"When a logician looks at two hats, they need to work out if their hat is the sum of the two visible hats, or the difference between the two vsible hats. The first insight is that if a logician can see two identical numbers, they know that their hat is the sum. Their hat cannot be the difference, since the difference between two identical numbers is 0, which is not possible by the statement of the question."
"A sees that B has 3 and C has 1. A deduces that he must have either 4 (the sum), or 2 (the difference). A doesn't yet know which, which is why A says I don't know the number on my hat. However, A's statement gives new public knowledge, namely that B = C. (If B = C, then A would know the number on his hat is the sum, and A would not have said I don't know the number on my hat.)"
Ade sees Binky=3 and Carl=1, so Ade's hat could be 4 (sum) or 2 (difference). Ade announces he does not know his number, which rules out the case of two equal visible numbers but does not resolve Ade's own choice. Binky then considers that if Ade were 2, Binky would see 2 and 1 and her own possibilities would be 3 or 1. Ade's earlier announcement implies Binky cannot equal Carl, so Binky would eliminate 1 and deduce 3. Binky's actual statement 'I do not know' contradicts that deduction, so Ade cannot be 2; Ade must be 4.
Read at www.theguardian.com
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