14:00
The n-queens problem asks how many ways there are to place n queens on an n×n chessboard so that no two queens can attack one another, and the toroidal n-queens problem asks the same question where the board is considered on the surface of a torus. Let Q(n) denote the number of n-queens configurations on the classical board and T(n) the number of toroidal n-queens configurations. The toroidal problem was first studied in 1918 by Pólya who showed that T(n)>0 if and only if n \equiv 1,5 \mod 6. Much more recently Luria showed that T(n)\leq ((1+o(1))ne^{-3})^n and conjectured equality when n \equiv 1,5 \mod 6. We prove this conjecture, prior to which no non-trivial lower bounds were known to hold for all (sufficiently large) n \equiv 1,5 \mod 6. We also show that Q(n)\geq((1+o(1))ne^{-3})^n for all n \in \mathbb{N} which was independently proved by Luria and Simkin and, combined with our toroidal result, completely settles a conjecture of Rivin, Vardi and Zimmerman regarding both Q(n) and T(n).
In this talk we'll discuss our methods used to prove these results. A crucial element of this is translating the problem to one of counting matchings in a 4-partite 4-uniform hypergraph. Our strategy combines a random greedy algorithm to count `almost' configurations with a complex absorbing strategy that uses ideas from the methods of randomised algebraic construction and iterative absorption.
This is joint work with Peter Keevash.