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# CP30710 - CP3 1 ECE 3331 COMPUTER PROBLEM 3 Spring 2007(due...

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CP3.1 ECE 3331 COMPUTER PROBLEM 3 Spring 2007 (due 3/8/07) An n × n matrix constructed from the integers from 1 to n in such a way that each integer occurs exactly once in each row and each column is called a Latin square. Two different 5 × 5 Latin squares are shown below. | 5 1 2 3 4 | |5 1 2 3 4| | 2 3 4 5 1 | |4 5 1 2 3| | 4 5 1 2 3 | |3 4 5 1 2| | 1 2 3 4 5 | |2 3 4 5 1| | 3 4 5 1 2 | |1 2 3 4 5| As esoteric as the definition first appears, Latin squares have found application in the field of experimental design. They are used to design experiments in order to reduce the effect of outside factors on the "treatments" being studied, which results in an decrease in the overall variance of results. The name Latin square originates from Leonhard Euler, who used Latin characters as symbols. Properties of specialized Latin squares have been studied extensively by mathematicians. The popular Sudoku puzzles are a special case of Latin squares; any solution to a Sudoku puzzle is a Latin square. The Latin squares shown above are rather simple, and the cyclic pattern used to produce them can be readily deduced. More complex patterns (and of a higher order than 5) can be calculated by other schemes, such as the one described here in terms of a 6 × 6 square, but applicable to any size: (1)Select all the elements of the top row by choosing a random permutation of the numbers from 1 to 6. (2)Suppose that the permutation was 631425. This sequence provides the first row of the Latin square. It can also provide the directions for fitting(in this case) all the 5's in the square as shown on the left in the figure below.

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