Room square

A Room square, named after Thomas Gerald Room, is an n-by-n array filled with n + 1 different symbols in such a way that:

  1. Each cell of the array is either empty or contains an unordered pair from the set of symbols
  2. Each symbol occurs exactly once in each row and column of the array
  3. Every unordered pair of symbols occurs in exactly one cell of the array.

An example, a Room square of order seven, if the set of symbols is integers from 0 to 7:

0,7 1,5 4,6 2,3
3,4 1,7 2,6 0,5
1,6 4,5 2,7 0,3
0,2 5,6 3,7 1,4
2,5 1,3 0,6 4,7
3,6 2,4 0,1 5,7
0,4 3,5 1,2 6,7

It is known that a Room square (or squares) exist if and only if n is odd but not 3 or 5.