Line Three: a possible win is not a forced win.
Read a complete five-route game tree, distinguish a draw you can secure from a win that needs a mistake, and work backward to the first choice.
A diagram can contain a route to victory without giving you a way to force that victory. The difference is who chooses the replies. If your plan needs the other player to cooperate, finding one successful continuation is not enough.
We will finish every legal continuation of one small Line Three position. Only three squares are empty, so the complete tree fits in a short ledger. The lesson is to read those results backward and distinguish what can happen from what a player can secure.
Fix the position and whose turn it is
Use the ordinary rules here: X starts, turns alternate, and the first completed row, column, or diagonal ends play. A full board without a line is a draw. Rows count from the top; columns count from the left. Dots are empty.
· · · X X O O X O
Each side has three marks and neither has won, so X moves next. One legal history is X R2C1, O R2C3, X R2C2, O R3C1, X R3C2, O R3C3. No earlier position in that history contains a completed line.
X has three choices: R1C1, R1C2, or R1C3. Before reading further, draw three branches on paper. For each branch, ask whether play ends immediately. If it does, stop that branch; the opponent does not get another move.
The middle choice ends the game now
X at R1C2 completes the middle column: R1C2, R2C2, R3C2 are all X. This is an immediate X win. There is no O reply, even though two squares are still empty.
Do not fill those empty squares to make a longer tree. The game's stopping rule matters just as much as alternating turns. A continuation after this win would be a different, illegal history.
The left choice lets O choose between two results
If X instead chooses R1C1, O can take either R1C2 or R1C3. Those replies have different consequences:
- O at R1C3: O completes the right column through R2C3 and R3C3. O wins immediately; the remaining empty square stays empty.
- O at R1C2: no one has a line. X fills the last square, R1C3. The board is full and every line is blocked, giving a draw.
The draw is a possible result of X's left choice. It is not a draw X can secure against every reply: O can take the right square and win. An opponent trying to win has no obligation to select the drawing branch.
The right choice preserves a draw but does not force a win
Now restore the original board on paper and try X at R1C3. That blocks O's right-column threat. The resulting diagram is:
· · X X X O O X O
O again has two choices, now R1C1 or R1C2:
- O at R1C1: X fills R1C2 and completes the middle column. X wins.
- O at R1C2: X fills R1C1. The full board is a draw.
The drawing finish is:
X O X X X O O X O
X can avoid losing after the right choice: both possible O replies lead to either an X win or a draw. But X cannot force a win, because O can select the drawing reply. “A win is possible” and “a win is unavoidable with correct play” are different statements.
Check the complete terminal ledger
The table lists every complete legal route from the starting board. A dash means that the game has already ended, so that player takes no further turn. Restore the original diagram for each row.
| First X | O reply | Last X | Result |
|---|---|---|---|
| R1C1 | R1C2 | R1C3 | Draw |
| R1C1 | R1C3 | — | O wins |
| R1C2 | — | — | X wins |
| R1C3 | R1C1 | R1C2 | X wins |
| R1C3 | R1C2 | R1C1 | Draw |
The two drawing rows reach the same final diagram through different move orders. They are separate routes, not two distinct finished boards. The ledger has five terminal routes: two X wins, two draws, and one O win.
Read the tree backward
At each finished branch, label the result from X's point of view: win, draw, or loss. Then move one decision toward the root. At an O choice, assume O selects the least favorable result for X. At an X choice, X selects the most favorable result still available.
- After X at R1C1, O can choose a draw or an O win. The backed-up result for X is a loss.
- After X at R1C2, play has ended with an X win. No further choice needs evaluating.
- After X at R1C3, O can choose an X win or a draw. The backed-up result is a draw.
At the original X decision, compare loss, win, draw. X should take the immediate win at R1C2. Blocking at R1C3 is safe against losing, but it passes up a victory already available.
This backward comparison is the basic minimax idea: each side chooses for itself, rather than letting the analyst select a convenient reply. A backed-up draw means that X can guarantee at least a draw here and O can prevent an X win. It does not mean that every continuation ends drawn.
Do not turn branch counts into a win probability
Two X-winning rows out of five do not establish a 40% chance of winning. Each row describes a possible sequence of choices, and no rule says those sequences are equally likely. Players choose moves at different points in the tree, and some routes end earlier than others.
To calculate an actual probability, you would need a specified rule for how each player chooses at every decision. Counting leaves is not that rule. The comparison above instead asks what each side can secure against a responding opponent.
Three paper checks
After X blocks at R1C3, which O move prevents an X win?
O must choose R1C2. X then has only R1C1, producing the printed draw. O at R1C1 allows X to complete the middle column at R1C2. One drawing reply is enough to refute a claim of a forced X win.
Why is R1C1 a losing X choice even though one branch draws?
Because O controls the reply and can win immediately at R1C3. The drawing branch requires O to pass up that win. It proves that a draw is possible, not that X can ensure it.
If both players play correctly after X chooses R1C3, must the original position have been a draw?
No. The original position contains the immediate X win at R1C2. Choosing R1C3 changes what can be secured afterward. Evaluate the position before a move separately from the position after that move.
These are paper positions, not diagrams that can be loaded into the live opponent. A permitted history need not be a history the computer would choose. All five routes and their backed-up results were checked under this site's rules. Use the same routine on a short continuation of your own: check the turn, stop at a terminal result, include every legal reply, then work backward without assigning the opponent your preferred choice.