Room Square
A Room square (named after T. G. Room) of order 
(for Even) is an arrangement in an 
SquareMatrix of objects such that each cell is either empty or holds exactly two different objects. Furthermore, each objectappears once in each row and column and each unordered pair occupies exactly one cell. The Room square of order 2 is shownbelow.
| 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 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.
.'' J. Combin. Th. 13, 306-314, 1972.
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