ç Index to Chessboard Dissection Problems

# 3-Square Pieces: Trominoes

Problem 151 in Chessics #23 was: With 144 matches one can delineate a chessboard (each match lying along a side of a square). Remove as few matches as possible to leave a pattern in which there are (a) no squares (b) no rectangles. [Case (a) with 40 matches in a 4×4 array is given by K. Fujimura in The Tokyo Puzzles (UK edition 1979, problem 7).] Problem (a) is solved by using mainly dominoes, with one 1×3 piece, 33 matches being removed. Problem (b) is solved by using L-shaped 3-pieces. The pattern of four L-shaped 3-pieces on which this is based solves the hoary old problem of the farmer with an L-shaped field to be shared equably between his four sons.

This arrangement of 3-pieces is an example of a more general result. If a single cell is removed from a large square whose side is a power of two (2^m) then the remaining area can always be dissected into L-shaped 3-pieces. The method of doing this is quite easy to visualise. Divide the square into quarters. One of these quarters contains the single-cell hole. Join the other three quarters together again. This three-quarters can then be dissected into L-shaped 3-pieces by repeated operation of the method for the farmer's L-shaped field. Exactly the same process can now be applied to the remaining quarter with the hole. And so on, until we reach a 2×2 square with the hole in a corner. I'm not sure who originated this result.