PreprintCC BY 4.0arXiv
Counting Reachable Positions in a Simplified Model of Connect Four
Abstract
By eliminating the win condition in the game of Connect Four and extending the board to infinite height, a rich state space of positions is obtained. We investigate the number of positions reachable on an n-column board after k color-alternating moves. For fixed k we demonstrate polynomiality, derive a partial formula for the polynomial coefficients, and precisely characterize the asymptotic behavior as n → ∞. We then turn our attention to the fixed-n case and show that, under a natural addition operation, positions reachable in an even number of moves form a monoid with a highly symmetric finite generating set; by examining certain free submonoids, we bound the exponential growth rate as k → ∞.
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BibTeX
@misc{folster2026counting,
title = {{Counting Reachable Positions in a Simplified Model of Connect Four}},
author = {Alistair Folster},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33068},
eprint = {2609.33068},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33068},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Counting Reachable Positions in a Simplified Model of Connect Four
AU - Alistair Folster
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33068
UR - https://arxiv.org/abs/2609.33068
AB - By eliminating the win condition in the game of Connect Four and extending the board to infinite height, a rich state space of positions is obtained. We investigate the number of positions reachable on an n-column board after k color-alternating moves. For fixed k we demonstrate polynomiality, derive a partial formula for the polynomial coefficients, and precisely characterize the asymptotic behavior as n → ∞. We then turn our attention to the fixed-n case and show that, under a natural addition operation, positions reachable in an even number of moves form a monoid with a highly symmetric finite generating set; by examining certain free submonoids, we bound the exponential growth rate as k → ∞.
ER -