Y (side 6)

Connection games · all families · data

Part of the Y connection family — see the family page for the scaling table and a playable explorer.

Also known as: Game of Y, Y-game.

First player wins. Strongly solved for board sides 1-6 — the game-theoretic value of every position is known — and the first player wins from the start at every size. Y can never draw (the Y theorem), confirmed: every position is a win or a loss.

Result	First player wins
Strength	Strong
Method	Retrograde analysis, Brute-force enumeration
Year	2026
Solved by	Brian Liou (open-source game-solver engine, 2026), sides 1-6; see the prior-art note.
Source	Liou 2026*
Board	Triangular board of hexagonal cells; side n has n rows and n(n+1)/2 cells (side 6 = 21 cells). Players alternately place a stone of their colour on an empty cell; a player wins by connecting all three sides of the triangle with one group. Hex reduces to Y. Strongly solved here for sides 1-6.
State space	2,173,243,128
Distance to result	Side-6 (2,173,243,128 positions) solved in ~13 minutes in ~1 GB of RAM on a single core, by slice-by-slice retrograde over the ply-layers.
Evidence	self-archived*

* Self-archived: the result is available only from its authors (a repository, a post, a deposit), not yet refereed, formally verified or reproduced by someone else. The mark comes off when one of those happens. It applies to this repo’s own results too.

Solution history

Date	Size or opening	Result	Strength	Credit	Source
1975	all sizes	first player wins (strategy stealing, names no moves)	Ultra-weak	Craige Schensted & Charles Titus	source
2026-06-19	sides 1-6	first player wins at every size; no position is a draw	Strong	Brian Liou	source

Superseded rows stay in the table.

Open problem: Y (side 7)

Status	Smaller board solved
Solved so far	Sides 1-6 strongly solved; first player wins at every size
Next target	Side 7 (28 cells), slice by slice
Barrier	memory, storage: Side 6 is 2,173,243,128 count-legal positions, solved in about 13 minutes on one core in about 1 GB of RAM. Side 7’s count-legal index exceeds 2^32, so it needs 64-bit indexing and a value table larger than RAM (disk-paged slices). An estimate of about 10^13 positions dates from 2026-06-19 and was not rechecked.
Latest result	Liou 2026 (2026-06-19)*
Last checked	2026-10-17
Record	data/open/y-side-7.yaml

References

• Brian Liou — strong solution of Y for sides 1-6 via the game-solver engine (2026): a full retrograde solve of every position. The first player wins at every size and no position is a draw. Side-6 has 2,173,243,128 count-legal positions (wins 1,352,692,412 / losses 820,550,716). Validated by three independent solve paths (reachable, dense-index, and slice-by-slice) that agree on every value, plus the no-draw invariant. Reproducible via cargo run --release --bin y_solve6 6.
• GamesCrafters (UC Berkeley), GamesmanClassic mgameofy.c (A. Esteban, 2023) — prior art: open-source Y solver code for board dims 4-7, but with no published game values, no paper, and kTieIsPossible = TRUE (a draw-handling flaw for a game that cannot draw). Only dim-5 is credibly confirmed computed.
• Craige Schensted & Charles Titus, Mudcrack Y and Poly-Y (Neo Press, 1975) — the game of Y; first player wins on any board by a Hex-style strategy-stealing argument (ultra-weak, non-constructive).

Resources

Code

• game-solver — generic retrograde engine (the game-solver repository); Y is a plugin (engine/src/games/y.rs), solved and tested in CI.

Notes

Y is a connection game on a triangular grid: place stones in turn, win by joining all three sides with one connected group. It generalizes Hex (Hex reduces to Y), and like Hex it cannot end in a draw — by the Y theorem a full board always has exactly one player connecting all three sides, and the two players can never connect at once. ULTRA-WEAK result, ALL board sizes: the first player wins on any board by the Hex-style strategy-stealing argument (Schensted & Titus 1975, after Nash). This is non-constructive — it names no moves. STRONG result, sides 1-6 (game-solver engine): it computed the game-theoretic value of every position by retrograde analysis. The first player wins from the start at every size, and no position is a draw — both asserted as correctness invariants (a wrong board model would surface a draw). Sizes and starting value: side 1 (2 positions), side 2 (13), side 3 (257), side 4 (16,505), side 5 (3,337,584 reachable), side 6 (2,173,243,128 count-legal index) — first player wins at every size. METRIC NOTE: sides 1-5 figures count reachable positions (forward discovery); the side-6 figure counts the tight count-legal index (a superset that also includes count-legal-but-unreachable positions), because side-6 was solved by the dense slice solver rather than reachable enumeration. The two metrics are not directly comparable; wins + losses equal the total exactly at every size, which is the no-draw invariant. PRIOR ART: the earlier solver located is GamesCrafters’ GamesmanClassic mgameofy.c (A. Esteban, 2023), which has Y code for dims 4-7 but publishes no game values, has no accompanying paper, and declares draws possible (kTieIsPossible = TRUE) for a game that cannot draw; an adversarial check could confirm only its dim-5 as credibly computed. The search was not exhaustive and no priority is claimed; this entry records the sides solved (1-6), with full credit to the earlier GamesCrafters work. Side-7 (28 cells) has a count-legal index beyond 2^32 and is the natural next target via external-memory (disk-paged) slices.