This Quantum World/Game

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A quantum game
Here are the rules:
 * Two teams play against each other: Andy, Bob, and Charles (the "players") versus the "interrogators".
 * Each player is asked either "What is the value of X?" or "What is the value of Y?"
 * Only two answers are allowed: +1 or −1.
 * Either each player is asked the X question, or one player is asked the X question and the two other players are asked the Y question.
 * The players win if the product of their answers is −1 in case only X questions are asked, and if the product of their answers is +1 in case Y questions are asked. Otherwise they lose.
 * The players are not allowed to communicate with each other once the questions are asked. Before that, they are permitted to work out a strategy.

Is there a failsafe strategy? Can they make sure that they will win? Stop to ponder the question.

Let us try pre-agreed answers, which we will call XA, XB, XC and YA, YB, YC. The winning combinations satisfy the following equations:


 * $$

X_AY_BY_C=1,\quad Y_AX_BY_C=1,\quad Y_AY_BX_C=1,\quad X_AX_BX_C=-1.$$

Consider the first three equations. The product of their right-hand sides equals +1. The product of their left-hand sides equals XAX<SUB>B</SUB>X<SUB>C</SUB>, implying that X<SUB>A</SUB>X<SUB>B</SUB>X<SUB>C</SUB> = 1. (Remember that the possible values are ±1.) But if X<SUB>A</SUB>X<SUB>B</SUB>X<SUB>C</SUB> = 1, then the fourth equation X<SUB>A</SUB>X<SUB>B</SUB>X<SUB>C</SUB> = −1 obviously cannot be satisfied.
 * The bottom line: There is no failsafe strategy with pre-agreed answers.

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