11 Jul 2013

Pentominoes in a Square: Middle Secondary Mathematics Competitions Question

It is easy to look up that there are 12 free pentominoes. The standard labelling is supposed to approximate to the shape of the tiles and is given in the diagram below. The term ‘free’ means that each tile can be both rotated and reflected. However, if the situation does not allow for reflections – such as Tetris clones – then tiles F, L, N, P, Y and Z have mirror-images that are not identical with themselves by any rotation, yielding 6 new tiles F’, L’, N’, P’, Y’ and Z’. We thus end up with 18 one-sided, or flat, pentominoes. We have already seen in our previous questions that some of these tiles have their own letters (such as P and Q), but I think it is less confusing to just label them as the chiral partners of the originals.

Pentomino Names [http://commons.wikimedia.org/wiki/File:Pentomino_Naming_Conventions.svg]

The Question

Take the 18 flat pentominoes and remove the tiles I, L, L’, P and P’; those that make very simple rectangles. Now, tile a 5x5 square with 5 different pentominoes so that there are no gaps or overlaps. How many solutions are there?

Solutions with rotational symmetry count as just one distinct layout, but reflections count as two solutions if they use different flat pentominoes.

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