Gravity four: what was worked out

Gravity four is a preset of the game Lines: a 7 × 6 board, a stone falls to the lowest free cell of its column, four in a row win. It has 42 cells, so the game's solver solved it exactly: with perfect play the first player wins by starting in the middle.^[1] The default budget of 300 million nodes is not enough (“not solved”); a question mark in the game's documentation is therefore a budget, not a result.^[2]

What holds with perfect play

• A first move in the middle column 4 wins, in column 3 or 5 it leads to a draw, and in column 1, 2, 6 or 7 to a loss.^[3]
• After the middle every answer of the second player loses; after column 3 the second player loses with an answer 1 or 2 and draws with an answer 3 to 6, and the other answers were not solved.
• On smaller boards with gravity and a line of 4 the second player wins on 6 × 4 and 6 × 6, elsewhere from 4 × 4 to 7 × 5 it is a draw; from 7 × 6 the first player wins.

Threats and the fork

A threat is an empty cell where a stone of the player would complete four in a row; only the lowest free cell of a column can be played at once, a higher threat is a future one: it cannot be blocked, and whoever plays under it gives the opponent the win.

• Two threats in the same column right above each other cannot be stopped: if the opponent had no threat, the first player with them won 39 of 39 sample positions, the second player 21 of 27.^[4]
• A fork is a move after which the opponent cannot win at once and after every answer the player has a win in one move.
• An explanation, not a measurement: when the columns are filled one stone at a time, the first player gets the odd rows and the second player the even ones. An odd threat of the first player without an even threat of the opponent won 94% of the sample positions, an even threat of the second player without an odd one of the first 61%; it is a tendency, not a law.
• A trap is a position in which exactly one move keeps the result. A block is always such a move; a quiet trap (the single move is not a block) is in 18 to 30% of the not lost positions with no threat to play at once.

Against the AI

All the numbers come from games of AI opponents; people were not measured.^[5]

• A worsened move is a move from a position that was not lost, after which the result with perfect play is worse. The Weak AI almost always turns a win into a loss, the Normal AI too, but from the 21st move on it errs in only 0.4% of its moves; the Strong AI turns a win into a draw and a draw into a loss.
• The Strong AI against the Strong AI draws 73% of the games, although the first player wins with perfect play: no level plays perfectly.