## A quantum gameEdit

Here are the rules:^{[1]}

- 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 **X _{A}**,

**X**,

_{B}**X**and

_{C}**Y**,

_{A}**Y**,

_{B}**Y**. The winning combinations satisfy the following equations:

_{C}

Consider the first three equations. The product of their right-hand sides equals +1. The product of their left-hand sides equals **X _{A}X_{B}X_{C}**, implying that

**X**. (Remember that the possible values are ±1.) But if

_{A}X_{B}X_{C}= 1**X**, then the fourth equation

_{A}X_{B}X_{C}= 1**X**obviously cannot be satisfied.

_{A}X_{B}X_{C}= −1- The bottom line: There is no failsafe strategy with pre-agreed answers.

- ↑ Lev Vaidman, "Variations on the theme of the Greenberger-Horne-Zeilinger proof,"
*Foundations of Physics*29, pp. 615-30, 1999.