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# 𝗛𝗔𝗥𝗗 𝗖𝗢𝗨𝗡𝗧𝗜𝗡𝗚 𝗣𝗥𝗢𝗕𝗟𝗘𝗠! 𝗣𝗟𝗘𝗔𝗦𝗘 𝗛𝗘𝗟𝗣!

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

Whoever answers, please give a thorough explanation, since I really want to understand how to solve this problem!

Thank you so much!

:)

Apr 26, 2020

#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.

Apr 26, 2020
#2
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How did you bold your question?

Apr 27, 2020
#3
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oh haha you go to this site: https://lingojam.com/BoldTextGenerator, and you just copy and paste!