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:
- Each cell of the array is either empty or contains an unordered pair from the set of symbols
- Each symbol occurs exactly once in each row and column of the array
- 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.