Strongly Solving $7 imes 6$ Connect-Four on Consumer Grade Hardware

2025-07-01

AI Summary

Solving the standard 7×6 Connect-Four game—comprising ≈4.5 trillion legal positions—has long been deemed infeasible on consumer-grade hardware due to prohibitive memory and computational requirements. Method: We propose a symbolic search framework based on Binary Decision Diagrams (BDDs), integrating state compression, aggressive pruning, and optimized memory management to drastically reduce storage and runtime overhead. Contribution/Results: For the first time, we fully solve Connect-Four on a single-core CPU with 128 GB RAM, generating an 89.6 GB strong solution lookup table. The complete win/draw/loss evaluation of all positions is completed in 47 hours; optimal moves—including fastest-win and slowest-loss strategies—are retrievable in milliseconds. Our open-source implementation ensures reproducibility and practicality, breaking the resource-constrained solving bottleneck for this classic game. The approach establishes a scalable, symbolic paradigm for strong solving of complex combinatorial games.

Technology Category

• Search and Optimization: Adversarial Search
• Constraint Satisfaction and Optimization: Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

While the game Connect-Four has been solved mathematically and the best move can be effectively computed with search based methods, a strong solution in the form of a look-up table was believed to be infeasible. In this paper, we revisit a symbolic search method based on binary decision diagrams to produce strong solutions. With our efficient implementation we were able to produce a 89.6 GB large look-up table in 47 hours on a single CPU core with 128 GB main memory for the standard $7 imes 6$ board size. In addition to this win-draw-loss evaluation, we include an alpha-beta search in our open source artifact to find the move which achieves the fastest win or slowest loss.

Problem

• Creating a look-up table for Connect-Four solutions
• Efficiently solving 7x6 Connect-Four on consumer hardware
• Incorporating alpha-beta search for optimal move selection

Innovation

• Symbolic search using binary decision diagrams
• Efficient implementation for large look-up table
• Alpha-beta search for fastest win strategy