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.

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.

See all of George Hart’s Math Monday columns

Gareth Branwyn

Gareth Branwyn

Gareth Branwyn is a freelance 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 to Boing Boing, Wink Books, and Wink Fun. And he has a new best-of writing collection and “lazy person’s memoir,” called Borg Like Me.

  • Cory Poole

    Ah.  I finally understand why these pentamino puzzles work so well.  I didn’t realize the area of them was 60 square units.  A nice number with gobs of factors.  Cool.  I’m glad I learned that.

  • Marcus Behrens

    But if you build the pentominos from cubes you get to fill a three-dimensional space with 60 of them. The smallest one has the size 3x4x5 inches. It is one of the toughest puzzles I ever found – after trying for 3 months I programmed an algorithm to solve it for me.