Number Theory Level 4

Alex and Bob stand apart from each other in a square lattice, \(101\) points long in each dimension. Alex stands at the point \((66, 66)\), and Bob stands at the point \((33, 33)\). They're both looking to triumph over the other, but there's a small handicap - each gunner can only shoot vertically and horizontally,

Alex moves first, taking a single, 1 unit step, either in a horizontal direction, or a vertical direction, positive or negative. He must take a step (he cannot 'pass' his turn), and he cannot leave the bounds of the lattice, \(0 \leq x, y \leq 100\). If he steps into the same row or column as Bob, though, he'll lose, because Bob will be ready to gun him down, and will get the first shot off.

Bob moves second, under the same constraints, and likewise, if he steps into the same row or column as Alex, he'll lose. Both players continue taking turns in this manner until the game ends.

Assume Alex and Bob play perfectly. What will be the outcome of the game?

Given: Alex and Bob both have perfect information at all times. They know all of the rules in the above paragraphs, and they always know where both players are positioned at the start of every turn.

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