The numbers 1, 6, 8, 13, 15, and 20 can be placed in the circle below, each exactly once, so that the sum of each pair of numbers adjacent in the circle is a multiple of seven.

In fact, there is more than one way to arrange the numbers in such a way in the circle. Determine all the different arrangements. Note that we will consider two arrangements to be the same if one can be obtained from the other by a series of reflections and rotations.

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Puzze Source: CEMC


1, 8 and 15 are all equal to 1 modulo 7 - call these Group A. 6, 13 and 20 are all congruent to -1 modulo 7 - call these Group B. So any arrangement where Group A are mutually not adjacent to each other will work, and any other arrangement will not work (it would have a sum equal to 2 modulo 7). In any such arrangement, you can pick an arbitrary element of Group B to go between 1 and 8. Then you can pick one of the remaining two elements of Group B to go between 1 and 15. This uniquely determines the configuration, up to rotations and reflections.

So there are 3*2 = 6 configurations, constructed as described above.

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