Using only the numbers 1, 2, 3, and 4, arrange four sets of these numbers into a four-by-four array so that no column or row contains the same two numbers. The result is known as a Latin square.
Here are two examples of Latin squares of order 4:
1 | 2 | 3 | 4 |
2 | 1 | 4 | 3 |
3 | 4 | 1 | 2 |
4 | 3 | 2 | 1 |
1 | 2 | 3 | 4 |
3 | 4 | 1 | 2 |
4 | 3 | 2 | 1 |
2 | 1 | 4 | 3 |
In effect, each row (and each column) is a permutation of four distinct numbers (or symbols).
Such arrays have proved useful for a variety of purposes. Suppose, for example, you wanted to check the resistance to wear of four brands of automobile tires. Putting one tire of each brand on the four wheels of one car isn’t good enough because the amount of wear may differ in those four positions and vary from week to week owing to different weather conditions. A better experiment would be to use four tires for four weeks and to interchange the four positions from week to week according to a four-by-four Latin square.