BDMO National Secondary 2018/7
 nahin munkar
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 Joined: Mon Aug 17, 2015 6:51 pm
 Location: banasree,dhaka
BDMO National Secondary 2018/7
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.
# Mathematicians stand on each other's shoulders. ~ Carl Friedrich Gauss
 samiul_samin
 Posts: 1007
 Joined: Sat Dec 09, 2017 1:32 pm
Re: BDMO National Secondary 2018/7
Here is a solution.But I didn't understand the last line.

 Posts: 21
 Joined: Sat Jan 28, 2017 11:06 pm
Re: BDMO National Secondary 2018/7
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 $\frac{2\times3\times3\times3\times2\times2\times2\times1\times1\times1}{3}=144$
so for all cases the number of ways are $\frac{2\times3\times3\times3\times2\times2\times2\times1\times1\times1}{3}=144$