Problem of the Month for January, 2006

Problem:

From five squares, one can build twelve different connected (along their edges) shapes called pentominoes, three of which are shown below. Will the other nine pentominoes cover the 5 x 9 rectangle shown below in such a way that the sums of the squares covered by each pentomino are all the same? (each pentomino covers 5 small squares…two pentominoes are considered to be the same if one of them can be rotated or reflected and moved so as to coincide with the other). Either find a way to do this covering or explain why it is impossible.

A possible generalization you might consider:

Eight of the twelve pentominoes can be thought of as looking similar, up to rotation and/or reflection, to the letters U, V, W, Z, I, L, P, and N (though the Z, P, and N take a bit of imagination…two pentominoes cannot be connected just at their corners). It can be shown that these eight pentominoes can tile a 5 x 8 rectangle. Find as many of these tilings as you can (which are all different up to reflections, rotations, and swaps).