Pentominoes: The Story of the 12 Pieces and Golomb

By Nesim Razon, independent Android developer · 2026-11-10

Take five squares and glue them edge to edge. You can do it in exactly twelve different ways, and those twelve shapes have kept mathematicians, puzzle makers and bored students busy for more than seventy years. This is the short story of the pentominoes.

What is a pentomino?

A pentomino is a flat shape made of five identical squares, where every square shares a full side with at least one other square. Two squares that only touch at a corner do not count as joined. If you turn a shape around or flip it over, it is still the same pentomino, and with that rule there are exactly 12 of them.

Shapes made of squares joined edge to edge have a general name: polyominoes. One square is a monomino, two squares are the familiar domino, three squares make a tromino (2 kinds), four make a tetromino (5 kinds), five a pentomino (12 kinds) and six a hexomino (35 kinds). The numbers grow quickly after that, which is why the pentominoes are a sweet spot: a small set, easy to hold in your head, but rich enough to make deep puzzles.

Golomb and the invention of the word

The person who gave these shapes their name was Solomon W. Golomb, an American mathematician. In 1953 he gave a talk at the Harvard Mathematics Club about the shapes you can build from squares, and he called them polyominoes. The word is a joke on the word domino. Golomb read the start of the word domino as if it were the prefix di-, meaning two, and swapped in other number prefixes: tromino, tetromino, pentomino and so on.

The first full paper on the subject, Checker Boards and Polyominoes, appeared in the American Mathematical Monthly in 1954. In it Golomb asked questions that sound like games. Can you cover a chessboard with certain polyominoes? Which shapes can fill a rectangle? Years later he collected the ideas in a book called Polyominoes (1965), which remains a classic introduction.

The twelve letters

The twelve pentominoes are known by the letters they look like: F, I, L, N, P, T, U, V, W, X, Y and Z. The set in alphabetical order is FILNPTUVWXYZ, and that is also the order the pieces are listed in the Pentolings app. The letters are shorthand: I is a straight line of five squares, X is a plus, V is a corner with two arms of three, and so on.

Some letters hide interesting facts. X is the only pentomino that looks the same in every direction you turn it, so it has just one position on a grid. I has two, because it can lie flat or stand up. T, U, V, W and Z have four positions each, and the remaining five (F, L, N, P and Y) have eight, because flipping them over gives new positions. Add them all and you get 63 positions in total. If you do not allow flipping, there are 18 different pieces, because six of the twelve (F, L, N, P, Y and Z) come in two mirror versions.

Martin Gardner and the 1957 spark

Polyominoes would have stayed a corner of mathematics if Martin Gardner had not noticed them. Gardner wrote the Mathematical Games column in Scientific American, and in 1957 he introduced pentominoes to a wide audience there. Readers loved them. They were simple enough to cut out of paper, and the puzzles that came with them were hard enough to keep people busy for weeks.

That column helped start a long habit. Puzzle makers began selling sets of the twelve pieces, and the shapes kept turning up in later puzzles and games. Golomb himself went on to write more about polyominoes and other recreational mathematics, and the term he coined in 1953 is the one everybody still uses.

The rectangle puzzles: 2339 ways to fill 6x10

The classic pentomino challenge is this: put all twelve pieces, each used once, into a rectangle with no gaps. The twelve pieces cover 60 squares, so the rectangle must have area 60. Four shapes are possible: 6x10, 5x12, 4x15 and 3x20. The number of different answers shrinks as the rectangle gets thinner. If we count answers that differ only by turning or mirroring the whole rectangle as one, the numbers are:

6x10: 2339 solutions. 5x12: 1010. 4x15: 368. 3x20: only 2. These counts were found by exhaustive computer searches, and they make a good sanity check for any pentomino solver. In Pentolings, the last level of the Fit mode is a 6x10 rectangle with the full set, so you are solving a puzzle that has thousands of valid answers in front of you, and the game accepts any one of them.

Beyond five: bigger and smaller families

The pentominoes are one member of a whole family, and the numbers show why five squares is such a comfortable size. With one turn-and-flip counted as the same shape, there is 1 monomino, 1 domino, 2 trominoes, 5 tetrominoes and 12 pentominoes. Then the count jumps to 35 hexominoes and 108 heptominoes. Past that, nobody can keep the whole set in their head, and the puzzles lose their friendly feel.

Each size has its own character. Tetrominoes are so few that you can learn them in a minute, but they cannot make interesting rectangle puzzles on their own. Hexominoes need a book to learn. Pentominoes sit in the middle: you can learn all twelve in an afternoon, and you can still spend years on them.

Try it with paper

You do not need an app to meet the pentominoes. Draw a grid on squared paper, cut out the twelve shapes using the letters as your guide, and try to build a 6x10 rectangle. Beginners often start with the easier 3x20 rectangle in mind and give up, because it has only two true answers and is the hardest of the four. Start with 6x10 or 5x12 instead, where there is more than one way to go right.

A good first goal is to place the X, which is the only piece that looks the same all round, then the long I, and then work on the pieces that hug the edges. If you get stuck, keep one shape you cannot place and try the others in its absence, because the last two or three pieces usually fall into place together.

Why they make such good puzzles

Pentominoes sit at a sweet spot. With 12 pieces you can still remember every shape and its name. With 63 positions every piece can face almost any direction, so the puzzles stay flexible. And the idea is easy to explain: fill the shape, no gaps. Beginners can solve small shapes with three or four pieces, and experts can spend an evening on a single 5x12 rectangle.

That was the reason we built Pentolings around them: a cheerful tile-fitting mode with 150 levels, plus a second mode where the same pieces fall from above and you pop colors. If you want to meet the twelve pieces gradually, the early levels use only three pieces at a time, and each new world adds one more.