Geodesic
An Existence Theorem for Room Squares*
Canadian mathematical bulletin, Tome 12 (1969) no. 4, pp. 493-497

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It is shown that if v is an odd prime power, other than a prime of the form 22n + 1, then there exists a Room square of order v + 1.A room square of order 2n, where n is a positive integer, is an arrangement of 2n objects in a square array of 2 side 2n - 1, such that each of the (2n - 1)2 cells of the array is either-empty or contains exactly two distinct objects; each of the 2n objects appears exactly once in each row and column; and each (unordered) pair of objects occurs in exactly one cell. 
Mullin, R. C.; Nemeth, E. An Existence Theorem for Room Squares*. Canadian mathematical bulletin, Tome 12 (1969) no. 4, pp. 493-497. doi: 10.4153/CMB-1969-063-6
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     title = {An {Existence} {Theorem} for {Room} {Squares*}},
     journal = {Canadian mathematical bulletin},
     pages = {493--497},
     year = {1969},
     volume = {12},
     number = {4},
     doi = {10.4153/CMB-1969-063-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1969-063-6/}
}
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