Set Matching: An Enhancement of the Hales-Jewett Pairing Strategy

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When solving k-in-a-Row games, the Hales-Jewett pairing
strategy [4] is a well-known strategy to prove that specic positions are
(at most) a draw. It requires two empty squares per possible winning
line (group) to be marked, i.e., with a coverage ratio of 2.0.

In this paper we present a new strategy, called Set Matching. A matching
set consists of a set of nodes (the markers), a set of possible winning lines
(the groups), and a coverage set indicating how all groups are covered
after every rst initial move. This strategy needs less than two markers
per group. As such it is able to prove positions in k-in-a-Row games to be
draws, which cannot be proven using the Hales-Jewett pairing strategy.
We show several ecient congurations with their matching sets. These
include Cycle Congurations, BiCycle Congurations, and PolyCycle
Congurations involving more than two cycles. Depending on conguration,
the coverage ratio can be reduced to 1.14.

Many examples in the domain of solving k-in-a-Row games are given,
including the direct proof (not based on search) that the empty 4 4
board is a draw for 4-in-a-Row.
Original languageEnglish
Title of host publicationAdvances in Computer Games
Subtitle of host publication15th International Conferences, ACG 2017, Leiden, The Netherlands, July 3–5, 2017, Revised Selected Papers
EditorsMark Winands, H. Jaap van den Herik, Walter Kosters
Publication statusPublished - 2017

Publication series

SeriesTheoretical Computer Science and General Issues

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