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

TL;DR

The paper introduces Set Matching, a new strategy for k-in-a-Row games that uses fewer markers than the traditional Hales-Jewett pairing, enabling the proof of draws in positions previously unprovable.

Contribution

It presents Set Matching, a novel strategy that reduces marker usage and proves more positions as draws in k-in-a-Row games compared to existing methods.

Findings

Set Matching requires less than two markers per group.

It can prove certain positions as draws that Hales-Jewett cannot.

The strategy is effective in configurations like Cycle, BiCycle, and PolyCycle.

Abstract

When solving k-in-a-Row games, the Hales-Jewett pairing strategy [4] is a well-known strategy to prove that specific 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 first 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 efficient configurations with their matching sets. These include Cycle Configurations, BiCycle Configurations, and PolyCycle Configurations involving more than two cycles. Depending on configuration, 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 (without further investigation) that the empty 4 x 4 board is a draw for 4-in-a-Row.