  The number of $n \times n$ matrices whose entries are either -1, 0, or 1,
whose row- and column- sums are all 1, and such that in every row and every
column the non-zero entries alternate in sign, is proved to be $[1!4! >...
(3n-2)!]/[n!(n+1)! ... (2n-1)!]$, as conjectured by Mills, Robbins, and Rumsey.
