertical
tube = 5.959737741
intersectingAngles = 137  totalTubeLength = 27.55662005413962  vertical
tube
= 6.0608952438
intersectingAngles = 138  totalTubeLength = 27.584259522386418  vertical
tube = 6.161359649
intersectingAngles = 139  totalTubeLength = 27.613340480028075  vertical
tube = 6.261153205
intersectingAngles = 140  totalTubeLength = 27.643853106856213  vertical
tube = 6.360297657
intersectingAngles = 141  totalTubeLength = 27.67578826639296  vertical
tube
= 6.4588142746
intersectingAngles = 142  totalTubeLength = 27.709137490836756  vertical
tube = 6.556723867
intersectingAngles = 143  totalTubeLength = 27.743892966890407  vertical
tube = 6.654046804
intersectingAngles = 144  totalTubeLength = 27.780047522436288  vertical
tube = 6.750803037
intersectingAngles = 145  totalTubeLength = 27.817594614026387  vertical
tube = 6.847012111
intersectingAngles = 146  totalTubeLength = 27.856528315156357  vertical
tube = 6.942693185
intersectingAngles = 147  totalTubeLength = 27.896843305295228  vertical
tube = 7.037865050
intersectingAngles = 148  totalTubeLength = 27.938534859643962  vertical
tube = 7.132546142
intersectingAngles = 149  totalTubeLength = 27.98159883959795  vertical
tube
= 7.2267545594
intersectingAngles = 150  totalTubeLength = 28.02603168389043  vertical
tube
= 7.3205080756
intersectingAngles = 151  totalTubeLength = 28.071830400394802  vertical
tube = 7.413824156
intersectingAngles = 152  totalTubeLength = 28.118992558566163  vertical
tube = 7.506719971
intersectingAngles = 153  totalTubeLength = 28.16751628250301  vertical
tube
= 7.5992124091
intersectingAngles = 154  totalTubeLength = 28.217400244612207  vertical
tube = 7.691318088
intersectingAngles = 155  totalTubeLength = 28.26864365986078  vertical
tube
= 7.7830533735
intersectingAngles = 156  totalTubeLength = 28.321246280600356  vertical
tube = 7.874434383
intersectingAngles = 157  totalTubeLength = 28.37520839195024  vertical
tube
= 7.9654770057
intersectingAngles = 158  totalTubeLength = 28.43053080772709  vertical
tube
= 8.0561969086
intersectingAngles = 159  totalTubeLength = 28.48721486690997  vertical
tube
= 8.1466095506
intersectingAngles = 160  totalTubeLength = 28.54526243063024  vertical
tube
= 8.2367301929
intersectingAngles = 161  totalTubeLength = 28.60467587967779  vertical
tube
= 8.3265739091
intersectingAngles = 162  totalTubeLength = 28.665458112514695  vertical
tube = 8.416155596
intersectingAngles = 163  totalTubeLength = 28.727612543789974  vertical
tube = 8.505489986
intersectingAngles = 164  totalTubeLength = 28.791143103348443  vertical
tube = 8.594591652
intersectingAngles = 165  totalTubeLength = 28.8560542357291  vertical tube

= 8.68347502412
intersectingAngles = 166  totalTubeLength = 28.922350900147922  vertical
tube = 8.772154390
intersectingAngles = 167  totalTubeLength = 28.99003857096223  vertical
tube
= 8.8606439169
intersectingAngles = 168  totalTubeLength = 29.059123238613566  vertical
tube = 8.948957647
intersectingAngles = 169  totalTubeLength = 29.129611411047154  vertical
tube = 9.037109518
intersectingAngles = 170  totalTubeLength = 29.201510115607714  vertical
tube = 9.125113364
intersectingAngles = 171  totalTubeLength = 29.274826901410314  vertical
tube = 9.212982931
intersectingAngles = 172  totalTubeLength = 29.349569842188355  vertical
tube = 9.300731880
intersectingAngles = 173  totalTubeLength = 29.425747539618733  vertical
tube = 9.388373798
intersectingAngles = 174  totalTubeLength = 29.503369127127957  vertical
tube = 9.475922207
intersectingAngles = 175  totalTubeLength = 29.582444274181274  vertical
tube = 9.563390570
intersectingAngles = 176  totalTubeLength = 29.662983191058892  vertical
tube = 9.650792305
intersectingAngles = 177  totalTubeLength = 29.7449966341255  vertical tube

= 9.73814078430
intersectingAngles = 178  totalTubeLength = 29.82849591159588  vertical
tube
= 9.8254493507
intersectingAngles = 179  totalTubeLength = 29.913492889807237  vertical
tube = 9.912731322
```
The plot is attached as antfarmplot.gif (Below). For those interested, here is an explanation of the source code: tubeLengths = {}; This is a List variable we will use to record the calculated tube lengths in order to create a nice little chart afterwards.

Do[ The Do Loop uses the intersecting angles of the diagonal tubes, relative to each other, as the iterating index value. The loop begins with an intersecting angle of 90 degrees, and increments by 1 until we get past 179, at which time it terminates.

farmSide = 10; I chose to use an ant farm height and width of 10 units for the sake of simplicity. Insofar as we're only concerned with finding the optimum angles, the actual ant farm size doesn't really matter since the best angle will always result in the least amount of tubing used, regardless of the ant farm size.

baseAngles = (180 - intersectingAngles)/2; The baseAngles would be the angles of the diagonal tubes relative to an imaginary line drawn between the ant colonies at the top and bottom of the ant farm. This is calculated quite easily since we know that the sum of all angles in any triangle will always be 180 degrees.

intersectingAngles *= Degree; baseAngles *= Degree; Here, we use a built-in constant value to convert from degrees to radians. The actual value of Degree is Pi/180.

diagonalTubeLengths = (farmSide * Sin[baseAngles])/ Sin[intersectingAngles]; This gives us the lengths of each of the diagonal tubes using a well-established trig formula.

verticalTubeLength = farmSide - (2*(farmSide/2 * Sin[baseAngles]/Sin[intersectingAngles/2]));

This is a little trickier. It calculates the length of the vertical tube. By splitting the triangle created by the intersecting diagonal tubes and an imaginary tube connecting the non-intersecting ends of the diagonal tubes together, we create a right triangle. With this new right triangle, we can measure the vertical distance covered by the diagonal tubes. We mutliply this value by 2 since we have 2 sets of diagonal tubes. Then, we subtract from the ant farm height (farmSide) to get the length of the vertical tube.

totalTubeLength = 4 * diagonalTubeLengths + verticalTubeLength; We calculate the total tube length by summing the 4 diagonal tubes with the 1 vertical tube.

AppendTo[tubeLengths, {intersectingAngles / Degree, N[totalTubeLength]}]; We append each new set of coordinates here for the plot we'll do later on. The N[totalTubeLength] designation forces Mathematica to save the data in a numerical format, rather than as an expression of Sin ratios.

Print[ "intersectingAngles = ", intersectingAngles, " totalTubeLength = ", N[totalTubeLength], " vertical tube = ", N[verticalTubeLength]]; Prints out the calculated values for comparison.

, {intersectingAngles, 90, 179} ]; Ends the Do Loop.

ListPlot[tubeLengths, PlotJoined -> True]; Creates the graph using the List of tubeLengths. The vertical bar in the graph gives the total tube lengths being used. The horizontal bar shows the intersecting angles. The graphical plot and the numerical data both agree that the optimal angle is 120 degrees.

Why is 120 degrees optimal? Symmetry. We are joining 3 tubes together in 2 different locations. At the top and bottom of the ant farm, we have 2 diagonal tubes connecting to the top and bottom of a single vertical tube. We have 3 angles at 120 degrees each on both the top and the bottom for a total of 360 degrees at both the top and bottom. Recall that there are 360 degrees in a circle. If we were to split a circle into 3 equal sections, we'd have the same angles (3 x 120), very much like the Mercedes Benz logo. If we intersect the diagonal tubes at any angle other than 120 degrees, we are forcing the angle of the vertical tube relative to the diagonal tubes out of symmetry. For example, if we have the diagonal tubes intersect each other at 100 degrees, we end up with a shorter vertical tube, because the diagonal tubes will be extending closer to the center of the ant farm, and we end up with 2 angles of 130 degrees each where the vertical tube connects with each diagonal tube to go with the 100 degree angle. In this case, goodbye symmetry means goodbye shortest tube length.

Mark Ganson

I had attached the file named antfarmplot.gif to an earlier message. I'll attach it to this one, too, since it is such a small file.

The graph shows the amount of tube used for the given intercepting angles. The plot is an arch resembling the Nike swoosh, with the low tube length point intersecting at the 120 degree mark.

There are a number of variations possible with the puzzle that come to mind. Your ant farm could be rectangular rather than square-shaped. You could have a pentagon-shaped ant farm with 5 different ant colonies, an octagon-shaped ant farm with 8 different ant colonies, etc. You could have a circular-shaped ant farm with 12 different colonies, each situated at a different hour position on the face of a clock: 1 o'clock, 3 o'clock, etc.

Another variation might be to keep the same ant farm shape, but lose one of the colonies, so that you have only 3 colonies, leaving one of the corners unoccupied. To me, for some odd unknown inexplicable reason, this seems the most intriguing variation. Assuming the upper right corner is unoccupied, which tube configuration would use the least amount of tubing? an "L" shape or a partial "X" without the upper right diagonal? or a different configuration altogether?

Solution: The "L" would take 10 + 10 = 20 tube units, whereas the partial "X" would take 1.5 * Sqrt[10^2 + 10^2] = 21.2132 tube units. The optimal configuration would be to extend a tube from each colony so that the 3 tubes intersect at a common point within the ant farm such that the angles of intersection are all at that magical 120 degree angle. If the upper right corner is unoccupied, the point of intersection would be, unless I'm wrong, right smack dab in the center of the lower left quadrant at coordinates (2.5, 7.5) in a 10 x 10 grid. (See 3antcolonies.gif, attached hereto.) It's all about symmetry. In such a tube configuration, we'd have 2 long diagonal tubes and 1 short diagonal tube, with the short tube extending from the lower left colony. We can calculate tube lengths in this configuration fairly simply using the Pythagorean Theorem:
```Long Diagonal Length = Sqrt[2.5^2 + 7.5^2] = 7.90569
Short Diagonal Length = Sqrt[2.5^2 + 2.5^2] = 3.53553
Total Tube Length = 2 x 7.50569 + 3.53553 = 19.3449
```
Of course, this is only the tip of the proverbial ice berg. I'm sure you can think of many other variations.

Mark Ganson

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