Introduction
The terms polyomino and pentomino were first used by Solomon Golomb in a talk
to the Harvard Mathematics Club in 1953 and a year later in an article in the American Mathematical Monthly. They were
coined by Golomb to describe a generalization of a domino. He defined a polyomino
as a set of equally-sized squares, each joined together with at least one other
square along an edge. The order of a polyomino is the number
of squares used to make it. An order five polyomino is called a pentomino. The
first pentomino problem was actually written much earlier in 1907 by the
English inventor of puzzles, Henry Ernest Dudeny, in his book The Canterbury
Puzzles. The popularity of the shapes, however, is attributed mainly to
Golomb from his book Polyominoes: Puzzles, Patterns, Problems, and Packings
and to Martin Gardner from his monthly articles in Scientific American.
The simpler polyominoes-all the possible shapes composed of fewer than five
connected squares-are shown below. It is assumed that two polyominoes are the
same if one can be rotated (turned 90, 180, or 270 degrees) and/or reflected
(flipped over) to get the second (the polyominoes are said to be free
in this case).
One Square |
Two Squares |
Three Squares |
Four Squares |
For five squares, the twelve pentominoes resemble certain letters of the alphabet, and are labeled as such.
Five Squares
The total number of squares used for each set of polyominoes is summarized
below.
Order |
Name |
Total Number of Shapes |
Total Number of Squares Needed |
1 |
Monomino |
1 |
1 |
2 |
Domino |
1 |
2 |
3 |
Tronomo |
2 |
6 |
4 |
Tetronomo |
5 |
20 |
5 |
Pentomino |
12 |
60 |
6 |
Hexomino |
35 |
210 |
7 |
Heptomino |
108 |
756 |
8 |
Octomino |
369 |
2952 |
The values in the above table have been calculated for pieces of much larger size using a computer (click here to see more). However, pieces of order 6 or larger have little practical value as the basis of a dissection puzzles due to the complexity of most of the pieces and their lack of assembling together as a complete set into square or rectangular shapes.
Pentominoes have some very interesting mathematical properties providing a nearly endless array of challenging puzzles. The most natural shapes to construct with the pentominoes are squares and rectangles. However, since the total area of the twelve pieces combined is 60 squares, constructing a square would require an 8 ´ 8 'checkerboard' that would leave four squares left over. This leads to some interesting patters where the four empty squares are arranged in some symmetric way about the board. (See patterns B-1 through B-9.)
Section A – Beginning Pentomino Problems
Section B – Intermediate Pentomino Problems
Section C – Problems Using Solid Pentominoes
Section D – Pentomino Games
Section E – Problems Using Pentominoes and Other Polyominoes
Section F – Advanced Pentomino Problems