Tic Tac Toe

• The board is a 3 × 3 grid of nine empty squares.
• X always goes first, then the players alternate. Nobody may pass a turn.
• A turn writes one mark on one empty square. Marks are never moved or erased.
• Three of your own marks in a straight line — a row, a column, or either diagonal — wins immediately.
• If all nine squares are filled with no line, the game is a draw.

There are exactly eight ways to make a line: three rows, three columns and two diagonals. That number is the whole geometry of the game, and it is why the centre square matters — the centre sits on four of those eight lines, a corner on three, and an edge square on only two.

Because the game is small enough to solve completely, and the solution is a draw. Counting every distinct sequence of moves, there are 255,168 possible games, played out across 5,478 reachable positions and ending on one of 958 distinct final boards — 138 of them once rotations and reflections are folded together. A computer can search all of it in a few milliseconds, which is why the Perfect level cannot be beaten on this board: it is not guessing well, it is reading the end of every line. On the two bigger boards below the same level searches deep rather than to the end, and there it can be beaten.

Those are counts of move sequences, not of likely outcomes — they weight a game in which O throws away the centre exactly as heavily as a game between two careful players. They do show the shape of the advantage, though: moving first is worth a great deal when your opponent is careless, and worth nothing at all when they are not.

Against a perfect opponent, none of them is better: all nine openings draw. Against a fallible one the centre and the corners are far more dangerous than an edge, because of how narrow they leave the reply. The table below is measured from the search on this page, not from folklore — it is the number of O's replies that still hold the draw.

So the corner is the sharpest opening in practice: it leaves your opponent exactly one saving square out of eight, and taking any of the other seven hands you a forced win. The centre looks strongest and is the safest reply, but as an opening it leaves four ways out rather than one.

With a fork: one mark that creates two winning squares at the same time. Your opponent has one turn and can block only one of them, so the other wins. Every win in this game against anyone who is paying attention is a fork, and the shortest one runs like this — open in a corner; if O answers on an edge, take the centre. You now hold two of the three squares on the long diagonal, so O must block the far corner, and your third mark on a free corner makes two lines-in-waiting at once.

The two corner replies need one adjustment each, and knowing which is the difference between a line you can run and a line you half-remember. If O answers on a corner beside yours, the centre still forces the same block — but the square that forks is then an edge, not a corner. If O answers in the corner opposite yours, the centre threatens nothing at all, because O is already sitting on that diagonal; take one of the two free corners instead, which threatens the side you share, and the fork arrives one mark later.

The defence is the mirror image. Answer every immediate threat, but before you do, check whether the square you were about to take lets your opponent fork next turn. That look-one-move-further is exactly the difference between the Sharp level here and the Perfect one — see how to play tic tac toe for the full worked line.

The 3 × 3 game is thin on its own once both players know it, so this page also offers two bigger boards where four in a row wins. They are genuinely different games rather than the same one stretched, because the number of ways to make a line changes completely.

The 4 × 4 board is tighter than it looks: sixteen squares but only ten lines, because a four-in-a-row there has to fill a whole row, a whole column or a full diagonal. The 5 × 5 board is the opposite — a run of four can start anywhere along a line, which gives 28 of them and makes double threats routine. Neither board is small enough to solve outright, so on those two the Perfect level searches as deeply as a browser tab allows and plays the last squares exactly. That is strong, but it is not a proof, and it can be beaten.