Using your Head is Permitted

June 2014 solution

This is a very old riddle.

The answer is that every one of the 64x64 squares can remain as the single untiled square.

All solutions this month used the following tiling pattern, involving four corner triominoes:

The classic solution to this riddle is to say that this arrangement creates a corner shape that is twice as large as the original corner in every dimension. Just like a regular corner can tile a 2x2 square leaving any 1x1 empty, a double-size corner can tile a 4x4 square leaving any 2x2 of its corners empty. This 2x2 can then be filled by a regular corner, leaving any of its 1x1s empty.

Furthermore, if we take four double-sized corner pieces, we can use the same tiling pattern to create a quadruple-sized corner piece, and we can do this recursively, all the way up to a giant corner piece that will tile a 64x64 square, leaving any chosen 32x32 of its corners empty. These can then be filled in by a smaller corner, and a smaller corner, etc.. until, ultimately, any single square can be left empty.

An equivalent construction described this month by many readers is to say that if we are able to tile n×n, leaving any single square untiled, then we can also tile 2n×2n leaving any single square untiled, in the following way: divide the 2n×2n into four n×n squares. One of these will contain the square you wish to leave empty. Tile it as usual. The other three, tile so that their missing squares are the ones closest to the centre of the 2n×2n square. These three missing pieces form the shape of a corner triomino, so with one additional triomino the tiling is complete.

Other readers showed that by using two corners, one can tile a 3×2 rectangle, which, in turn, allows one to tile 3×2n rectangles, for any n. One can, using just such strips, tile any (6k+4)×(6k+4) square so as to leave a 4x4 square empty, where the position of the square can be moved arbitrarily in jumps of 3 squares in any direction. In particular, the missing area can be made to contain any chosen square. Once placed so that it contains the square which is to remain empty, the basic 4x4 tiling can be used to complete the rest.

Readers are welcome to try and generalise this 6k+4 result even further.

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