# Can you solve this numbers-on-foreheads puzzle?

Originally published at: Can you solve this numbers-on-foreheads puzzle? - Boing Boing

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could be wrong (because of a linguistic uncertainty) but i think it’s something akin to: if Alice’s number is 5 then due to addition/multiple uncertainty Bob doesn’t know if his is 45 or 10 and if it’s 10 then Alice doesn’t know if her’s is 40 or 5

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I like that better than mine - (Bob could have solved mine with logic - I said it had to be 25 because it needed to be the same number on both heads, for us to be able to say which was on Alice’s, but bob would have known his number as soon as Alice said she didn’t know hers.
EDIT: Nope. Alice spoke first. We don’t know Bob’s number, but we can deduce Alice’s.
)

But doesn’t it work just as well if Bob and Alice’s numbers are reversed? In which case, we, like them, have no way of choosing between two possible numbers.

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I quickly narrowed it down to 2, 5, 10, or 25, but I’m stuck there. I don’t see how to figure out which of those it is

eta: nevermind I got it; it’s 25
There are five possible numbers on Bob’s head from which Alice couldn’t deduce her own (1,2,5,10, and 25).
When Alice says she doesn’t know her number, Bob knows he has one of those five possibilities.
There is only one value that Alice could have that Bob can’t determine which of the 5 he has because 25 could sum to 50 (with another 25) or multiply to 50 (with 2).
When Bob says he doesn’t know his number, Alice immediately deduces she has 25.

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It has to be a factor of 50 because if it was a non factor, say 9, it would immediately be obvious that the other number would be 50 - 9 =41

so 1 x 50, 2 x 25, 5 x 10

It can’t be 50 because it would be obvious the addition condition didn’t apply and the other factor would be 1.

I’m taking a leap here, but it can’t be 25 because (assuming they can’t both have the same number) it would be obvious the other factor would have to be 2

So this leaves 5 and 10

5 has a possible partner as 45 or 10 and 10 can have 40 or 5 - every other possibility is already rulled out

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But the question is which did ALICE have?

EDIT: And why assume they can’t have the same number?

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That assumption isn’t stated in the puzzle. Alice’s number is indeed 25. Bob’s number is either 2 or 25, but he (and we!) can’t get enough info from Alice’s info to determine which of those two numbers he has.

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Dunno. Alice is an infinitely intelligent logician and she was unable to figure that either!

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After Bob’s answer, she did know!

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Does that rule out 5 and 10?

Edit for clarity - does that same logic not apply to 5 and 10?

No. If Bob had a 10, Alice wouldn’t know whether her number is 5 or 40. So Bob sees either a 40 and immediately knows he has a 10, or he sees a 5 and knows he must have either a 10 or a 45. But if he’d had a 45, he knows Alice would have known her own number, so he’d still correctly deduce that he had a 10.

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Yes, but the question here is important: what is Alice’s number? The person posing the quiz thinks we must know the answer, and that neither Alice or Bob would know it.

This can be true only if both have the same number, and the only one that satisfies this is 25. Neither knows if they have 25 or 2

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Well played!

That’s not entirely true. Bob may very well have a 2. We know Alice’s number, but we still don’t know Bob’s.

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Alice goes first and says she doesn’t know what her number is. We know she can see Bob’s number. This means Bob’s number has to be 1, 2, 5, 10, or 25, because any other number would tell Alice what her number is.

Bob knows this as well, so Bob knows his number is either 1, 2, 5, 10, or 25. He can see Alice’s number, and then says he doesn’t know what his number is either. This can only happen if Alice’s number is 25. If Bob sees that Alice’s number is 25, he only knows that his number is either 2 or 25. Any other number on Alice’s forehead would tell Bob exactly what his number is. So the answer is 25.

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Alice tells Bob what his number is and Bob tells Alice what hers is and they each finish their drink and go wash this silly game off their face.

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What?!

Each knows that both numbers are positive integers.

But the “answer” is… (posted elsewhere). Did I miss something?

If they were intelligent, they wouldn’t both be confused after the first said they didn’t know their number. This is because the only ambiguous answer that isn’t resolved with just one one view is 25+25, so as soon as one said “I don’t know”, the other should have recognized this unique case. There is no reason for the second intelligent player to echo “I don’t know mine either”.

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I think you might have misread the puzzle

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Yes, that’s it. Alice must see either 1, 2, 5, 10, and 25, each which can have more than one “partner”. So then Bob must be looking at either 2, 5, 10, 25, 40, 45, 48, 49, or 50.

A: B
2: 25
5: 10
10: 5
25: 2 or 25
40: 10
45: 5
48: 2
49: 1
50: 1

Of the Alice:Bob combinations, only Bob seeing 25 is ambiguous. But Alice doesn’t know her own number until Bob says that he doesn’t know. And we still don’t know Bob’s number, only Alice knows that, but it will be 2 or 25.

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