Bridg-It one edge short of two trees
David Gale's game (1958). Red bridges neighbouring red dots to join left to right, Blue joins top to bottom; bridges may not cross. Click a crossing point to build there. Moves marked ✓ win against any defence, ✗ do not, by Lehman's theorem (1964): the player who just moved has a won game exactly when some part of their graph holding both goals carries two edge-disjoint spanning trees.
who wins from here
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The Shannon switching game
Look at the board from Red's side only. Red's dots are vertices, the left column is one vertex s, the right column another t, and each crossing point is an edge. A red bridge claims that edge (glue its two ends together); a blue bridge across it deletes it. Red wins when s and t meet, Blue when they fall apart. That is Shannon's switching game, and Lehman solved it for every graph: with the opponent to move, you win exactly when some vertex set containing s and t carries two spanning trees that share no edge. When the opponent deletes an edge of one tree, you claim an edge of the other that reconnects it, and both trees survive.
One edge short
On the n × (n+1) board Red's graph has n(n−1)+2 vertices and n² + (n−1)² edges, while two spanning trees need 2(V−1) = 2n² − 2n + 2: the board is exactly one edge short. Every opening gluing two vertices removes one vertex and one edge, and then the edges split into two spanning trees exactly, for all 41 openings on the 5 × 6 board (and for every board up to 14 × 15). So the first player wins, with any first bridge.
The pebble game
The hints do not search. Each vertex starts with two pebbles; an edge is accepted when three pebbles can be gathered on its two ends (moving pebbles back along accepted edges), and then covers one of them. This (2,2) pebble game of Lee and Streinu builds a maximal set of edges no part of which is denser than two trees, and s, t lie in a common two-tree part exactly when three pebbles can no longer be gathered on them. The two trees on the board are drawn by Edmonds' matroid partition.