CombinatoricsDifficulty 5.6AIME, harderProve itUnited States
Problem:
A lame king is a chess piece that can move from a cell to any cell that shares at least one vertex with it, except for the cells in the same column as the current cell. A lame king is placed in the top-left cell of a 7×7 grid. Compute the maximum number of cells it can visit without visiting the same cell twice (including its starting cell).
Solution
Solution:
Color the columns all-black and all-white, alternating by column. Each move the lame king takes will switch the color it's on. Assuming the king starts on a black cell, there are 28 black and 21 white cells, so it can visit at most 22+21=43 cells in total, which is easily achievable:
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Source: MathNet,
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