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Published online by Cambridge University Press: 27 June 2025
This paper features a problem that was composed to illustrate the power of combinatorial game theory applied to Go endgame positions. The problem is the sum of many subproblems, over a dozen of which have temperatures significantly greater than one. One of the subproblems is a conspicuous four-point ko, and there are several overlaps among other subproblems. Even though the theory of such positions is far from complete, the paper demonstrates that enough mathematics is now known to obtain provably correct, counterintuitive, solutions to some very difficult Go endgame problems.
1. Introduction
Consider the boards in Figures 1 and 2, which differ only by one stone (the White stone a knight's move south-southwest of E in Figure 1 is moved to a knight's move east-southeast of R in Figure 2). In each case, it is White's turn to move. Play will proceed according to the Ing rules, as recommended by the American Go Association in February 1994. There is no komi. Both sides have the same number of captives when the problem begins.
We will actually solve four separate problems. In Figure 1, Problem 1 is obtained by removing the two stones marked with triangles, and Problem 2 is as shown. Does the removal of the two stones matter? In Figure 2, Problem 3 is obtained by removing the two marked stones, and Problem 4 is as shown. (Problem 3 appeared on the inside cover of Go World, issue 70.)
In each case, assume that the winner collects a prize of $1,000. If he wins by more than one point, he can keep the entire amount, but if he wins by only one point, he is required to pay the loser $1 per move.
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