I've just thought of an even simpler formulation. Imagine that Alice tells you: "I have three coins, each of them is either heads or tails. You can ask me about

I've just thought of an even simpler formulation. Imagine that Alice tells you: "I have three coins, each of them is either heads or tails. You can ask me about one of them, but then the other two will disappear." Bob, who is far away from Alice, tells you the same thing. They also claim that their sets of coins are "identical", though you're not sure what that means. 1) If you ask Alice and Bob about the same coin, their answers always agree. 2) If you ask Alice about coin 1 and Bob about coin 2, they are different about 5% of the time. 3) If you ask Alice about coin 2 and Bob about coin 3, they are different about 5% of the time. 4) If you ask Alice about coin 1 and Bob about coin 3, they are different about 20% of the time! Why is it shocking you ask? Well, if coin 1 is different from coin 3, then at least one of them must be different from coin 2, which you didn't ask about, but could have. So the probability of the last situation can't be larger than the sum of the previous two. But here it is, almost twice larger. Quantum magic :-)