Earlier today I set you this logic puzzle. Here is is again with the solution.
(If you found it too simple. Here’s a harder version.)
Head sums
Ade, Binky, and Carl are honest perfect logicians. A hat is placed on each of their heads. Each hat has a whole number larger than zero written on it, which the other two logicians can see but which the wearer cannot. One of these numbers is the sum of the other two (so for example, the three numbers could be 3, 7, 4, or another possibility is 6, 6, 12). All of this is public knowledge.
Ade sees that Binky has a 3 and Carl has a 1.
Ade says for all to hear: “ I do not know the number on my hat”
Binky then announces: “I do not know the number on my hat”
Ade then announces: “I know the number on my hat!”
What number is on Ade’s hat?
Solution 4
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.”)
B says “ I don’t know the number on my hat”. This statement tells everyone that A ≠ C.
However, this statement also allows A to deduce that A ≠ 2.
Let’s assume that A = 2. B therefore sees C = 1 and A= 2, so B knows that either B = 3 (the sum) or B = 1 (the difference). But B has learned that B ≠ C. And since C = 1, B cannot be 1. So if A = 2, B would have deduced that she was 3 and she would not have said “I don’t know the number on my hat.”
So A = 4. It’s the number on Ade’s hat.
The puzzle was devised by Timothy Chow, inspired by a puzzle by Dick Hess. Chow had originally formulated a harder question, which can be read here on Puzzling Stack Exchange.
I’ve been setting a puzzle here on alternate Mondays since 2015. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
