# 6

# Mutually Orthogonal Latin Squares

### 6.1 Introduction

Two latin squares L _{1} and L _{2} of order n are said to be orthogonal if for each $(x,y)\in \{1,2,...,n\}\times \{1,2,...,n\}$ there is exactly one ordered pair $(i,j)$ such that cell $(i,j)$ of L _{1} contains the symbol x and cell $(i,j)$ of L _{2} contains the symbol y. In other words, if L _{1} and L _{2} are superimposed, the resulting set of n ^{2} ordered pairs are distinct. The latin squares L _{1}, L _{2}, . . ., L _{ t } are said to be mutually orthogonal if for 1 ≤ a ≠ b ≤ t, L _{ a } and L _{ b } are orthogonal.

The easiest way to show that a pair of latin squares are orthogonal is to use the “ famous” two -finger rule.

The Two-Finger Rule Let L and M be latin squares ...

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