Please note: The wonderful results of this challenge are now published in my book The Zen of Magics Squares, Circles, and Stars (An Exhibition of Suprising Structures Across Dimensions) (Princeton Unviversity Press, 2002). If you want to see how it all started, see below.

For this fun but difficult Internet math challenge, you don't have to know more than the mathematics of addition. If you can find an elegant solution to one of these, I'll try to put your solution in a book and credit you so you will be forever remembered in the annals of history. So very few of these have been solved.

(For all these problems, I prefer you use consecutive numbers starting with 1, and that no numbers repeat, but if this is not possible, you are free to relax the constraint. For example, start at a number other than one.)

Durga yantra image by Yantras and Mandalas by PennyLea Mackie . Please visit this wonderful site as soon as possible. You may purchase the exquisite fine art reproductions.

Click on each image for magnification.

Can you place numbers on some of the vertices of these incredible Nagle mandala images by Nagle Design ? Please visit this wonderful site as soon as possible. You may purchase the exquisite fine art reproductions.

Can you place numbers on these Rosicrucian symbols?

Can you place numbers on some of my own designs?

Solution!

Solution!

Solution!

Rules: Place an integer in each cell so that when the numbers in every row and column are added together, the sum is the same constant.

Cube with cutout Tortured cube Cube with cutout 2 Dissected Cube Hyperdimensional frame cube CrossTwisted world

Rules: 1) Look at the small squares. Place an integer in each square. The sums of the integers forming a diagonal must always be the same. Truncate the figure as you wish in order to remain sane.

2) Another set of rules for the same figure. You see two
kinds of cells. Cells A are small squares. Cells B are larger polygons
with a stairscase edge. Place an integer in each of the cells A so that
the sum of these integers adds up to the number in each Cell B. Be creative.
If you don't like these rules, make your own.

Hypercheckerboard

Each central star must contain an integer that is equal to the sum of 6 numbers in each surrounding hexagon. Truncate the figure as you wish in order to remain sane. (If you don't like these rules, make up your own.)

Islam

Each petal has two spirals that intersect it. Place an integer in each petal so that each spiral has the same sum.

Flower from Ganymede

Solution by the brilliant and enigmatic Joseph DeVincentis. We do not currently know how many solutions exist.

Place an integer at each of the vertices so that each diagonal sum is the same. This sum must also be the same as the sum of the numbers along the outer perimiter. What other interesting symmetries can you create?

A higher dimension

If these problems are too difficult, or do not seem elegant, make up your own problem that has similar rules to those in magic squares in which sums in various directions are the same. Send your figure to me, and I'll try to publish it with your name.

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