Positions and Games on 2x2
Number of positions
On a 2x2 board, there are 3^4 = 81 possible positions:
Of these, there are 16 that have 4 stones and are illegal, and another 8 that have three stones and are illegal, which leaves 57 that are legal. The illegal positions have been crossed out.
This gives an overal percentage of 70% for valid positions, on bigger boards this percentage is much lower.
Note: The first row or column gives all the positions for the 1x2 board.
Number of games
John Tromp had calculated the number of possible games on a 2x2 board (under the Tromp-Taylor rules of go) as being 386,356,909,593