2024 SSMO Relay Round 3 Problems/Problem 3

Problem

Let $T = TNYWR-1.$ Riley and Boris are playing a game on a $T\times T$ grid of dots. The game alternates turns and starts with Riley. Each turn, a player draws a line connected two different random dots, exactly 1 unit apart. The first person to complete the first unit square loses the game. Given that Riley plays optimally and Boris plays randomly, the probability that Riley wins can be expressed as $P.$ Find the least positive integer $a$ such that $aP$ is an integer.

Solution

Let the "score" of an edge be the number of unit squares it is part of. Note the $4T$ edges on the edge of the square are part of one unit square and the other $2T^2-2T$ edges all lie on two unit squares. Now, note that the first player that places an edge such that the total score of all edges placed exceeds $3T^2$ loses. Note that if Riley places all the edges, then after $4T$ turns, there will be a score of $12T.$ Afterwards, there will be an odd score after Boris's turn each time and an even score after Riley's turn each time. Since $3T^2$ is even, this means that Boris places the edge that increases the total score to $3T^2+1,$ in other words, Boris completes the first unit square. So, $P = 1,$ meaning $a = \boxed{1}.$

~SMO_Team

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2024 SSMO Relay Round 3 Problems/Problem 3

Problem

Solution