Quantum TicTacToe is played on the same board as Classical
TicTacToe. It consists of a 3x3 square array of 9 squares labeled 1
through 9, left to right, and top to bottom. The game starts with an
empty board.
There are two players, X and O. They take turns indicating their
moves with X going first. As in Classical TicTacToe, player X marks
his moves with X's and player O marks her moves with O's. However, in
Quantum TicTacToe, each player must place marks in two different
squares. The marks are subscripted with the number of the current
move, so all of X's moves are subscripted with odd numbers, and all of
O's moves are subscripted with even numbers. The moves in Quantum
TicTacToe are "mixed state" moves, half in one square, half in
another. One only finds out later in the game which square each move
was "actually" in.
Mixed state moves can share squares, becoming "entangled". There is
no limit on the number of mixed state moves that can be in a square
(well, nine). However, at some point, these entanglements will always
become circular, and then a new type of move is required  the
collapse. Collapses are how the quantum moves are converted to
classical moves. Half of each pair of the involved mixed state moves
are eliminated. The one that is left is the classical move, there can
be only one per square, and further quantum moves into such collapsed
squares are forbidden.
It really helps to have an example in an attempt to explain this core
feature of Quantum TicTacToe. Consider the following four moves in
a game; X1 in squares 1&2, O2 in squares 2&5, X3 in squares 5&9, and
O4 in squares 5&1. X1 depends on O2 which depends on O4 which depends
back onto X1. This selfreference is another way to look at the
cyclic entanglement. It is now required to collapse the quantum
moves, the mixed state moves, to classical moves, since if more
quantum moves were allowed in just these four squares, one would have
five moves trying to fit into only four squares. There wouldn't be
room for all of them. Since O made the move that caused the cyclic
entanglement, X gets to choose how it settles out; he gets to choose
the collapse.
There are three moves (X1, O2, O4), and three squares (1, 2, 3),
involved in the cycle. The fourth move and square (X3 and square 9)
are entangled with the cycle, but not actually a part of it. No
matter how the cycle is collapsed, X3 must end up in square 9. It is
called a stem. The players have no choices in how stems collapse.
They do have lots of choices of how to go about specifying the
collapse, but in the end, all collapsing entanglements have only two
possibilities. To specify a particular collapse, a player merely
selects one subscripted mark from among those mixed state moves
involved in the cyclic entanglement to be the classical move in that
square. This forces all the other entangled moves to settle out to
classical states. Once the collapse has been indicated, X gets to
make his next regular mixed state move, X5. Note, that mixed state
moves cannot be played in squares that have collapsed to classical
moves. Also, mixed state moves cannot be played in the same square
(selfcollapse), as this would allow Quantum TicTacToe to degenerate
directly to Classical TicTacToe. The exception to this rule is if
on the last move, there is only one uncollapsed square left on the
board, then X may play both halves of his mixed state move, X9, into
the single remaining square.
Because classical moves only occur from collapses, the game cannot end
until at least one collapse occurs. A 3row only counts if it
consists entirely of classical moves. 3rows of quantum moves, (mixed
state moves) don't count. Since multiple squares are involved with
each collapse, it is possible for both players to get 3rows from a
single collapse. This could be regarded as a tie, something that
can't happen in Classical TicTacToe, since the only tie in that game
is no 3rows for either players, the "cat's" game. However, since all
moves are subscripted, it is reasonable to consider that one 3row
occurred "earlier in classical time" than the other. The earlier one
gets 1 point, the later one gets ½ point. The earlier 3row is the
one with the smaller largest subscript. If X's 3row consisted of
(X1, X3, X7), and O's 3row consisted of (O2, O4, O6), then O's
largest subscript, 6, is less than X's largest subscript 7, and so O
gets 1 point, and X gets ½. X gets partial credit for his tactical
success in preventing O from getting a classical 3row until he could
get one also as a part of the final collapse of the game.
As a curiosity, X can actually get two 3rows,
that is, if O is an exceptionally poor player. Depending on the
structure of the collapse, these can be either simultaneous, giving
X a stunning 2 point victory, or sequential, giving him 1½
points.
