The story of pentominoes

A pentomino is a shape made of five equal squares joined edge to edge. There are twelve of them, and the puzzle is to pack all twelve into a board with no gaps. The pieces are simple, but how many ways there are to do it depends enormously on the shape of the board.

Why twelve?

If you count shapes that match after turning or flipping as the same, there are exactly 12 pentominoes. Treat a shape and its mirror image as different and you get 18; also tell rotations apart and you get 63. In the puzzle you may turn and flip freely, so 12 is the number of pieces.

This site's own tests confirm that there are 12 pieces of five squares each, with the right number of orientations.

The board decides the count

The twelve pieces cover 60 squares in total, and this site offers five boards of 60 squares. Here is how many solutions each has:

Board	Solutions
6×10	2,339
5×12	1,010
4×15	368
3×20	2
8×8 with the middle 2×2 removed	65

We checked these numbers by running this site's own search over every possibility. The counting rule: turning or flipping the whole board gives the same solution, but turning or flipping just some of the pieces inside gives a different one. The thinner the board, the fewer the solutions: 3×20 has only two.

Where the name comes from

“Pentomino” joins pente, Greek for five, with “-omino”, taken from “domino”. The mathematician Solomon W. Golomb used the word “polyomino” for shapes built from squares, treating the “do” in domino as if it meant “two”. That is how the story is usually told. His 1954 paper is recognised as the earliest appearance in print. Three squares make a tromino, four a tetromino.

The twelve pieces are named after letters they resemble: F, I, L, P, N, T, U, V, W, X, Y and Z. A popular memory aid is “FILiPiNo” followed by TUVWXYZ.

Before and after the name

Golomb is said to have given a talk on the idea to the Harvard mathematics club around 1953, but the shapes were known earlier. In his 1907 book The Canterbury Puzzles, the English puzzle writer Henry Dudeney included “The Broken Chessboard”, which asks you to build a chessboard from 13 pieces. It is regarded as the earliest puzzle containing all twelve pentominoes, although the name did not exist yet.

An early example of making rectangles from the twelve pieces appeared in a chess-problem magazine in the 1930s.

The puzzle reached a wide audience when Martin Gardner wrote about it in his “Mathematical Games” column in Scientific American, in May 1957, and readers responded enthusiastically. In 1965 Golomb published his book Polyominoes, which came with a set of twelve plastic pieces.

Early computers did the counting

The 2,339 solutions of the 6×10 rectangle are said to have been counted by computer in 1960 by C. B. Haselgrove and Jenifer Haselgrove. The 65 solutions of the 8×8 board with a hole in the middle are said to have been counted in 1958 by Dana Scott. Both are remembered as early examples of letting a computer try possibilities and back up when stuck.

How to think about it

• Start with the awkward pieces. The X has arms in four directions around its centre square, so that centre can never sit on the edge of the board. Deciding the pieces with the fewest possible spots first cuts down the guessing.
• Fill corners first. Only a few pieces fit a corner neatly. Leave one open and you will get stuck later.
• Never leave a tiny hole. A gap of one or two squares fits no piece. After each placement, look at the shape of what remains.
• When stuck, back up one step. The computer does exactly this: it tries each piece that fits the first empty square, and undoes the last move if none works.

If you get stuck part-way, the Hint and Watch the computer buttons on the puzzle page can finish the job.