Maker–Breaker · the Snaky hexomino · 758 proof cards · improved
The same game as Snaky in 21, but Black plays an improved strategy: the paper's cards plus 30 new ones, found by computer search and checked with the paper's own verifier. Black now always wins within 20 moves, and after a diagonal or knight's-move first reply within 18. How long can you make it take?
Copy this game, or paste an SGF and load it. Only White's moves are read; Black replays its own.
The paper proves the result with cards. A card is a promise of the form: if Black already owns the required cells A and White has no stone in the envelope T, Black wins within h more moves. Cards 0–5 are the six ways to be one stone from Snaky. Every later card plays one pivot stone and holds a hand of child cards whose envelopes have nothing in common except cells Black already owns. So whatever White replies, at least one child card survives, and its height is smaller.
The final card here is 757. Like the paper's 727 it needs nothing (A = ∅) and fits in a 17×17 square, but it has height 20. Its pivot is tengen, K10. Of its children, single-stone cards of height 17 (new cards 738 and 749, in all their rotations) answer every diagonal and knight's-move reply, so those games end by move 18. Only a reply directly next to tengen keeps Black busy until move 20.
Black follows its plan to the letter. It only ever plays the next stone of its current card and never looks for other ways to finish, so it can leave a ready win on the board for many moves. A perfect defence exploits this; the improved certificate adds cards that fit the extra stones Black happens to have, which is where the saved moves come from. In glass-box mode, a win Black has already passed over gets a green dashed ring.
The paper's original 21-move strategy is at Snaky in 21.
Card 757 is a hand of placed cards: the paper's height-15 cards plus the new height-17 ones. Step through them and watch their common envelope shrink until only the pivot is left. That is why Black needs nothing to start.
Open any of the 758 cards (728 onward are new). Black stones are required cells, squares are the envelope, the red ring is the pivot. Click a child to follow it down.