Y — the connection family

Y is a connection game (Schensted–Titus, 1953) played on a triangular board of side n — n rows, n(n+1)/2 cells. Each move places one stone; you win by connecting all three sides of the triangle with a single group. Because a filled board always has exactly one winner, Y can never draw, and a strategy-stealing argument shows the first player wins at every size. Hex reduces to Y, which is what makes Y worth studying in its own right.

This project strongly solved Y from side 1 through side 6 with its open-source retrograde engine — computing the game-theoretic value of every position, not just the opening. The explorer below plays perfect moves for sides 3–5, solving the whole board live in your browser.

First verified strong solution

No published constructive solution of Y existed at any board size. This is the first documented, draw-correct strong solution — sides 1–6 — with every value cross-checked by three independent solve paths. Prior art and citations →

The family, by size

Side	Cells	Positions	Result
1	1	2	First player wins
2	3	13	First player wins
3	6	257	First player wins
4	10	16,505	First player wins
5	15	3,337,584	First player wins
6	21	2,173,243,128 †	First player wins

All sides above are strongly solved by this project. Y is a first-player win at every size, with no draws.

† Sides 1–5 count reachable positions; side 6 is the dense-index space (reachable positions were not separately enumerated). The explorer solves over the dense index, so its side-5 count (3,492,117) is slightly higher than the reachable count shown here.

Frontier — side 7

Side 6 is the largest Y solved here. Side 7 (28 cells) is the open next target, and the first that outgrows the current solver: its dense index exceeds 2³² (an estimated ~10¹²–10¹³ positions), so it needs 64-bit indexing and a value table far larger than RAM. The plan is a disk-paged (external-memory) slice retrograde — the extension the engine is already being built toward — which would extend the first verified strong solution of Y one more rung. See the open problem.

Play the solver

Every legal move is coloured by its value under perfect play — green wins, red loses. Switch board size with the toggle; each size is solved live in your browser when you select it.

More

• Y data sheet — result, method, citations, and prior art.
• The engine — the generic retrograde solver Y is a plugin of.