# 3 travelers and 9 diamonds

I think this is the most appropriate place to share this. I haven't figured out the answer yet, but I think I'm close. I couldn't find anything like this on the Internet, so I'm posting it here.

There are three travelers. They are inside a cave. Deep down in this cave, there are three doors, each door containing at least one diamond, and there are a total of 9 diamonds. Each traveler picks a door and gets the diamonds behind it.

The travelers loot their diamonds, and, before exiting the cave, all three must say simultaneously if they can deduce the number of diamonds that each of the other two have. They never lie, and all of them said they couldn't deduce it.

After these statements were said, one of the travelers realized that now he knows the answer.

So the question is:

How were the diamonds split between the three travelers? (the order doesn't matter)

I was going to write what I've managed to do so far, but I think it isn't a good idea to give away too much information even if its concealed by spoilers. This riddle was suggested by my discrete mathematics professor. It isn't homework and she didn't mention where it came from.

• Welcome to Puzzling.SE! May I ask where you found this riddle? – Riley Sep 1 '18 at 0:28
• Unfortunately I don't know either. – AtilioA Sep 1 '18 at 0:35
• I have a question : When one of the travelers knew the answer, did he say that out loud? @AtAn – Kevin L Sep 1 '18 at 1:10
• Yes. sorry, I thought that was explicit enough. – AtilioA Sep 1 '18 at 1:22
• Yes he can immediately deduce. I think that's what OP means by "the order doesn't matter". – sedrick Sep 1 '18 at 1:47

I think it isn't possible and here is my reasoning.

First we know that nobody has 6 or 7 because they could immediately deduce the other two.

Now let's analyze this from the perspective of the traveler who figured it out. Here are the possibilities given the number he knows.

1: 35, 44
2: 25, 34
3: 15, 24, 33
4: 14, 23
5: 13, 22

The only way a traveler could figure out the quantities later is if there was only one combination but all possibilities have 2 or 3. Therefore I don't think it's possible.

• Note that you didn't discard the repeated ones (135 and 513, 144 and 414, etc). If you didn't because you're analyzing one's traveler outcome then it is fine, but at this point we don't get anything from it... – AtilioA Sep 1 '18 at 2:03

Someone has shown why the puzzle does not seem to have a solution. however...

maybe your professor gave this puzzle to you in writing, and also has bad handwriting, and wrote an 8 which looked like a 9. which is not totally implausible. however I don't know whether they actually gave it to you in writing.

like with other analyses, we can rule out a few possibilites:

we know that no traveller can have 6, because they would deduce the others have 1 each, and also not 5, since then they would deduce the others have 1 and 2. from here, every traveller has between 1 and 4 diamonds, and knows this of each other traveller, and nows the diamonds sum to 8 when a traveller has a number of diamonds, there are only a few possibilities for what the other travellers hold, due to the summing to 8. I list the possibilities here

4: 13, 22
3: 23, 14
2: 24, 33
1: 34

1 only has 1 possibility for what the other travellers hold, which means that the solution was deduced by a traveller who looted 1 diamond

First off:

If you have 6 or 7 diamonds, you know that the combination is 1,1,7 or 1,2,6 right away.

Now, since no one said that, it's clearly not true. However:

The only time that this gives you new information is if you have either 1 or 2 diamonds

This means:

Anyone who looks interested in that fact has either 1 or 2 diamonds - but they still don't know how many the other people have

This means that someone else has to make a guess:

Now knowing that at least 1 person has either 1 or 2 diamonds:
If they have 5 diamonds, then the combinations available are 1,3,5 with 1 interested person or 2,2,5 with 2 interested people
If they have 4 diamonds, then the combinations available are 1,4,4 with 1 interested person, or 2,3,4 also with 1 interested person
If they have 3 diamonds, then the combinations available are 1,3,5 with 1 interested person, or 2,3,4 also with 1 interested person
If they have 2 diamonds, then the only combination available is 2,2,5 with 1 other interested person
If they have 1 diamond, then no one else will show any interest - so we discount this option

Since we know that someone must make a correct guess here, we know that

The guesser does not have 1, 3, or 4 diamonds

Meaning

The combinations are 1,3,5 or 2,2,5

Furthermore! We know that one person guesses, hence:

The combination is 1,3,5, and the guesser is the person with 5 diamonds, because if the combination was 2,2,5 then everyone would be able to guess it at this point.

• Your solution seems compelling, but I don't know what you mean by "interested person". – AtilioA Sep 5 '18 at 23:06
• @AtAn There are 3 cases going into the first declaration: You know what the numbers are, you don't know what the numbers are, and you know that no one knows what the numbers are. After everyone says "I don't know", everyone in the second group moves to the third, and has to re-think based on that information - the new fact is of interest to them. Anyone already in the third group will not show any interest in the fact, as it was already known to them. (i.e. the 3rd spoiler) – Chronocidal Sep 6 '18 at 2:52
• how does someone determine that someone else is "interested" in the fact that no one has 6 or 7? – SilverCookies Sep 6 '18 at 7:33
• @SilverCookie Facial expressions / cold reading? If their reaction to finding out that no one knows how many diamonds they each have is to fall into deep thought, then they must not have already known/realised that that was going to be the case. (As other answers have pointed out, we need at least 1 more piece of info to answer the puzzle - I just tried to come up with one you could determine by observing the other ) – Chronocidal Sep 6 '18 at 7:41
• @ ok I understand your point – SilverCookies Sep 6 '18 at 8:14

I think I may have the solution (I hope)

I will write down all the possibilities first :

117 126 135 144 153 162 171
216 225 234 243 252 261
315 324 333 342 351
414 423 432 441
513 522 531
612 621
711

From this, I will only write the numbers that don't have similar numbers with different arrangements :

333

As you can see, this is the only possibility where the travelers will be able to know the number of diamonds because it is the only number with a unique arrangement

• Can you explain why 3,3,3 is easier to figure out than 3,2,4? – Reibello Sep 1 '18 at 1:25
• @AtAn is this the answer u had in mind or am I completely off track :D ? – Kevin L Sep 1 '18 at 1:25
• @Reibello Well because 324 has other similar numbers with different arrangements (342, 423, 432) – Kevin L Sep 1 '18 at 1:26
• But how does the traveler know that? – Reibello Sep 1 '18 at 1:28
• Order doesn't matter I take to mean "126=162=216=261=612=621", so your argument doesn't hold. Also in say "144" the first traveller could possibly deduce the others' diamonds since there is no ordering in "44" – boboquack Sep 1 '18 at 1:29

If the one guy is a little slow then...

The one person who knows received seven diamonds, and the other two each received one. I have no explanation for why the traveler with seven couldn't immediately figure this out. The one person who knows received six diamonds, and one other person received one, and another received two.

• Well, that's not possible because they couldn't deduce the quantity at first right? – Kevin L Sep 1 '18 at 1:16
• Yeah, it seems unlikely that this is the answer a math professor is going for. I'm seeing if I can work something better out. – Reibello Sep 1 '18 at 1:18
• Yup, I just posted my answer (I think I may be wrong also) :) – Kevin L Sep 1 '18 at 1:21
• @Kevin L is right; they couldn't deduce. – AtilioA Sep 1 '18 at 1:24
• Isn't this the same. They still couldn't deduce – Kevin L Sep 1 '18 at 1:38

The statements:

• all three say simultaneously,

so noone has any information before they call it. and

• they couldn't deduce it.

with their own diamond.

moreover,

• each door containing at least one diamond.

so There are a few possibilities for their loots:

| $1,1,7$ |, | $1,2,6$ |, | $1,3,5$ |, | $1,4,4$ |, | $2,2,5$ |, | $2,3,4$ |, |3,3,3|

and out of these, we can easily eliminate some of them since noone could deduce the number of diamonds;

| $1,1,7$ | -> If this was the case, whoever had 7 diamonds would figure out the number of diamonds for each person since at least one diamond per person and only 2 diamonds are left to be divided into two people.

and after they said they could not

"one of the travelers realized that he knew the answer". Actually he/she knew the answer but could not realize before they all call for it. So he didnt lie just he could deduce the number of the diamonds of others, but he realized it after they call for their deduction. Sometimes you know the answer, but you could not realize it before something happens.

| $1,1,7$ |

• They all know there are 9 diamonds. If one picked up 7 diamonds, then there's only 2 left, 1 for each of the others. This way he would've said he could deduce, when exiting. – AtilioA Sep 1 '18 at 14:27
• @AtAn I think u didnt read my explanation. you say "he realizes", not "then he knows" ... "one of the travelers realized that he knew the answer." – Oray Sep 1 '18 at 14:28
• Ok, I'm editing the post. I put it that way to make it less clear that the fact of none of them knowing the answer is crucial to the puzzle. He didn't knew the answer before saying. – AtilioA Sep 1 '18 at 14:32

The traveller who realized the answer did 9 minus the sum of the two numbers other traveller said. You know your quantity, you know how many the other is missing, so you can figure out the third one (if you didn't pay attention to what he said). That's how he knew the answer. Otherwise you can't know the specific combination. You can only know the answer after the statements are made.

EDIT: Fixed spoiler

• You don't know the quantities involved. That's what you're trying to figure out. – AtilioA Sep 4 '18 at 23:44
• What do you mean? Don't they know there are 9 diamonds in total? – Nicolás Magno Sep 4 '18 at 23:45
• Yes. But you gotta find how many diamonds each traveler has. – AtilioA Sep 5 '18 at 0:15
• I know. I'm saying it's impossible unless someone takes a guess first. – Nicolás Magno Sep 5 '18 at 0:16
• Unless... rot13(Gurl nyy cvpx gur fnzr qbbe). – Nicolás Magno Sep 5 '18 at 0:23