OK, my searcher has now found matrices of the form Toeplitz*Hankel, where * denotes Hadamard elementwise matrix product (C=A*B means C_ij=A_ij B_ij) where both A and B are sign matrices, which achieve the maximum possible |determinant| among ALL matrices of size N with entries in [-1,1], for EVERY N=1,2,3,...,13 (and N=16). See http://oeis.org/A003433 . I doubt it will be possible for me to resolve the first open case N=14 by brute force, though; I do not have enough compute power. The largest det it has found so far is 68853760, not reaching 77635584. Richard Brent points out that my property of Toeplitz*Hankel doesn't seem to be invariant under Hadamard equivalence. Indeed, this kind of matrix is rather mysterious to me. Are there fast algorithms for doing stuff (like computing determinant) for matrices of this form? I do not know. This kind of matrix arose naturally in an application of mine (which is too much trouble to tell you about right now). They do have the nice property that any sub-block matrix of a Toep*Hank matrix, also is one. Here are example matrices: (Toeplitz specified by stating leftmost column going up, then continuing along top row going right, i.e. "clockwise"; then Hankel specified by stating top row going right, then continuing along rightmost column going down, also "clockwise." N det example matrix 1 1.00 -- 2 2.00 +---+- 3 4.00 ++----+--- 4 16.00 ++--+---+----- 5 48.00 -+++-+++-++------- 6 160.00 +--+++-++---+--------- 7 576.00 +--+---+-++---+----------- 8 4096.00 ++-+++--+---+---+-+-+--------- 9 14336.00 ---+---+--+++-++--+-+-+-+--------- 10 73728.00 ++-++-+++---+--+---------------------- 11 -327680.00 +-+++--++++-+---+------------------------- 12 2985984.00 ---++--+--+++-++--+++----+---+-+-+-+---+------ 13 14929920.00 +--+-+------++--+-+------------------------------- For N=16, you can use a Circulant*Backcirculant (top rows given) 16 4294967296.00 circ=++---++-+++-+++- back=-+-+------------ which is a special case of Toep*Hank. (Note, this special case, and the plain Toeplitz special case, both are inadequate to get max determinant for N=9. I have not yet investigated the other interesting case where we demand the Toep and Hank be the same.) I actually am quite impressed with this. I had suspected my searcher was not going to find a maxdet example for N=12, and indeed had fantasies that maybe I could prove it impossible using the known fact the size-12 Hadamard matrix is unique with amazing symmetry group (Mathieu). However it duly found it after 9 billion random trials. So now, does anybody dare to conjecture that this very restricted matrix form always suffices to reach max determinant? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)