Combinatorics of Go

We present several results concerning the number of positions and games of Go. We derive recurrences for L(m,n), the number of legal positions on an m× n board, and develop a dynamic programming algorithm which computes L(m,n) in time O(mnλ) and space O(mλ), for some constant λ < 5.4. An implementation of this algorithm lets us list L(n, n) for n ≤ 17. For bigger boards, we prove existence of a base of liberties limm,n→∞ mn p L(m,n) of 2.9757341920433572493 . . .. Based on a conjecture about vanishing error-terms, we derive an asymptotic formula for L(m,n), which is shown to be highly accurate. We also study the Game Tree complexity of Go, proving an upper bound on the number of possible games of (mn) and a lower bound of 2 n2/2−O(n) on n × n and 2 n−1 on 1 × n boards, in addition to exact counts for mn ≤ 4 and estimates up to mn = 9. We end with investigating whether one game can encompass all legal positions.