Complete Latin Squares and Related Experimental Designs

A description is given of complete latin squares, which are squares with all ordered pairs of elements occurring next to each other once each in rows and columns, and of their uses as experimental designs. The existence of complete latin squares for all even numbers has been known for more than 30 years. Squares with various types of symmetry are considered here, all possible squares being described with side 4, 6, 8 and 10, together with some of side 12, 14, 16 and 18. These are put into genera whose members can all be obtained from each other by one or more of the processes of interchanging rows and columns, rearranging rows or reading the square in reverse. Where no complete latin squares exist, quasi-complete latin squares are considered in which all unordered pairs occur twice in rows and twice in columns. Such squares exist for all odd numbers, as well as even numbers greater than 4, and examples are given up to size 9.