In fact, this is the best winning strategy classically. We can only satisfy a maximum of 3 out of 4 winning conditions. Let a 0 , a 1 be Alice's response to question 0 and 1 respectively, b 0 , b 1 be Bob's response to question 0, 1, and c 0 , c 1 be Carol's response to question 0, 1. We can write all constraints that satisfy winning conditions as a 0 + b 0 + c 0 = 0 mod 2 a 1 + b 1 + c 0 = 1 mod 2 a 1 + b 0 + c 1 = 1 mod 2 a 0 + b 1 + c 1 = 1 mod 2