Crossways

Rules

Crossways is played on two overlapping dot grids — Red's dots and Blue's dots, each shifted half a step from the other so they interlock.

Red is trying to connect the top edge to the bottom edge with an unbroken chain of red lines. Blue is trying to connect the left edge to the right edge with an unbroken chain of blue lines.

On your turn, draw one line between two of your own neighboring dots (straight up/down/left/right — no diagonals). Every possible line on the board has exactly one line of the other color crossing it. Once either line is drawn, the other is blocked for good — that's the whole game: build your own path while cutting off theirs.

Whoever completes an unbroken path first wins. There's no way for both players to finish, and no way for the game to end in a draw — someone always gets through eventually.

Controls

Dots are named by column letter and row number, counted within your own color's grid, row 1 at the bottom -- "a1" is your own grid's bottom-left dot. Clicking or tapping a highlighted line on the board plays it, same as always.

To play by keyboard instead, type your two dots dash-joined into the move box below the board and press Enter or Play -- "a1-b1" draws the line between them. You never need to say which color: it's always inferred from whose turn it is. Type "status" (or just "s") at any time to have the board's connection progress read back.

Prefer no on-screen keyboard at all -- especially handy on phones, where it eats a lot of the screen? A "Keyboard move-entry" checkbox in Settings hides the move box and its label. That choice is remembered per device, not per game, so you can have it on for a mouse/trackpad and off for touch.

Roots

Crossways is a straight rebuild of Bridg-It, also called Gale's game — invented by mathematician David Gale and sold as a board game by Hasbro starting in 1960. Mathematically it's a matched pair of Shannon Switching Games: Red's grid and Blue's grid are each other's planar dual, which is the technical way of saying every possible line one player could draw crosses exactly one possible line the other could draw, and drawing yours permanently rules theirs out. It landed in this collection while chasing down a half-remembered childhood dot-grid game; whether that childhood version was this exact game, a variant of it, or something invented independently before ever crossing paths with Gale's original is still an open question, pending an old marked-up book turning up to settle it.

Under the hood

Easy plays a purely random legal move every turn — no lookahead, no preference. Medium and Hard both run the same core measurement: for each legal move, simulate playing it, then count the fewest additional turns still needed to finish a connected path, where a line already drawn is free and a line that's still legal to draw costs one future turn. Medium just plays whichever open move shrinks its own count the most. Hard does that too, but also rewards moves that lengthen the opponent's matching countdown — weighing "how much closer does this get me" against "how much does this set them back," where Medium only ever asks the first question.

That's a genuinely useful heuristic, but it's one move of lookahead, not solved play. Bridg-It has a fully known answer: whoever moves first has a provable winning strategy, by a strategy-stealing argument — if the second player could force a win, the first player could just make a throwaway opening move and then steal that exact strategy, which is a contradiction, so no such second-player strategy can exist. An explicit, constructive version of that strategy — not just a proof it exists — was later worked out by Oliver Gross, building on the two-edge-disjoint-spanning-trees idea Alfred Lehman used to characterize this whole family of games. None of Crossways' three difficulty levels play that strategy; they're all still greedy one-move lookahead, so a player who's internalized Gross's actual method has real room to beat even Hard consistently from the first move.

Self-play testing backs up that the edge is real against these heuristics too, not just in theory: on a 5×5 board, whichever side moved first won roughly 60% of games between two AIs of the same difficulty. That's why who goes first rotates automatically every game here instead of always starting Red — there's no way to make a single game of Bridg-It fair, so the fairness has to live across games instead.