Ultimate Quantum Tic-Tac-Toe
Ultimate Quantum Tic-Tac-Toe is a quantum variant of the classic game of Tic-Tac-Toe which combines the superpositional moves of Quantum Tic-Tac-Toe with the multi-board structure of Ultimate Tic-Tac-Toe. See this paper for details.
Rules of (classical) Ultimate Tic-Tac-Toe
- This is played on a 3×3 grid of 3×3 mini-boards.
- X plays odd-numbered moves, O plays even-numbered moves.
- X begins the game by placing an X in any square on any mini-board.
- Players then take turns placing their symbol on the mini-board corresponding to the position of the previous move, within its mini-board.
- For example, if X plays in the top-right square of a mini-board, then O must play in the top-right mini-board on the next turn.
- If that mini-board is already won or full, the next player may play in any mini-board.
- A mini-board is won by three symbols in a line, and the game is won by three won mini-boards in a line.
Rules of Quantum Tic-Tac-Toe
- This is played on a 3×3 grid of squares, all of which are initially open.
- A square is open if it has no classical symbol on it (even if it is full of spooky symbols).
- X plays odd-numbered moves, O plays even-numbered moves.
- On each move, the current player marks two distinct open squares on the board with their symbol (X or O), subscripted with the move number, and connects them with a line, entangling them.
- These marks are spooky (not classical), so these squares remain open to play.
- Players take turns making moves until a cycle (of three or more squares) is formed, at which point a collapse is triggered.
- The player who did not cause the collapse chooses which of the two possible classical outcomes is realised for the cycle (and stems hanging off of it).
- A square with a classical symbol becomes closed to further moves.
- Three classical symbols in a line wins the game.
- If both players have three symbols in a line, it is a shared win. A full board with no win is a draw.
Rules of Quantum Tic-Tac-Toe Squared
- This is played on a 3×3 grid of 3×3 mini-boards.
- Players may entangle squares within the same mini-board or across mini-boards, subject to the rule that there is at most one entanglement per pair of mini-boards.
- Cycles and cascades work as above, but may span mini-boards.
- Open squares on won mini-boards remain open. In particular, a mini-board won by one player may be turned into a shared win later.
- Three won mini-boards in a line wins the game. A shared win on a mini-board counts for both players.
Rules of Ultimate Quantum Tic-Tac-Toe
- Same as Squared, with one additional rule: the endpoint positions of the pair just played specify the pair of mini-boards (or single mini-board, when the endpoint positions are the same) for the next player.
- If there are no possible moves under the constraint, the next player can play freely.
- Note: as in Squared, open squares on won mini-boards remain open, which is different from (classical) Ultimate Tic-Tac-Toe.