<p>Each vertex of a regular octagon is independently colored either red or blue with equal probability. The probability that the octagon can then be rotated so that all of the blue vertices end up at positions where there had been red vertices is <img alt="$\tfrac{m}{n}$" height="18" src="https://latex.artofproblemsolving.com/c/8/5/c85ea1ad780a799bdaf8f327e664f65eb6f3546b.png" width="14" />, where <img alt="$m$" height="8" src="https://latex.artofproblemsolving.com/f/5/0/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png" width="15" /> and <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> are relatively prime positive integers. Find <img alt="$m+n$" height="12" src="https://latex.artofproblemsolving.com/0/2/d/02dbab3601fa6cce490d54263017048f69e1a35d.png" width="48" />.</p>

Question No: 425

Each vertex of a regular octagon is independently colored either red or blue with equal probability. The probability that the octagon can then be rotated so that all of the blue vertices end up at positions where there had been red vertices is $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

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Tags: Combinatorics

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