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Solved Games

Hex (7×7)

Connection games · all families · data

Also known as: Nash, John, Con-Tac-Tix, Polygon.

First player wins with perfect play. All 49 opening moves are resolved: central openings (and the obtuse-corner short diagonal) win for the first player, and a band of 14 losing openings clusters near the two acute corners (Yang et al. derived them by symmetry), so the first player wins by opening on a central cell (e.g. the centre cell d4).
Result First player wins
Strength Weak
Method Strategy-stealing, Knowledge-based, Depth-first PNS, Threat-space search
Year 2003
Solved by Jing Yang, Simon Liao & Mirek Pawlak (first explicit hand strategy, 2001–2002); first full computer solution of all 49 openings by Ryan B. Hayward, Yngvi Björnsson, Michael Johanson, Morgan Kan, Nathan Po & Jack van Rijswijck (University of Alberta, 2003)
Source Yang et al. 2003
Board 7×7 rhombus of hexagonal cells (49 cells); each player owns two opposite sides, first player connects their pair; no-swap (no pie rule) ruleset
Distance to result Longest line of play is given for each of the 49 openings; the hardest single opening took ~110 hours of search, ~615 hours total for all 49 (single-processor AMD Athlon, 2003). No single uniform ply count — proof depth varies by opening.

Solution history

Section titled “Solution history”
Date Size or opening Result Strength Credit Source
2003 7×7 first player wins Weak Jing Yang, Simon Liao & Mirek Pawlak 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”

Explorers

Write-ups

Notes

Section titled “Notes”

WEAK solution, not strong — the most common error here is to call the explicit Yang strategy “strong.” Finding a winning strategy from the start (and from each of the 49 openings) is a constructive WEAK solution; it does NOT store a game-theoretic value for every reachable position, so there is no full tablebase. Attribution (important): the hint attributes the CG 2002 paper “Solving 7×7 Hex: Virtual Connections and Game-State Reduction” to “Yang, Liao & Pawlak,” but that paper is by Hayward, Björnsson, Johanson, Kan, Po & van Rijswijck (Univ. of Alberta) and is the first FULL COMPUTER solve of all 49 openings. It was presented at Advances in Computer Games 10 (the CG/ACG 2003 line; IFIP proceedings carry a 2004 imprint) and the expanded journal version appeared in Theoretical Computer Science 349 (2005). Yang, Liao & Pawlak are the authors of the SEPARATE, earlier hand solutions: a decomposition-method explicit first-player winning strategy (ADCOG21 2001; “A New Solution for 7×7 Hex Game”, 2002) and “New Winning and Losing Positions for 7×7 Hex” (CG 2002). Yang’s hand proof used ~41 local game patterns in a ~12-page case analysis; the Alberta team’s program independently verified and completed the result for every opening. The headline year is 2003 (first complete computer solve); the Yang hand work is 2001–2002. How the computer solve works: a recursive-descent game-tree search (after Van Rijswijck) that proves a winning virtual connection for the winner via fixed/dynamic connection-composition rules (the “win-link”/“win-prelink” H-search lineage of Anshelevich and Berge), pruned by new move-domination and cell fill-in reductions plus mustplay-region narrowing. For each of the 49 openings the search emits an explicit proof-tree for the winning player; all 49 trees plus a tree viewer were published at the hex7trees site. Runtime: ~1 minute for the easiest openings up to ~110 hours for the hardest, ~615 hours total, on a single AMD Athlon 1800+ (gcc 3.1.1, Slackware Linux), 2003. Result: first player wins. Central openings (including the centre cell d4) and the obtuse-corner short diagonal are wins; the 14 losing first moves cluster near the two ACUTE corners (Yang et al. derived them by board symmetry). Under the no-swap ruleset reported here. Ruleset caveat: this is the no-swap (no pie-rule) game. Under the swap/pie rule the second player can flip a winning first move, changing the practical game but not the cell-by-cell opening values. Ultra-weak result, ALL board sizes: independently of any search, Hex on any n×n board is a first-player win by John Nash’s strategy-stealing argument (c. 1949) combined with the Hex theorem (Hein 1942 / Nash / Gale) that Hex can never be a draw — exactly one player connects. That argument is non-constructive (it proves first player wins without giving a move); the 7×7 results here are the constructive WEAK upgrade for this size. Deciding an arbitrary generalized-Hex position is PSPACE-complete (Even & Tarjan 1976; Reisch 1981), which is why explicit solving stalls around 9×9–10×10. Lineage of forward-Hex solves: 7×7 — Yang/Liao/Pawlak hand (2001–2002) and Hayward et al. full computer solve (2003); 8×8 — Henderson, Arneson & Hayward (IJCAI 2009, all openings); 9×9 — Pawlewicz & Hayward (2013, all openings); 10×10 — only the central opening (2013). Complexity: no 7×7-specific state-space/game-tree figure is cited in the primary paper, so both fields are left null rather than guessing. For reference, surveys give 11×11 Hex a state-space complexity of roughly 10^57 and game-tree complexity of roughly 10^98; 7×7 is correspondingly far smaller.