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BDMO National Secondary 2018/7

Posted: Tue Jan 08, 2019 12:22 pm
The vertices of a regular nonagon ($9$-sided polygon) are labeled with the digits $1$ through $9$ in such away that the sum of the numbers on every three consecutive vertices is a multiple of 3. Two acceptable arrangements are considered to be indistinguishable if one can be obtained from the other by rotating the nonagon in the plane. Find the number of distinguishable acceptable arrangements.

Re: BDMO National Secondary 2018/7

Posted: Mon Jan 21, 2019 10:55 pm
Here is a solution.But I didn't understand the last line.

Re: BDMO National Secondary 2018/7

Posted: Wed Feb 20, 2019 1:55 pm
Take $mod 9$ to all number. The given argument is only possible if the sequence is $0,1,-1$ and $0,-1,1$. For the 1st case the number of ways are $\frac{3\times3\times3\times2\times2\times2\times1\times1\times1}{3}$(divided by $3$ because of the rotational symmetry.)
so for all cases the number of ways are $\frac{2\times3\times3\times3\times2\times2\times2\times1\times1\times1}{3}=144$