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By George Hart for the Museum of Mathematics


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.

pent 1 Math Monday: Pentominoes

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.

pentominos 5x12 Math Monday: Pentominoes

The 5×12 rectangle above is another challenging puzzle to try once you make your set. This one is three feet across.

pentominos 4x15 Math Monday: Pentominoes

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.

pentominos 3x20 Math Monday: Pentominoes

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.

See all of George Hart’s Math Monday columns

Gareth Branwyn

Gareth Branwyn is a freelancer writer and the former Editorial Director of Maker Media. He is the author or editor of over a dozen books on technology, DIY, and geek culture. He is currently a contributor for Boing Boing and WINK Books. And he has a new best-of writing collection and “lazy man’s memoir,” called Borg Like Me.

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