Lino vs. Calvin

Discrete Mathematics Level 5

A two player game is played on a \(5 \times 5\) grid. A token starts in the bottom left corner of the grid. On each turn, a player can move the token one or two units to the right, or to the leftmost square of the row immediately above it. The last player who is able to move wins.

Lino and Calvin decide that they want to make the game more interesting and instead of playing with a single token, they will play with two tokens, one red, and one blue, and on a turn a player moves either of the tokens. They also decided that the tokens will start in random positions on the board. Of the \(25 \times 25 =625\) possible starting positions for the 2 tokens, how many of these are winning positions for the first player if he plays optimally?

Details and assumptions

Clarification: Both tokens are allowed to be on the same square at any point during the game.

For the edge case where both tokens start in the top right corner, we declare that the second player wins.


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