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# counting question

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286
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Find the number of ways of arranging the numbers 1, 2, 3, ... 9 in a circle, so that the sum of any three adjacent numbers is divisible by 3. (Two arrangements are considered the same if one arrangement can be rotated to obtain the other.)

Nov 20, 2019

#1
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Each number is a multiple of 3, or 1 more than a multiple of 3, or one less than a multiple of 3.  These number we will call x's, y's, and z's.

If we add x + y + z, then we get x, because y and z will even each other out, since y is 1 more than multiple of 3 and z is one less.

Because of this, we can conclude that we have to arrange the numbers in a repeating pattern, like x, y, z, x, y, z, ... or z, y, x, z, y, x, ...

In this set 1, 2, 3, ..., 9, there are 3 x's, 3 y's, and 3 z's.

To fill up a pattern, there are 3 choices for the first x, y, z, 2 choices for second, then 1 choice.

3^3 * 2^3 * 1^3 = 27*8*1 = 216

This is equal for both patterns, so there are 216 * 2 = 432 ways to arrange the numbers.

Nov 20, 2019
#2
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This question has been asked many times in the recent past.

The earlier answers are easy to find  Sinclairdragon428  posted them.  (He likely posted this time too)

Nov 20, 2019
edited by Melody  Nov 20, 2019