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Math Magic Archive

Problem of the Month
(February 2016)

Consider a chess piece that has some possible moves that are not necessarily symmetric, each possible move being one square horizontally, vertically, diagonally, a knight move. What is the longest loop that piece can make on an n×m chessboard, visiting each square no more than once? How does the size of the longest loop change as n, m, or both approach infinity?

Up to symmetry, there are 17 different collections of such 3 moves, where loops of length more than 2 are possible. The longest-known loops (and paths) are known for some of these. Can you extend these results? Can you find any patterns? What about collections of 4 such moves? Which n×m chessboards can be completely visited?


You can see all the best known results here.

Submit your answers here.

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