Room Square

A Room square (named after T. G. Room) of order $n$ (for $n$ Even) is an arrangement in an $(n-1)\times(n-1)$ Square Matrix of $n$ objects such that each cell is either empty or holds exactly two different objects. Furthermore, each object appears once in each row and column and each unordered pair occupies exactly one cell. The Room square of order 2 is shown below.

1,2

The Room square of order 8 is

1,8			5,7		3,4	2,6
3,7	2,8			6,1		4,5
5,6	4,1	3,8			7,2
	6,7	5,2	4,8			1,3
2,4		7,1	6,3	5,8
	3,5		1,2	7,4	6,8
		4,6		2,3	1,5	7,8

References

Dinitz, J. H. and Stinson, D. R. In Contemporary Design Theory: A Collection of Surveys (Ed. J. H. Dinitz and D. R. Stinson). New York: Wiley, 1992.

Gardner, M. Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 146-147 and 151-152, 1988.

Mullin, R. C. and Nemeth, E. ``On Furnishing Room Squares.'' J. Combin. Th. 7, 266-272, 1969.

Mullin, R. D. and Wallis, W. D. ``The Existence of Room Squares.'' Aequationes Math. 13, 1-7, 1975.

O'Shaughnessy, C. D. ``On Room Squares of Order $6m+2$.'' J. Combin. Th. 13, 306-314, 1972.

Room, T. G. ``A New Type of Magic Square'' (Note 2569). Math. Gaz. 39, 307, 1955.

Wallis, W. D. ``Solution of the Room Square Existence Problem.'' J. Combin. Th. 17, 379-383, 1974.