The wizards Albus and Brian are playing a game on a square of side length 2n+1 metres surrounded by lava. In the centre of the square there sits a toad. In a turn, a wizard chooses a direction parallel to a side of the square and enchants the toad. This will cause the toad to jump d metres in the chosen direction, where d is initially equal to 1 and increases by 1 after each jump. The wizard who sends the toad into the lava loses. Albus begins and they take turns. Depending on n, determine which wizard has a winning strategy.
Solution
Solution:
Brian wins, irrespective of n. Suppose Brian plays with the following strategy: for every move Albus makes, Brian makes a move in the opposite direction. If Brian were to lose, this would mean the toad was at the edge of the board two jumps prior, before Albus's final move, as otherwise Brian's final move could not possibly have sent it off the board. Now, because every two jumps the total displacement is 1, it would take at least 2n jumps for the toad to be on the edge at the end of Brian's turn specifically. But this means Albus's next move is of length 2n+1; which always sends the toad off the board, contradiction.
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