Room square

Room square

A Room square, named after professor Thomas Gerald Room (1902–1986), is an "n" × "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:

: egin{bmatrix} 7,0 & & & 1,5 & & 4,6 & 2,3 \ 3,4 & 7,1 & & & 2,6 & & 5,0 \ 6,1 & 4,5 & 7,2 & & & 3,0 & \ & 0,2 & 5,6 & 7,3 & & & 4,1 \ 5,2 & & 1,3 & 6,0 & 7,4 & & \ & 6,3 & & 2,4 & 0,1 & 7,5 & \ & & 0,4 & & 3,5 & 1,2 & 7,6 \end{bmatrix}.

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

See also

* Combinatorial design

References

*cite book
author = Dinitz J. H. (ed.), Stinson D. R. (ed.)
title = Contemporary Design Theory — A Collection of Surveys
publisher = John Wiley & Sons
date = 1992
pages = 137–204
isbn = 0471531413

*cite web |url=http://www.asap.unimelb.edu.au/bsparcs/aasmemoirs/room.htm
title=Thomas Gerald Room 1902–1986 |accessdate=2008-08-05 |work= |publisher=Australian Academy of Science |date=


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