2016 AIME II Problems/Problem 12

Problem

The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.

$[asy] draw(Circle((0,0), 4)); draw(Circle((0,0), 3)); draw((0,4)--(0,3)); draw((0,-4)--(0,-3)); draw((-2.598, 1.5)--(-3.4641, 2)); draw((-2.598, -1.5)--(-3.4641, -2)); draw((2.598, -1.5)--(3.4641, -2)); draw((2.598, 1.5)--(3.4641, 2)); [/asy]$

Solution

Assume, WLOG, the left ring is color $1$ . Now, let $f(a,b)$ be the number of ways to satisfy the conditions where there are $a$ sections ending on color $b$ . We make a table on $f(a,b)$ .

$\begin{tabular}{c|c|c|c|c }&1 & 2 & 3& 4 \\ \hline 1& 1 & 0 & 0 & 0\\ 2 & 0 & 1 & 1 & 1 \\ 3& 3 & 2 & 2 & 2 \\ 4 & 6 & 7 & 7 & 7 \\ 5 & 21 & 20 & 20 & 20\\ 6& 0& 61 & 61 & 61\\ \end{tabular}$

Note that $f(6, 1)=0$ because then $2$ adjacent sections are both color $1$ . We also have to multiply by $4$ by symmetry for other colors in the left ring, so the desired answer is $(61+61+61) \cdot 4 = \boxed{732}$ .

Solution by Shaddoll

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2016 AIME II Problems/Problem 12

Problem

Solution