Solution

Congratulations to Michael Keith, a software engineer from Salem, Oregon, the first person on Earth to solve the mysteries of my Dharmakaya Rings. He receives a free autographed copy of my novel Spider Legs for his solution that uses consecutive numbers from 1 to 15. The magic sum is 40. This means that the numbers at the intersection points of the circles sum to 40 as you travel around the circles! Additionally, the four points along the triangle edges sum to 40!

The middle triangle contains the numbers 3 (top left vertex), 14 (right), 9 (bottom). The leftmost circle contains the numbers (starting at the bottom left intersection point with the bottom circle and going clockwise): 1, 6, 11, 4, 5, 13. The rightmost circle contains the numbers (starting at the bottom right intersection point with the bottom circle and going clockwise): 2, 7, 5, 8, 12, 6.

How many other solutions do the Dharmakaya Rings have?

Mike points out that the answer is...79,824. This is the number of solutions counting every one as distinct; it's probably more natural to call two numberings the same (isomorphic) if they are equivalent under one of the six-fold symmetries of the figure. In that case the number of distinct solutions is 13304. Of course, all of these solutions are perfect numberings, using 1...15.

Note that the number of possible numberings using 1....15 is 15! = 1307674368000, so while that may sound like a lot of solutions it really isn't that many.

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