Explorations of arrangements, combinations and structures of dominoes and domino sets – in which the numbers of pips may or may not be significant – provide opportunities for students to encounter various mathematical ideas, and to act mathematically.

A shape that consists of two squares of equal size joined along complete edges to form a unit is called a domino.

During the 1950s Solomon Golomb explored the more general idea of a shape composed of any number (n) of squares of equal size joined along complete edges, which he called a Polyomino. Martin Gardner brought to a worldwide audience Professor Golomb’s findings and puzzles about polyominoes. Polyominoes for n = 1, 2, 3, 4, 5, 6, 7 and 8 have so far been named – as monomino, domino, triomino, tetronimo, pentomino, hexomino, heptomino, and octomino respectively.

It is likely that dominoes were originally invented in China around 1120 A.D. as small bone tiles with inset round ebony pips. They seem to have been derived from cubic dice, which had been introduced into China from India. Originally each domino represented one of the possible results of throwing two dice together – the pips on one half of the domino being the pips on the top face of one die, and the pips on the other half being the pips on the top face of the other die.

Without showing students the following diagram, challenge them to explain how from the information above they can work out the number of dominoes in a full set of 12th century Chinese dominoes.

Dominoes and dice composed by the author from images of dominoes by Jelte, and dice by Alexander Dreyer

Dominoes reached Italy from China during the 18th century, and then spread throughout the rest of Europe. Europeans enlarged the Chinese set of 21 dominoes – that represented the possible combinations of two die scores – to include representation of a score of zero:

Dominoes and dice composed by the author from images of dominoes by Jelte

Ask students to explain how the European set of dominoes up to (6,6) – shown above on the right – can be thought of as containing 6 sets of 7 dominoes and yet consists of only 28 (4 x 7) dominoes rather than 42 (6 x 7) dominoes.

At the present time, in the 21st century, people everywhere in the world are familiar with dominoes. The Chilean miners who were trapped underground recently for more than two months in the San Jose mine played domino games to help maintain their good spirits!

And playing dominoes is the ‘second national sport’ in Cuba. As you can see in this film, Cubans play dominoes using tiles with up to 9 pips on each half – they play with double-9 domino sets. The double-9 domino set contains all the pairs taken from 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Generally, the double-2 set contains all the pairs taken from 0, 1, 2, the double-3 set contains all the pairs taken from 0, 1, 2, 3, the double-4 set contains all the pairs taken from 0, 1, 2, 3, 4, and so on... This is how the sequence of double-n domino sets begins:

Dominoes and dice composed by the author from images of dominoes by Jelte

Can your students work out mentally the number sequence created by the total number of dominoes in each set as the sequence progresses? What is the nth term of the sequence?

You can help students establish mental images of the structure of domino sets by asking questions such as ‘what is the probability of a domino picked at random from a double-2, double-3, double-4, … set having just one pip on at least one half of it?’

What value of n has this double-n set of dominoes?

Dominoes have inspired mathematicians to devise many puzzles. Some domino puzzles are, or you can easily extend them to be, starting-points for mathematical investigation. For example Amusements in Mathematics by Henry Ernest Dudeney, published in 1917, includes these three ‘investigational’ puzzles:

378 Dominoes in progression

It will be seen that I have played six dominoes, in the illustration, in accordance with the ordinary rules of the game, 4 against 4, 1 against 1, and so on, and yet the sum of the spots on the successive dominoes, 4, 5, 6, 7, 8, 9, are in arithmetical progression; that is, the numbers taken in order have a common difference of 1. In how many different ways may we play six dominoes, from an ordinary box of twenty-eight, so that the numbers on them may lie in arithmetical progression? We must always play from left to right, and numbers in decreasing arithmetical progression (such as 9, 8, 7, 6, 5, 4) are not admissible.

379 The five dominoes

Here is a new little puzzle that is not difficult, but will probably be found entertaining by my readers. It will be seen that the five dominoes are so arranged in proper sequence (that is, with 1 against 1, 2 against 2, and so on), that the total number of pips on the two end dominoes is five, and the sum of the pips on the three dominoes in the middle is also five. There are just three other arrangements giving five for the additions. They are: —

Now, how many similar arrangements are there of five dominoes that shall give six instead of five in the two additions?

380 The Domino Frame Puzzle

It will be seen in the illustration that the full set of twenty-eight dominoes is arranged in the form of a square frame, with 6 against 6, 2 against 2, blank against blank, and so on, as in the game. It will be found that the pips in the top row and left-hand column both add up 44. The pips in the other two sides sum to 59 and 32 respectively. The puzzle is to rearrange the dominoes in the same form so that all of the four sides shall sum to 44. Remember that the dominoes must be correctly placed one against another as in the game.

This is another well-known domino puzzle. In this ‘hollow’ square of four dominoes the sum of the dots on each of the four sides is 9:

composed by the author
from images of dominoes
by Jelte

Arrange the 28 dominoes of a complete double-6 set into seven hollow squares so that in each square the sum of the dots on each of the four sides is equal.

Many domino puzzles are about arranging dominoes to form ‘solid’ rectangles, as in this Dudeney ‘magic square’ puzzle. Complete this 6x6 square formed with dominoes so that the sum of the pips in every row, column and diagonal is 13.

composed by the author
from images of dominoes
by Jelte

This puzzle can be extended to an investigation: is there just one way of arranging 18 dominoes to make a magic square with a row, column and diagonal total of 13? Can 18 dominoes be arranged to make other 6x6 number squares in which the sum of the pips in every row, column and diagonal is the same but not 13?

Students can explore double-n (n ≤ 9) domino sets, and investigate puzzles, using the really lovely NRICH Dominoes Environment. The full screen version on the classroom IWB makes a ‘WOW’ impact!

At NRICH you will also find more good ideas for activities using an ordinary double-6 set of dominoes.

Some interesting investigations and phenomena involve dominoes in which the pips are ignored.

These include explorations of dominoes covering squares on grids. Dominoes on a Chessboard Any rectangular n x m chessboard can be covered with dominoes if, and only if, at least one of n and m is even. The squares on an n x m chessboard are created by a set of evenly spaced horizontal lines intersecting a set of equally spaced vertical lines. A domino piece that covers exactly two squares has to cross one of these lines. For example in this diagram the domino crosses the vertical line that is shown red.

We can call a line that is not crossed by any domino a fault line. For example, in this covering of the board with dominoes there is just one fault line, which is shown green.

Is it possible to cover the whole board with dominoes so that every one of the lines is crossed by at least one domino – so that no fault lines are created?

Do these arrangements have fault lines?

Students could explore this problem on boards with various different numbers of squares using this Cut the Knot applet. Warning – on this page, general facts are stated and explained that you will probably prefer students not to read until they have had plenty of opportunity to experiment, think for themselves and reach their own conclusions.

Colin Wright is available to give a mathematical talk that is based on dominoes on a chess board, and which goes on to explore pattern, possibility and proof, looking at how we can be sure that something really is impossible.

On Dr Ron Knott's multimedia website, hosted by the Mathematics Department of the University of Surrey, you will find a description of an exploration of Fibonacci numbers and Brick Wall Patterns using dominoes.

Guillermo Bautista is a professor at the University of Waterloo in Canada. Dominoes and Mathematical Induction, which was posted recently in his Mathematics and Multimedia Blog, also relates a proof by induction to toppling dominoes.

This Cut the Knot applet may help students understand Solomon Golomb's inductive proof that if a unit square is removed from a 2^{n} x 2^{n} board the rest of the board can be tiled with L-shaped triominos.

If you give students a set of dominoes they may ‘automatically’ start trying to build balanced structures with them! On his web page All a matter of balancing dominoes, a senior lecturer in the School of Mathematics at the University of Birmingham, Christopher J. Sangwin, shows three interesting ways of balancing many dominoes one on another with no cheating – no gluing!

Your students may also enjoy seeing someone balance 17 dominoes on one domino!

The Station House Opera company was the winner in 2009 of the Bank of America CREATE Art Award with their creation Dominoes 2009. Thousands of breeze block dominoes toppled their way southwards through East and South East London, finishing at dusk in Greenwich.

Out of Order is a piece of installation art by David Mach, which is known as Kingston’s phone box dominoes. These toppling domino phone boxes were ‘installed’ in 1989. Are they still there?