Pentominoes: Twelve Shapes and 2,339 Answers

Quick answer: a pentomino is a shape made of five squares joined edge to edge. There are exactly twelve of them, they were named by Solomon Golomb in 1953, and the classic puzzle of fitting all twelve into a 6 by 10 rectangle has 2,339 distinct solutions.

Best for: anyone who liked Tetris and wants to know where the shapes came from.

Twelve is not an approximation and not a convention. Five squares can be joined edge to edge in exactly twelve distinct ways, and you can prove it to yourself in an afternoon with squared paper by working systematically and discarding rotations and reflections.

What Is a Pentomino?

Five unit squares joined along full edges. Corner-to-corner contact does not count, and shapes that are rotations or mirror images of one another are treated as the same.

They belong to the polyomino family: one square is a monomino, two a domino, three a tromino, four a tetromino, five a pentomino. The tetrominoes are the seven shapes used in Tetris, which is why pentominoes look so familiar to anybody who has played it.

Twelve pentominoes of five squares each gives a total area of 60 squares, which is the number that makes the classic puzzles work.

Who Named the Pentominoes?

Solomon Golomb, in 1953, in a talk to the Harvard Mathematics Club. He coined the term and gave the first full description of polyominoes in general.

The twelve shapes are labelled by the letters they resemble: F, I, L, N, P, T, U, V, W, X, Y and Z. The scheme is not arbitrary and it is genuinely useful, because the shapes are hard to describe in words and everyone who works with them uses the same names.

Golomb's work turned a recreational curiosity into a studied object, and polyominoes have since appeared in tiling theory, computational complexity and, eventually, a video game.

How Many Ways Can Pentominoes Fill a Rectangle?

The famous number is 2,339, which is the count of distinct ways to tile a 6 by 10 rectangle with all twelve pentominoes, ignoring rotations and reflections of the whole arrangement.

Sixty squares can be arranged as 6 by 10, 5 by 12, 4 by 15 or 3 by 20, and the number of solutions collapses as the rectangle gets narrower. The 3 by 20 case has famously only a couple, which makes it the one worth attempting if you want a puzzle rather than an exercise.

That collapse is the interesting part. The same twelve pieces and the same total area produce thousands of answers in one shape and almost none in another, purely because a narrow rectangle leaves the awkward pieces nowhere to go.

What Is the Chessboard Problem?

The most elegant pentomino puzzle, and it comes from the arithmetic.

A chessboard has 64 squares. Twelve pentominoes cover 60. Remove a 2 by 2 block from the board and you have exactly 60 squares left, which the twelve pieces can fill.

The question that makes it a puzzle is where to put the hole. The 2 by 2 square can be placed in many positions, and the tiling is possible for some and not others. Working out which is a far better problem than filling a plain rectangle, because the constraint is invisible until you have failed at it.

Golomb used exactly this kind of argument to show results about what polyominoes can and cannot tile, and colouring arguments of that sort are now standard in combinatorics.

Why Are Some Pentominoes Harder to Place?

Because of how much they constrain their neighbours.

The X pentomino, the plus sign, is the worst offender. It is the only one with no straight edge longer than one square on any side, so it creates awkward notches wherever it sits and can never be tucked against a wall neatly.

The I pentomino, the straight line of five, is the opposite: easy to place, and the piece most likely to be left over because it needs a clear run of five.

The practical rule is the same as in any dissection: place the constrained pieces first. In a pentomino tiling that means X, then the other awkward ones, leaving I and L until the end. The same logic applies to tangrams and to the dissections in assembly puzzles explained.

What Is the Three-Dimensional Version?

Polycubes, and the most famous of them is the Soma Cube.

Where a pentomino is squares joined edge to edge, a polycube is unit cubes joined face to face. The Soma set is every polycube of three or four cubes containing at least one inside corner, which comes to seven pieces totalling 27 cubes, and so a 3 by 3 by 3 cube.

We make that one. Cube Alchemy is the seven Soma pieces with a booklet of 180 further shapes, and the flat tetromino version is Galette, five tetrominoes into a frame.

Can You Buy Pentominoes?

Yes, widely, and not from us. The set is public domain and sold in wood and plastic everywhere.

What we make is the neighbouring shelf: flat tray puzzles with fewer pieces and one solution rather than thousands. Trivial and Fish Tank by Goh Pit Khiam are the clearest examples, and the family is in packing puzzles. The wider range is mechanical puzzles.

The three-dimensional version of the same idea is the Bedlam Cube, which uses pentacubes rather than pentominoes.

Frequently Asked Questions

What is a pentomino?

A shape made of five unit squares joined along full edges. There are exactly twelve distinct pentominoes once rotations and mirror images are treated as the same shape.

Why are there exactly 12 pentominoes?

Because five squares can only be joined edge to edge in twelve distinct arrangements. It is a complete enumeration rather than a convention, and you can verify it yourself on squared paper by building systematically and discarding duplicates.

How many solutions does the 6x10 pentomino puzzle have?

2,339 distinct solutions, excluding rotations and reflections of the whole rectangle. Narrower rectangles have far fewer: the 3 by 20 case has only a couple, which makes it much the harder puzzle.

Are pentominoes related to Tetris?

Closely. Tetris uses tetrominoes, the seven shapes made from four squares, which belong to the same polyomino family. Pentominoes are the next size up and were named and studied first.

Which pentomino is hardest to place?

The X, the plus-shaped one. It has no long straight edge on any side, so it leaves awkward notches wherever it goes and cannot sit flush against a boundary. Place it first.