Lawrence (abangaku) wrote in moppers,

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Interesting geometry problem I just thought up

Consider the hexagon made by placing a 2*2 square and a 1*1 square flush together (the P-pentomino). This hexagon, and another individual unit square, can be fitted together as two separate pieces in two distinct ways to make the same final figure (an octagon known as the R-hexomino).

What I wonder is, is there any pair of geometrical pieces, planar or no, fractal or no, such that they can be fitted together into the same shape in at least *three* different ways? And if not, can you prove why not? (Different ways are, of course, not counted as different if they are reflections or rotations of each other, though the pieces themselves may be rotated and reflected.)

Any ideas???
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