Brainstorming with Hats
Jane, Kate, and Lily are perfect logicians and always tell the truth.
They are each wearing a hat marked with a distinct positive integer. They know that one of the hats has a single-digit integer , N , written on it, and that the other two hats each have a divisor of N written on them. Each person can only see the two numbers on the others’ hats.
- Jane: "I don't know my number."
- Kate: "I don't know my number."
- Lily: "But I know my number!"
- Kate: "Now I know my number, too!"
- Jane: "I still don't know my number."
What is the sum of the numbers written on Kate and Lily’s hats?
The answer is 9.
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4 solutions
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I must say that the word "factor" in the question should instead be "divisor". Factors are terms of a product and implies two of the hats multiply to give the other.
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No. Actually I found "divisor" to be confusing and would have preferred "factor".
Saying two of the numbers are factors of the third number in no way implies that their product is the third number.
I think There are some mistakes. After Kate's first statement, the rest posibilities are {1,2,4}, {1,2,8}, {1,4,8}, {2,4,8}, 126, 162, 261, 621, 136, 631, 236, 326, 216, 316. Kate cannot know her number if she sees 2 and 6 or 3 and 6 on Jane and Lily. so the posibilities are 48-6-4-4=34.
And after Lily's statement, the posibilities become 162, 261, 621, 136, 631, 236, 326, 316.
Then, since Kate find her own number, they becomes 162, 261, 136, 236.
Finally, since Jane cannot determin her number, the rest posibilities are 136, 236.
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You are right; I ruled out the possibilities J K L = 2 1 6 and J K L = 3 1 6 too soon. I have made the changes.
The numbers written on Kate's and Lily's hats are 3 and 6 respectively, whereas the number written on Jane's hat can be either 1 or 2 .
The reasoning is as follows:
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Jane looks at the numbers on Kate's and Lily's hats, which are 3 and 6 respectively, and says she doesn't know her number. 6 is probably the largest. However, 6 is the only multiple of 6 which is single-digit. So, 6 is the largest. Knowing that, Jane deduces that her number is either 1 or 2 , which she can't confirm.
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Kate looks at the numbers on Jane's and Lily's hats, which are either 1 and 6 or 2 and 6 respectively, and says she doesn't know her number. We have just shown that 6 is the largest. Knowing that, Kate deduces her number is either 2 or 3 or 1 or 3 in respective cases. Therefore, she can't confirm her number.
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Lily looks at the numbers on Jane's and Kate's hats, which are either 1 and 3 or 2 and 3 respectively.
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When the numbers on Jane's and Kate's hats are 1 and 3 , Lily deduces that her number can only be 6 or 9 . If her number is 9 , Kate would be looking at 1 and 9 on Jane's and Lily's hats respectively. Kate would be able to deduce that her number is 3 in her first attempt, which contradicts with the description in our original problem. In other words, the number on Lily's hat can only be 6 .
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When the numbers on Jane's and Kate's hats are 2 and 3 , Lily deduces that her number can only be 6 . This is because l c m ( 2 , 3 ) = 6 .
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As soon as Lily deduces her number to be 6 , Kate also knows her number.
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If the number on Jane's hat is 1 , Kate's initial deduction for her own number is 2 or 3 . However, if Kate's number turns out to be 2 , Lily could not figure out her number in the first try because Lily's number could be 4 , 6 or 8 . So, Kate's number has to be 3 .
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If the number on Jane's hat is 2 , Kate's initial deduction for her own number is 1 or 3 . However, if Kate's number turns out to be 1 , Lily could not figure out her number in the first try because Lily's number could be 4 , 6 or 8 . So, Kate's number has to be 3 .
Summing up, the sum of numbers written on Kate's and Lily's hats = 9
Please take note that any other combinations and permutations apart from the mentioned ones do not work here. The justification is left for the exercise of the readers.
There's a flaw here, at least in Jane's logic. If Lily could see 2 & 3, she would know immediately that she had 6. As it is, Lily is only able to determine she wears 6 after Jane is unable to determine her number. Once Lily declares this, Kate knows she has 3 but Jane should therefore figure out she has 1.
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See @Arjen Vreugdenhil comment
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I think that answer still relies on Lily's response being purely of a Know/Don't Know nature and strictly speaking the use of the word "now" arguably means nothing.
But why add it? If Jane and Kate were wearing 2 and 3, a simple Know/Don't know would suffice. As a result Jane should be able to infer what number she has.
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@Malcolm Rich – Mathematically speaking, the now word is strict. It means now at the point of time of speech and not any other point of time
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@Donglin Loo – I'm inclined to agree from a strict logical viewpoint. Yet the sense conveyed when Lily says "Now" still implies a prior sense of "not knowing". In Kate's case this is true although we know that from her earlier statement.
@Donglin Loo – " It means now at the point of time of speech and not any other point of time"
Well....in this context it means "I didn't know this fact previously but I know it now".
"not any other point in time" is misleading. Kate's knowledge doesn't disappear after she makes her statement. It's understood that knowledge isn't as ephemeral as that.
@Malcolm Rich – I thought I've read "Now" in Lily's statement as well, but I see that it reads:
Lily: "But I know my number!"
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@Darko Simonovic – It has been edited by moderators of Brilliant. Only they could do so.
I agree here. Lily claims she could only deduce her number once Jane and Kate couldn’t, so 2,3,6 (J,K, L) isn’t an option. Jane would know she has 1. Surely 3,1,6 is an option?
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Do note that 3,1,6 would eventually make Jane know that her number is 3.
Wow, this logic problem is very complicate, learned from it! Good question.
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@Kelvin Hong haha yeah... I am quite fascinated by logic problems like this. First time tackled problems like this was in Singapore Asian Schools Mathematics Olympiad(SASMO). You can go look up "Albert Cheryl Bernard SASMO problem". It's something very similar to mine which went viral on the Internet.
I came to another solution and need help on figuring out, why it's wrong:
If Jane has 2 or 3 (undetermined. In the explanation she will have a 2), Kate has 6 and Lily has 1, the reasoning should still hold:
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Jane sees a 6 and a 1 and determines, she must have a 2 or a 3.
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Kate sees a 2 and a 1 and knows that she has a 4, 6 or 8.
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Liliy sees a 2 and a 6 and knows, she does not have a 3. Because otherwise Kate would have seen a 2 and a 3 and would have known that she must have a 6. Since 1, 2, 3 and 6 are the only divisors of 6, Lily knows that she must have a 1.
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Kate now reasons if she had a 4, Lily would have seen a 2 and a 4 and could not have known that she has a 1, because she could have had an 8 also. If she had an 8, Lily would have seen a 2 and an 8 and could not have known that she has a 1, because she could have had a 4 also. Hence, Kate knows, she must have the 6.
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Jane now does still not know if she has the 2 or the 3.
EDIT: Nevermind. Jane now knows that she has the 2, because if she had the 3, Kate would have known that she has the 6 in the second step.
Why can the answer not be 12? Let me try to explain using a scenario: J: 2 K: 4 L: 8
J: sees 4 and 8. Therefore knows the integer is 8 and a divisor is 4. She knows this because 8 is larger than 4 (thus is not a divisor of 4) and that 4 is not a divisor for numbers 5-9 other than 8. Thus 8, must be the integer. She must also be a divisor and she knows this and can then deduce that she is either 1 or 2 (The only other divisors left).
K: sees 2 and 8. Therefore knows the integer is 8 and a divisor is 2 using the same basic logic that J used. K knows she must be 4 or 1.
L: Sees 4 and 2. Those are only divisors of 4 and 8. All numbers must be distinct. Therefore L must be 8.
K: If L knows her number then I could not have been 1 because the integers 1 and 2 are divisors, of 2, 4, 6, 8. Thus I had to be unique to 8 and am 4. Therefore K is 4.
J: I am 1 or 2. If I am 1, L would have seen 4 and 1 and known she 8. If I am 2 then L would have seen 4 and 2 and known she was 8. If I am 1 K would have seen that I was 1 and still could have been 4 or 2. Now knowing that L knows, K must have figured out she was 4 because if she was 2 it would have been impossible to tell that L was 8. Therefore, it would have been impossible, I must have been 2 or 1. If I was 2 then K could be 4 or 1 and still would have known that she was not 1 because then I must have been 2 and therefore could not have been 1. So I am still either 1 or 2 and unsure.
Is this logic not right? Sorry if the grammar was bad, I was piecing it all together without a lot of confidence.
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The problem is when L sees 4 and 2, 4 could also potentially be the integer, she would think that she is either 1 or 8, which doesn’t help her at all.
Just because J and K know that the integer is 8 doesn’t mean that L automatically knows it too.
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L sees 4 and 2 thus 4 cannot be the integer since all of the integers have to be distinct. Therefore L knows that L is 8 at that moment.
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@Benjamin Cornell – With that I can also say that, if L sees 1 and 6 instead, N cannot be 6 since 'all integers are distinct'.
By "all integers are distinct", that only refers to the three numbers on the three hats (that includes the integer N , since N is on one of the hats), you are treating the integer N as another separate distinct digit.
Does that answer your question?
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@Kenneth Tan – Ah, I see what your saying. L could think that she is 1! Makes sense... tough problem, but a great one.
@Benjamin Cornell – As Kenneth explained. If Lily sees 4 and 2, the three numbers could still be 4,2,1.
Jane sees 1,2 => Doesn't know number Kate sees 1,4 => her number could be 2 or 8 Lily sees 2,4 => her number could be 1 or 8
The possible values of N are 9,8,6,4 the only ones with enough divisors to work.
Two of the players will see N.
9 is out because at least one of the first two will see a (9,x) combination then know immediately that her number is the other divisor on x=3 is 1, or on x=1 is 3. It will not get to a third player as written. The is important because it means that (1,3) must indicate N = 6 not 9 for the third player.
4 and 8 are out because there are no combinations that can give the third player a certain result.
Therefore N = 6. There are the following pairs that a player might see:. (6.1) (6,2) (6,3) (3,2) (3,1) (2,1), and two with 6 must be present.
There are two arrangements that satisfy the given statements. 1,3,6 and 2,3,6.
The first player sees 3,6 and cannot decide between 1 and 2.
The second player sees 1,6 or 2,6 and cannot decide between 2 and 3, or 1 and 2 respectively.
The third player sees either 1,3 or 2,3 (and since 9 is not an option) knows that N = 6, and it must be on her hat.
The second player now knows that the third must have seen either 1,3 or 2,3 since 1,2 cannot be decided between 4,6, or 8. The second player now knows that her hat must be 3 to complete the pair regardless of whether player one's hat shows 1 or 2.
The first player still can't decide between 1 and 2.
In both cases the final two hats are 3 and 6 with a sum of 9.
Since there are only single digit numbers under each hat there can be 1,2,3...,9 any of them under any hat.0 will not be there under any hat(for obvious reasons).There can be 3,4,6,8,9 under the hats.Lily has the largest number under her hat,since she first came to know about her no.If she saw 2 and 1 under Kate and Jane's hat then she may have 3,4,6,8 so she can't tell what her no is.If she saw 3 and 1 still she is uncertain about it but if she saw a 3 and a 2 it is definitely 6 under her hat all other combinations will give possibilities of more than one number under her hat and she will not be able to know her number.jane saw 6 and 3 under the other two hats,so she may have a 2 or a 1 under her's so she could not figure it out.Hence the sum of the numbers is 9 and not 8.
There are 48 possible scenarios, which are permutations of { 1 , 2 , 4 } ; { 1 , 2 , 8 } ; { 1 , 4 , 8 } ; { 2 , 4 , 8 } ; { 1 , 2 , 6 } ; { 1 , 3 , 6 } ; { 2 , 3 , 6 } ; { 1 , 3 , 9 } . In the last case, either Jane or Kate (or both) would see the 9 and know the numbers. This is not the case, so we rule out these six possibilities. Also, if either Jane or Kate saw the numbers 2 and 3, she would know her own number to be 6. This rules out another four possibilities.
After Jane's first statement, Kate knows that there is no 9 involved. If she saw the numbers 1 and 3, she would therefore know her own number to be 6. Also, Kate knows that Jane did not see the combination of 2 and 3. Therefore, if Kate sees a J L = 6 2 or 6 3 , she knows her own number to be 1. None of these is the case, so we rule out four more possibilities. There are now 4 8 − 6 − 4 − 4 = 3 4 possibilities left: { 1 , 2 , 4 } ; { 1 , 2 , 8 } ; { 1 , 4 , 8 } ; { 2 , 4 , 8 } ; J K L = 1 2 6 , 1 6 2 , 2 6 1 , 6 2 1 , 6 1 2 ; 1 3 6 , 6 3 1 , 6 1 3 ; 2 3 6 , 3 2 6 .
Lily is now able to know her number. This rules out any of the twenty-four combinations of the first row. (If Lily sees two of the numbers 1,2,4,8, then there are always at least two possibilities for her own hat.) We also rule out 1 2 6 because Lily cannot distinguish it from 1 2 4 or 1 2 8 . That leaves us with only eight possible scenarios: J K L = 1 6 2 ; 2 6 1 ; 6 2 1 , 6 3 1 ; 1 3 6 ; 2 3 6 ; 3 1 6 , 3 2 6 . In each case, the combination J K is unique so that Lily can determine the number on her hat.
But now Kate can also uniquely determine her number. This rules out the two pairs of scenarios 6 2 1 , 6 3 1 and 3 1 6 , 3 2 6 , because these are indistinguishable for Kate. This leaves four scenarios: J K L = 1 6 2 ; 2 6 1 ; 1 3 6 , 2 3 6 . At this point, Jane is still unable to decide. That means that we have one of the scenarios 1 3 6 , 2 3 6 : Jane sees that Kate has 3 and Lily has 6, but she does not know whether her number is 1 or 2.
Therefore the answer is 3 + 6 = 9 .