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Encyclopaedia of DesignTheory: A complete Latin square

The ordering

(0,1,5,2,4,3)

of the group of integers modulo 6 is called a sequencing of the group, because the list

(1,4,3,2,5)

of differences of consecutive elements contains each non-zero group element once. (This list is called a directed terrace, and the sequencing can be recovered from it, up to translation).

The translates of the sequencing form the rows of a Latin square, which is row-complete: each ordered pair of distinct symbols occurs just once as consecutive elements in the same row. If we order these rows so that the first column is also a sequencing, then the Latin square is also column-complete (that is, the same property holds for columns), and hence complete:

 0 1 5 2 4 3 1 2 0 3 5 4 5 0 4 1 3 2 2 3 1 4 0 5 4 5 3 0 2 1 3 4 2 5 1 0

If we add a new symbol 6 at the end of the first row, then develop that row cyclically mod 6 (with 6 fixed), and regard the rows as cyclic (so that 6 is also adjacent to the symbol at the beginning of the row), we obtain a complete-block neighbour design; that is, given any two distinct symbols, the second occurs to the right of the first precisely once. In the first three rows of the table, any pair of symbols occur in adjacent positions (in some order) once.

 0  1  5  2  4  3  6   1  2  0  3  5  4  6   2  3  1  4  0  5  6 3  4  2  5  1  0  6   4  5  3  0  2  1  6   5  0  4  1  3  2  6

Such a design is called a "round-dance neighbour design", in view of its connection with the problems of Lucas and Dudeney of arranging for an odd number of girls to dance round dances so that each pair of girls hold hands once (or once on each side, respectively).

Peter J. Cameron
28 October 2002