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Running it in Reverse
OK, I confess.
I got stubborn, and wouldn't publish the next challenge
until I found a solution to
December's.
Finally Allan took pity on me and showed me how to approach it.
It's actually not that hard ... once you've bent your mind a certain way.
So here's what you do.
Start with an ordinary QT3 game where the first two moves are not entangled
(let's say X moves in squares 1 and 2, O in squares 4 and 5,
as shown to the right).
This situation has a classical ensemble consisting of four real games.
Now, if the two moves are entangled in one square
(X in 1 and 2, O in 2 and 5),
the classical ensemble consists of three games.
And if both moves occupy squares 1 and 2,
the entanglement collapses and the
classical ensemble consists of just one game.
We missed a step somewhere. We went from four real games, to three, to one.
Where's the ensemble of two real games?
That turns out to be the key.
If X and O both put their spooky marks in squares 1 and 2, then you do get an ensemble
of two real games; but immediately X has to choose which one of them
becomes real (the other one vanishes).
Now, let's take that cyclic entanglement and untangle it so it doesn't have to collapse.
In each of the real games, take the spooky O mark and move it down a row (from square 1
to square 4, or from square 2 to square 5); leave the spooky X mark where it is.
We still have two real games in the ensemble:
one has an X in square 1 and an O in square 5,
the other an X in square 2 and an O in square 4.
This doesn't have to collapse, because we have two moves in four squares.
But ... there is no way to set up a quantum game that has this ensemble!
And that is what we were looking for: a classical ensemble that doesn't
have a quantum game to go with it.
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