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