Dodgem Colin Vout's car game
Blue drives right, up or down and leaves over the right edge; Red drives up, left or right and leaves over the top. Get all your cars off to win — but a move that leaves your opponent no legal move loses. Blue moves first. On 3×3 the first player wins; on 4×4 and 5×5 perfect play never ends. The computer has solved every position; with hints on, each move is marked win, draw or loss with the plies left.
solved boards
1996 claims, re-checked
How the computer knows
A position is where every car stands plus whose turn it is. Cars never come back, so the number of cars on the board only goes down: positions split into layers by (Blue cars, Red cars), and each layer only moves into itself or into a layer with one car fewer. Solving the layers from the emptiest up, each one is a small retrograde analysis: start from what is already known, walk moves backwards, mark a position won as soon as one move reaches a lost position and lost once every move reaches a won one. Whatever is never marked is a draw — both sides can shuffle forever and neither can force a finish. The 5×5 board has 240 million positions (counting the side to move) and takes a few minutes in Node; the page solves 3×3 and 4×4 on the spot.