Abstract
The authors give a condensed proof of the existence of Room squares for positive odd sides except 3 and 5. Some areas of current research on Room squares are also discussed.
Similar content being viewed by others
References
Anderson, B. A.,A perfectly arranged Room square, Proceedings of the Fourth Southeastern Conference on Combinatorics. Graph Theory and Computing, March 1973, pp. 141–150.
Archbold, J. W.,A combinatorial problem of T. G. Room, Mathematika7 (1970), 50–55.
Archbold, J. W. andJohnson, N. L.,A construction for Room squares and application in experimental design, Ann. Math. Statist.29 (1958), 219–225.
Berlekamp, E. R. andHwang, F. K.,Constructions for balanced Howell rotations for bridge tournaments, J. Combinatorial Theory, Series A,12 (1972), 159–166.
Bruck, R. H.,What is a loop? In Studies in Modern Algebra. Mathematical Association of America, 1963, pp. 59–99.
Byleen, K.,On Stanton and Mullin's contruction of Room squares, Ann. Math. Statist.41 (1970), 1122–1125.
Chong, B. C. andChan, K. M.,On the existence of normalized Room squares, Nanta Math.7 (1974), 8–17.
Collens, R. J. andMullin, R. C.,Some properties of Room squares - a computer search. Proceedings of the First Louisiana Conference on Combinatorics. Graph Theory and Computing, Baton Rouge, 1970, pp. 87–111.
Constable, R. L.,Positions in Room squares, Utilitas Math.5 (1974), 57–64.
Dillon, J. F. andMorris, R. A.,A skew Room square of side 257, Utilitas Math.4 (1973), 187–192.
Gross, K. B., Mullin, R. C., andWallis, W. D.,The number of pairwise orthogonal symmetric Latin squares, Utilitas Math.4 (1973), 239–251.
Gross, K. B.,Equivalence of Room designs I, J. Combinatorial Theory16 (1974), 264–5.
Gross, K. B.,Equivalence of Room designs II, J. Combinatorial Theory17 (1974), 299–316.
Gross, K. B.,Some new classes of strong starters, (to appear).
Gross, K. B.,On the maximal numbers of pairwise orthogonal Steiner triple systems, (to appear).
Horton, J. D.,Variations on a theme by Moore. Proceedings of the First Louisiana Conference on Combinatorics. Graph Theory and Computing, Baton Rouge, 1970, pp. 146–166.
Horton, J. D.,Quintuplication of Room squares, Aequationes Math.7 (1971), 243–245.
Horton, J. D.,Room designs and one-factorizations, Aequationes Math., (to appear).
Horton, J. D., Mullin. R. C., andStanton, R. G.,A recursive construction for Room designs, Aequationes Math.6 (1971), 39–45.
Hwang, F. K.,Some more contributions on constructing balanced Howell rotations. Proceedings of the Second Chapel Hill Conference on Combinatorial Mathematics and its Applications, Chapel Hill, 1970, pp. 307–323.
Lawless, J. F.,Pairwise balanced designs and the construction of certain combinatorial systems. Proceedings of the Second Louisiana Conference on Combinatorics. Graph Theory and Computing, Baton Rouge, 1971, pp. 353–366.
Lindner, Charles,An algebraic construction for Room squares, SIAM J. Appl. Math.22 (1972), 574–579.
Lindner, Charles C. andMendelsohn, N. S.,Construction of perpendicular Steiner quasigroups, Aequationes Math.9 (1973), 150–156.
Mendelsohn, N. S.,Latin squares orthogonal to their transposes, J. Combinatorial Theory, Series A,11 (1971), 187–189.
Mendelsohn, N. S.,Orthogonal Steiner systems, Aequations Math.5 (1970), 268–272.
Mullin, R. C.,On the existence of a Room design of side F 4, Utilitas Math.1 (1972), 111–120.
Mullin, R. C. andNemeth, E.,A counter-example to a multiplicative construction of Room squares, J. Combinatorial Theory7 (1969), 264–265.
Mullin, R. C. andNemeth, E.,On furnishing Room squares, J. Combinatorial Theory7 (1969), 266–272.
Mullin, R. C. andNemeth, E.,An existence theorem for Room squares, Canad. Math. Bull.12 (1969), 493–497.
Mullin, R. C. andNemeth, E.,On the non-existence of orthogonal Steiner triple systems of order 9, Canad. Math. Bull.13 (1970), 131–134.
Mullin, R. C. andNemeth, E.,A construction for self-orthogonal Latin squares from certain Room squares. Proceedings of the First Louisiana Conference on Combinatorics. Graph Theory and Computing, Baton Rouge, 1970, pp. 213–226.
Mullin, R. C. andSchellenberg, P. J.,Room designs of small side. Proceedings of the Manitoba Conference on Numerical Mathematics. University of Manitoba, 1971, pp. 521–526.
Mullin, R. C. andWallis, W. D.,On the existence of Room squares of order 4n, Aequationes Math.6 (1971), 306–309.
Nemeth, E.,A study of Room squares, Thesis, University of Waterloo, 1969.
O'Shaughnessy, C. D.,A Room design of order 14, Canad. Math. Bull.11 (1968), 191–194.
O'Shaughnessy, C. D.,On Room squares of order 6m+2, J. Combinatorial Theory, Series A,13 (1972), 306–314.
Parker, E. T. andMood, A. N.,Some balanced Howell rotations for duplicate bridge sessions, Amer. Math. Monthly62 (1955), 714–716.
Room, T. G.,A new type of magic square, Math. Gazette39 (1955), 307.
Schellenberg, P. J.,On balanced Room squares and complete balanced Howell rotations, Aequationes Math.9 (1973), 75–90.
Shah, K. R.,Analysis of Room's square design, Ann. Math. Statist.41 (1970), 743–745.
Stanton, R. G. andHorton, J. D.,Composition of Room squares. Colloquia Mathematica Societatis János Bolyai, 4: Combinatorial Theory and its Applications, North-Holland, 1970, pp. 1013–1021.
Stanton, R. G. andHorton, J. D.,A multiplication theorem for Room squares, J. Combinatorial Theory12 (1972), 322–325.
Stanton, R. G. andMullin, R. C.,Construction of Room squares, Ann. Math. Statist.39 (1968), 1540–1548.
Stanton, R. G. andMullin, R. C.,Techniques for Room squares. Proceedings of the First Louisiana Conference on Combinatorics. Graph Theory and Computing, Baton Rouge, 1970, pp. 445–464.
Stanton, R. G. andMullin, R. C.,Room quasigroups and Fermat primes, J. Algebra20 (1972), 83–89.
Wallis, W. D.,Room squares. Invited and contributed papers, Australasian Statistical Conference, Sydney, 1971.
Wallis, W. D., Street, A. P., andWallis, J. S.,Combinatorics: Room squares, sum-free sets, Hadamard matrices. Lecture Notes in Mathematics, 292. Springer Verlag, Berlin, 1972.
Wallis, W. D.,Duplication of Room squares, J. Austral. Math. Soc.14 (1972), 75–81.
Wallis, W. D.,A construction for Room squares. A Survey of Combinatorial Theory, J. Srivastava (ed.), North Holland, 1973, pp. 449–451.
Wallis, W. D.,A doubling construction for Room squares, Discrete Math.3 (1972), 397–399.
Wallis, W. D.,On one-factorization of complete graphs, J. Austral. Math. Soc.16 (1973), 167–171.
Wallis, W. D.,A family of Room subsquares, Utilitas Math.4 (1973), 9–14.
Wallis, W. D.,On the existence of Room squares, Aequationes Math.9 (1973), 260–266.
Wallis, W. D.,On Archbold's construction of Room squares, Utilitas Math.2 (1972), 47–54.
Wallis, W. D. andMullin, R. C.,Recent advances on complementary and skew Room squares. Proceedings of the Fourth Southeastern Conference on Combinatorics. Graph Theory and Computing, March 1973, pp. 521–532.
Wallis, W. D.,A Room square of side 257. Proceedings of the Fourth Southeastern Conference on Combinatorics. Graph Theory and Computing, March 1973, p. 533.
Wallis, W. D.,Room squares with sub-squares, J. Combinatorial Theory15 (1973), 329–332.
Wallis, W. D.,Room squares of side five, Delta3 (1973), 32–36.
Weisner, L.,A Room design of order 10, Canad. Math. Bull.7 (1964), 377–378.
Author information
Authors and Affiliations
Additional information
Aequationes Mathematicae launches a systematic program of expository papers. We will endeavour to publish at least one in every volume.
Rights and permissions
About this article
Cite this article
Mullin, R.C., Wallis, W.D. The existence of Room squares. Aeq. Math. 13, 1–7 (1975). https://doi.org/10.1007/BF01834113
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01834113