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Solved Games

Pentago

Alignment & m,n,k games · all families · data

First player (white) wins with perfect play
Result First player wins
Strength Strong
Method Retrograde analysis, Brute-force enumeration
Year 2014
Solved by Geoffrey Irving
Source Irving 2014
Board 6×6 with four rotatable 3×3 quadrants
State space 3,009,081,623,421,558

Solution history

Section titled “Solution history”
Date Size or opening Result Strength Credit Source
2014 standard game first player wins Strong Geoffrey Irving source

One row, derived from the record’s year, solver and primary source; no fuller history is recorded yet.

Engines

Section titled “Engines”

References

Section titled “References”

Resources

Section titled “Resources”

Tablebases

Code

Explorers

Write-ups

Notes

Section titled “Notes”

Pentago: a 6×6 board split into four 3×3 quadrants; each turn a player places one marble and then rotates one quadrant 90°, aiming for five-in-a-row. Strongly solved by Geoffrey Irving in 2014 — perfect play is efficiently computable for any position, the defining property of a strong solution. The first player wins. Computed by exhaustive parallel in-core retrograde analysis in under 4 hours on 98,304 threads of NERSC’s Cray Edison (≈80 TB RAM at peak). At 3.0×10^15 states (3,009,081,623,421,558 positions with symmetries removed), it is described as the largest divergent game solved to date by two orders of magnitude, and the only nontrivial divergent game solved via retrograde analysis. The solution is stored as a 3.7 TB database holding full results for all positions with ≤18 stones; positions with 18+ stones are evaluated on the fly. Ties are possible (e.g. both players complete five-in-a-row on the same move, or the board fills with no winner), but the game-theoretic value from the empty board is a first-player win. The three sources above are distinct artifacts (peer-style paper, project website, open-source code+dataset) of a single research effort by Irving — one independent primary result, multiply corroborated. No conflicting solution exists in the literature. The state_space figure is post-symmetry-reduction reachable positions per the project; distance to the result (ply count of the forced win) is not published as a single figure.