From: "James Van Buskirk" Subject: Re: Maximum value of determinant Date: Tue, 28 Nov 2000 18:22:42 -0700 Newsgroups: sci.math Summary: [missing] Derek Ross wrote in message <3A22A14F.5B7B@iders.ca>... >I recall reading in on of Ian Stewart's books, that the maximum >determinant for a {0,1} matrix is an unsolved problem, and has only been >solved for matrices up to 14x14. Yes, that's right. There are three degrees of "unsolvedness:" 1) A good upper bound to the upper bound is known, but a matrix corresponding to the upper bound or a proof of the existence of such a matrix is not yet known. The most well-known case of this is when n = 3 (mod 4) for sufficiently large n. 2) A matrix giving a good lower bound to the upper bound is known, but has not yet been proved to have the maximal determinant. I believe that the smallest such matrices are: 0 0 1 0 0 0 0 1 1 1 0 1 1 1 1 0 1 1 1 0 0 0 0 0 1 0 1 1 1 1 1 1 0 0 1 0 0 1 0 0 0 1 1 1 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 1 0 1 0 0 1 1 1 0 1 0 1 0 1 0 1 1 0 1 1 1 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 0 1 0 1 0 1 0 1 1 1 0 1 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 0 0 1 0 0 0 0 1 0 1 1 1 1 1 0 0 1 1 0 0 0 0 1 1 1 0 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 0 0 1 0 1 1 1 1 0 0 0 1 1 0 0 0 1 0 1 1 0 1 0 1 0 0 1 1 0 0 1 1 1 1 1 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 1 1 1 1 0 1 1 0 1 1 0 1 1 0 0 1 1 1 0 1 1 0 0 1 0 0 0 1 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 0 0 1 0 0 0 1 0 1 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 1 1 0 0 1 0 0 1 0 1 1 1 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 0 1 0 0 0 1 1 1 0 1 1 1 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 0 0 0 0 0 1 1 1 1 0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 1 1 1 0 0 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 1 0 1 1 1 1 0 0 1 0 1 0 0 0 0 1 0 1 1 1 1 1 1 0 1 0 0 0 0 1 0 1 1 0 0 1 0 0 0 1 0 0 1 1 1 0 0 0 0 1 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 1 0 0 0 0 1 0 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 1 0 0 0 1 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 0 1 0 1 0 0 1 1 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 0 0 1 1 0 0 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1 1 1 1 1 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 1 1 0 0 1 0 1 1 0 0 1 1 0 0 1 1 0 1 1 1 0 0 0 0 1 1 0 1 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 0 1 1 0 0 0 0 0 1 0 0 1 0 1 1 1 1 1 0 0 0 0 0 1 1 0 1 1 0 1 0 1 0 0 0 1 0 0 1 0 1 1 1 1 1 1 1 0 0 1 1 0 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 1 1 1 0 0 0 1 1 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 0 1 1 1 0 0 1 1 0 1 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 0 1 1 1 1 1 1 0 0 0 0 1 0 0 0 1 1 1 1 0 0 1 1 1 0 0 1 1 1 0 1 0 0 0 1 1 1 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 1 1 0 0 1 1 0 1 0 0 0 0 1 1 0 1 1 0 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 0 1 0 1 0 0 1 1 0 0 0 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 0 1 1 0 1 0 1 1 1 1 0 1 0 0 0 1 1 0 0 1 1 0 1 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 1 0 0 0 1 0 0 0 1 1 0 1 0 0 1 0 1 0 0 1 1 1 1 0 1 1 0 1 0 1 0 1 0 0 1 1 1 0 0 0 0 0 1 1 0 0 1 1 1 0 0 1 1 1 1 0 1 1 0 1 1 1 0 1 0 0 1 0 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 0 1 1 0 1 1 1 1 1 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 1 0 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 1 0 0 1 0 1 0 1 0 0 0 1 0 0 0 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 1 0 1 1 0 1 1 0 1 1 0 0 1 1 1 0 0 1 0 0 1 1 1 1 1 1 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 1 1 1 0 0 0 1 0 1 1 1 1 1 0 0 0 0 1 0 1 0 0 0 1 1 1 1 0 0 0 1 3) Cases where nobody really has a clue, e.g. 0 1 1 1 0 0 1 1 0 1 0 0 1 0 0 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 0 1 1 1 0 1 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 1 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 0 1 1 1 1 0 0 1 0 1 1 0 1 0 1 0 0 0 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 0 0 0 1 1 0 1 0 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 1 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 1 0 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 1 1 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 1 0 1 1 1 0 0 0 0 1 0 1 1 1 1 0 1 1 0 1 0 0 1 1 0 1 1 1 0 1 0 1 1 1 0 1 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 1 0 0 0 1 1 1 1 1 0 1 0 1 1 1 0 1 0 1 1 0 1 0 0 1 1 1 0 1 0 0 0 1 1 0 1 1 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 1 1 1 1 1 0 0 1 0 1 0 0 1 1 1 0 0 0 1 1 1 0 0 1 0 1 0 1 0 0 1 1 1 1 0 0 0 0 0 1 1 0 1 1 0 1 0 1 0 0 1 0 1 1 1 0 1 1 0 1 1 1 0 1 0 0 0 0 0 0 0 1 1 1 0 1 0 1 1 0 0 1 0 1 1 1 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 0 0 1