Vtyjkyuk

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.

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

Warning: Skewey logic ahead.

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.

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

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.

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.

so the answer is actually

| $1,1,7$ |

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