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Introduction
Quantum tic tac toe is a strategy game for two players. It is an extension of classical tic tac toe, with a new feature borrowed from quantum mechanics. The goal is the same &mdash; get three marks in a row &mdash; but play is rather different, as can be seen in the brief sequence to the right.

The first difference is that X places marks in two squares on his turn. The second difference is seen when O places two marks also, and one of them is in teh same square that X already marked. X and O have, in fact, placed one mark each in the grid, but their marks are in a superposition of locations: X's mark may be in square 1 or square 2, but at the moment it is not really in either square. Likewise O's mark may be in square 2 or in square 5, but is not yet really in either square; this is why she can get away with sharing square 2 with X's mark. These are called spooky marks

Rules
Quantum tic tac toe is played very much like classical tic tac toe: two players (called X and O) take turns placing marks in a 3&times;3 grid, and the winner is the first player to place three marks in a row. However, in quantum tic tac toe each player's mark is placed in two different squares; the mark is in both squares, in a sense, but is not really in either square yet. In the language of quantum mechanics, the mark is in a superposition of locations. A pair of "spooky marks" are used to indicate this superposition.

Spooky marks may not be placed in squares containing real marks, but they may be placed in squares that contain other spooky marks; there is, after all, nothing really there yet. Moves that share a square are entangled: several such moves can link together like a chain, and anything that affects the state of one such move, affects them all.

A chain can become a closed loop, called a cyclic entanglement. When an entanglement becomes cyclic, it immediately must collapse to a group of real marks. There are always two ways that a cycle can collapse, and the results can be quite different. To keep the game competitive, the player who creates a cyclic entanglement does not get to collapse it. His opponent chooses how it collapses, and then proceeds to make his own next move. The animation to the right show the two possible collapses of the game opening shown above. In the top image, move 1 settles into square 1, forcing move 3 out; it settles into square 3, which in turn forces move 2 into square 2. In the bottom image, move 3 is in square 1, forcing move 1 into square 2 and move 2 into square 3.

Strategy
Winning in quantum tic tac toe involves pursuing four strategic goals (and preventing one's opponent from achieving any of them):


 * 1) Complete three real marks in a row, to win.
 * 2) Threaten a win by placing two marks in two different rows, so that only one threat can be blocked.
 * 3) Threaten a win by placing two marks in a row.
 * 4) In case of multiple completions (or the threat thereof), make sure that one's own is earlier.

The first three goals are the same as in classical tic tac toe, and are often pursued similarly even though no real marks may be yet present on the grid. For instance, in classical tic tac toe, if X marked squares 1 and 7, O would have to mark square 4 (or X would do so next, and win). In quantum tic tac toe, X could mark squares 1 and 4, and then 4 and 7, creating an entanglement that covered all three squares. O must respond, but she may place her marks in any two of the three squares; her marks will entangle with X&#146;s marks and create a cycle. Even though X gets to collapse this, the result will always be two real X marks and one real O mark, and so X is blocked from winning.

The fourth goal is the result of something that cannot happen in classical tic tac toe. Since real marks may not appear on the board until late in the game, it is possible that both players will complete three in a row. In such case, the players look at the completed rows to see which row was completed earlier. If X&#146;s completed row consists of the marks from moves 1, 3 and 7, while O&#146;s completed row consists of marks from moves 2, 4 and 6, then O is regarded as having completed her row earlier (one move prior to X). O thus wins the game.

Classical Ensemble
Quantum tic tac toe (and quantum mechanics) represents a very different way of looking at the world. Ordinary experience dictates that a thing is always in a definite place, even if one does not know the place. The mind balks at the idea that a thing can be in several places, or that its location is truly undetermined (not merely unknown). One wants some way of thinking about the situation that makes sense. Quantum tic tac toe provides such a way: the quantum game can be thought of as an ensemble of classical tic tac toe games, all being played simultaneously. The games are not independent of each other, however, because each game represents one possible state of the quantum game.

To illustrate this, consider X&#146;s first move, in the example game above; this can be viewed as an ensemble of two classical games where X marked square 1 in one game and square 2 in the other. When O makes her move in squares 2 and 3, the ensemble is duplicated; her mark goes in square 2 in half of the games and in square 3 in the other half. The ensemble now represents all states of the quantum game &mdash; including an impossible state with two marks in square 2. Such impossible states are "pruned" from the ensemble and are not part of the further development of the game.

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