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# 16: Latin Squares

• • Joy Morris
• Professor (Mathematics) at University of Lethbridge
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• 16.1: Latin Squares and Sudokus
You can think of a Latin square as a Sudoku puzzle that can be of any (square) size, and does not have the requirement that every value appear in each of the outlined smaller subsquares. A Latin square of order n is an n×n array whose entries are elements of a set N of cardinality n, with the property that every element of N appears exactly once in each row and each column.
• 16.2: Mutually Orthogonal Latin Squares (MOLS)
Most of design theory is concerned with creating nice structures in which different combinations of elements occur equally often. This is the general structure of all of the design theory we will be covering here, and in this context, orthogonal Latin squares are the natural thing to learn about.
• 16.3: Systems of Distinct Representatives
Suppose we start filling in a Latin square, one row at a time, at each step ensuring that no element has yet appeared more than once in a column (or in a row). Under what conditions will it be impossible to complete this to a Latin square? Although it may not be immediately obvious, the answer to this question can be found in a well-known theorem published by Philip Hall in 1935, about systems of distinct representatives.
• 16.4: Summary
This page contains the summary of the topics covered in Chapter 16.