Line Three: rotate the whole position, not just the winning square.
Transfer a winning move through rotations and reflections, and learn when similar-looking squares are no longer interchangeable.
You recognize a winning pattern, but it faces a different direction from the diagram you remember. Should you still choose the top-right corner? Only if the entire position matches. A useful move belongs to the marks around it, not to a fixed address on the screen.
This paper exercise shows how to transform a complete Line Three board and its answer together. It uses the ordinary rules here: X starts, turns alternate, and the first completed row, column, or diagonal ends the game. It does not rotate an ongoing live game or load these diagrams into the computer opponent.
Start with a position whose answer you can check
Rows R1–R3 run from top to bottom. Columns C1–C3 run from left to right. A dot is empty. In Board A, X and O each have two marks, neither has won, and X is next.
X X · O · · · · O
One legal history is X at R1C1, O at R2C1, X at R1C2, then O at R3C3. X can now win by taking R1C3. That completes the top row. No other empty square gives X an immediate win on this turn.
Write the board and the answer on paper. You will move every X, every O, and the empty answer square using the same transformation. Do not exchange the players or give either side another turn.
A clockwise quarter-turn moves rows into columns
Imagine turning the paper 90 degrees clockwise, then naming rows and columns from the viewer's new top-left corner. The old top-left becomes the new top-right. The old top-right becomes the new bottom-right. The center stays in the center.
Apply that turn to all four occupied squares of A. The complete rotated position is Board B:
· O X · · X O · ·
The answer R1C3 has moved to R3C3. Placing X there completes the right column, just as the original move completed the top row. The original screen address R1C3 already contains X in B, so copying only the old address would not even be a legal move.
A coordinate recipe makes the turn reproducible. An old square at row r, column c moves to row c, column 4 − r. For example, R2C1 becomes R1C2. Use row and column numbers 1, 2, and 3; this formula is not written for zero-based array indices.
Rotate again, then compare a reflection
A half-turn is two clockwise quarter-turns. It moves an old square to row 4 − r, column 4 − c. Board A becomes C:
O · · · · O · X X
The winning square is now R3C1, completing the bottom row. You can obtain the same result by turning B once more. Checking the two-step route is a useful way to catch a reversed rotation.
A left-to-right reflection uses a different rule: the row stays r and the column becomes 4 − c. Reflecting A produces D:
· X X · · O O · ·
Now X wins at R1C1. A reflection is not a quarter-turn: compare where the O from A's R2C1 went. It is at R2C3 in D, but R1C2 in B. Following an opponent mark as well as your own marks helps distinguish the transformations.
Why the reasoning survives the transformation
Line Three has eight winning lines: three rows, three columns, and two diagonals. Each rotation or reflection carries every complete winning line to another complete winning line. It also preserves empty squares, each player's mark count, and whose turn it is.
Consequently, a legal continuation can be transformed move by move. A winning move still completes a line; an occupied square does not suddenly become available. Applying the inverse transformation returns the same continuation to the original board. This is why the result transfers, rather than merely looking plausible in one picture.
There are eight board symmetries: the unchanged board, three rotations, and four reflections. Different transformations can produce the same diagram when that position is symmetric. Eight transformations do not mean every position has eight different appearances.
Use symmetry to reduce work, without grouping too much
On an empty board, all four corners are interchangeable under rotations and reflections. The four edge-middle squares form another group, and the center is a group by itself. Nine possible opening squares therefore fall into three positional types. This is a way to organize analysis, not a claim that the types have identical threats or that a particular opening guarantees victory.
Once marks are present, use only transformations that preserve the entire current position if you want to treat two candidate moves as interchangeable within that position. A transformation that changes the board instead explains a corresponding move on a different, transformed board.
Board A shows the danger. R1C3 and R3C1 are both empty corners. The first wins immediately; the second does not. No rotation or reflection that leaves all of A's X and O marks in place exchanges those two squares. “Both are corners” is no longer enough.
Try three checks before revealing the answers
Turn A 270 degrees clockwise. Where is X's immediate win?
· · O X · · X O ·
The answer is R1C1, completing the left column. The original R1C3 answer has traveled through R3C3, then R3C1, then R1C1 over three quarter-turns. Four quarter-turns would restore A and R1C3.
In B, are the empty corners R1C1 and R3C3 interchangeable?
No. X at R3C3 completes the right column. X at R1C1 completes no line. The current O and X placements matter, even though both candidate squares belonged to the corner group on an empty board.
Can you swap any two rows and transfer the same argument?
No. An arbitrary row swap need not preserve diagonal wins. For example, the diagonal squares R1C1, R2C2, R3C3 become R2C1, R1C2, R3C3 when rows 1 and 2 are exchanged. Those three squares are not a winning line. Use an actual board symmetry, not any rearrangement that happens to keep some rows together.
A short paper routine
- Check the original mark counts, turn, and whether play has already ended.
- Choose one named rotation or reflection.
- Transform every occupied square and the candidate answer square.
- Read all eight lines again on the resulting diagram.
- Undo the transformation and confirm that every mark and the answer return to their starting addresses.
The examples and all eight line-preserving transformations were checked independently under the site's rules. The exercise helps reuse a verified line of reasoning across orientations. It does not reconstruct the history actually played, rank every strategic option, or promise that the live computer will allow these teaching positions. Other games can have different symmetry rules: rotating a numbered sliding puzzle without rotating its fixed goal, for example, changes the problem.