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2023 WSMO Team Round Problems/Problem 13

Problem

Gerry the Squirrel is at a corner of a $9\times9$ grid. 1 acorn $A_1$ is placed. There are four coins placed randomly on the $9\times 9$ grid. Gerry will go the shortest path to $A_1,$ and if there are multiple shortest paths, Gerry will pick one randomly. Given that Gerry does not start on top of a coin, the acorn is not on a coin, find the expected value of coins Gerry passes during his trip to $A_1$ is $\frac{m}{n},$ for relatively prime positive integers $m$ and $n.$ Find $m+n.$

Solution

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