The caret (^) symbol typically means XOR (exclusive OR) in programming and computer science. XOR is associative and commutative: 1 XOR 2 XOR 3 XOR 4. In binary: 01 XOR 10 = 11, 11 XOR 11 = 00, 00 XOR 100 = 100. But XOR has a special property: a XOR a = 0, and 0 XOR b = b. So pairs cancel out: 1 XOR 2 XOR 1 XOR 2 = 0.