Math Monday: Pentominoes

by George Hart



A pentomino is like a domino, but with five connected squares instead of two. There are twelve ways to connect squares edge-to-edge in the plane, not counting rotations and flips. A set of all twelve can be cut from scraps of plywood using any kind of saw. Large sets are fun to play with, so I based these on three inch squares. It is made from half-inch plywood, cut on a band saw and lightly sanded.As the area of all twelve pieces totals sixty squares, a natural puzzle is to try to fill a 6×10 rectangle. There are over 2000 solutions! But even though you are allowed to flip and rotate the pieces however you wish, it is harder to solve than you might think.The 5×12 rectangle above is another challenging puzzle to try once you make your set. This one is three feet across.

You can also make a 4×15 rectangle, as shown above. And the same twelve pentominoes can make the 3×20 rectangle, below, which is five feet long.It is interesting that as we consider longer, skinnier rectangles, the number of possible solutions is drastically reduced. These four rectangles have 2339, 1010, 368, and 2 solutions, respectively.



This article first appeared on Make: Online, October 17, 2011.

Return to Math Monday Archive.