Pentominos Page

Pentominos are shapes that use five square blocks joined together with at least one common side. There are at least twelve shapes in the set of unique pentominos, named T, U, V, W, X, Y, Z, F, I, L, P, an N respectively. As a mnemonic device, one only has to remember the end of the alphabet (TUVWXYZ) and the word, FILiPiNo. Pentominos are said to have been "invented" by Solomon W. Golomb in 1953 at a talk he gave to the Harvard Mathematics Club. Although he coined the name, Pentominos have been around since a much earlier time. The first pentomino problem, written by the great English inventor of puzzles, Henry Ernest Dudeny, was published in 1907 in the Canterbury Puzzles.

Pentominos can be hand made with ease. Just go to a craft store such as Michael's or Hobby Lobby and buy several packages of cubes. You will need a total of 60. The best cubes to use are the 3/4" or larger. Do not use anything smaller as there is much more room for error in their proportion and the resulting pentominos may not work well together.

Manipulating the various pentominos into figures can be very amusing and challenging. Throughout this page there are many shapes which can be made with pentominos, some are easy and others are very difficult. Try your hand at making the following figures, and then think of some of your own. There are many possibilities! When you ar ready for strategic competition, there is a game that can be played with them, which will be explained later. First, familiarize yourself with the basics.

The twelve pentominos are often referred to by the letters they resemble.

Level I - 3/1 Scale Models (two-dimensional)

Each of the twelve pentominos can be modeled two-dimensionally with nine of the remaining pentominos. To do this, first pick up a piece to model and put it aside as this piece will not be necessary for the scaled model. Second, try to construct a duplicate on a 3/1 scale. The new model will only need nine pieces.

Level II - Fun Figures

Here are some interesting figures. The 3x20 rectangle has only two solutions whereas the 4x15 rectangle has many. The two jagged figures can be constructed by dividing them into congruent halves. The cross is an intermediate puzzler.

Level III - Checkerboards

The figures shown are known as pentomino checkerboards and are constructed two dimensionally (lying the pieces flat). Try your hand at these. They are more difficult than you think!

Level IV - 2/1 Scale Models (three-dimensional)

Ten of the twelve pentominos can be modeled three-dimensionally with all twelve pieces. Construct the pentomino figures on a 2/1 scale, but this time around a new dimension is added -- they must be three stories tall! This technique is fiendishly difficult as the pentomonos can be used any which way that works. Keep in mind that there are two pentominos that cannot be constructed in this manner -- the W and the X.

Here is a list of possible combinations for constructing three-dimensional, 2/1 scales of the pentominos. You may assume that they are in order from easiest to most difficult.

Level V - Geometric Delights

Try to make these geometric figures. The "fence may be folded into shape by first making a long, flat figure. The pyramid is constructed using only 11 pentominos, allowing your to omit the T or the X. Best of luck!

The Game of Pentominos

Besides its intrigue as a puzzle, the placement of pentominos on a checkerboard also makes it an exiting competitive game of skill. Played by two or three players, the object of the game is to be the last player to place a pentomino piece on the checkerboard. Players take turns choosing a piece and placing it on the board. The pieces must not overlap or extend beyond the boundary of the board, but they do not have to be adjacent. The game will last at least five, and at the most twelve moves.