# 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 $( x , y ) ∈ { 1 , 2 , . . . , n } × { 1 , 2 , . . . , n }$ 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 ≤ abt, 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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