By Daniel J. Velleman
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Additional info for American Mathematical Monthly, volume 117, May 2010
Vecchi, Optimization by simulated annealing, Science 220 (1983) 671–680. 671 46. N. Metropolis, The beginning of the Monte Carlo method, Los Alamos Science 15 (1987) 125–130. 47. N. Metropolis and F. H. Harlow, Computing and computers: Weapons simulation leads to the computer era, Los Alamos Science 12 (1983) 132–141. 412 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 48. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equation of state calculations by fast computing machines, Journal of Chemical Physics 21 (1953) 1087–1092.
Gilks, D. G. Clayton, D. J. Spiegelhalter, N. G. Best, A. J. McNeil, L. D. Sharples, and A. J. Kirby, Gibbs sampling in medicine, J. Roy. Statist. Soc. Ser. B 55 (1993) 39–52. 30. W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, Markov Chain Monte Carlo in Practice, Chapman & Hall/CRC, London, 1996. 31. R. Glauber, Time dependent statistics of the Ising model, J. Math. Phys. 4 (1963) 294–307. doi:10. 1703954 32. J. E. Gubernatis, Marshall Rosenbluth and the Metropolis algorithm, Phys. Plasmas 12 (2005) 1–5.
Conjecture. Let k be a positive integer. Then a k-by-n array in which each column has k distinct integers has the following properties: 1. If n ≤ k − 1, there is a transversal (of length n). 2. If k ≤ n ≤ 2k − 2, there is a near-transversal (of length k − 1). 3. If n ≥ 2k − 1, there is a transversal (of length k). A. Drisko  in 1998 established 3. In Section 4 we present a slightly different but simpler proof of his result. E. T. Parker, in personal correspondence, in 1989 showed by a construction that in case 2 there need not be a transversal.
American Mathematical Monthly, volume 117, May 2010 by Daniel J. Velleman