Episode 9: A Dragon & 2 Scientists Cont.

So last week, I proposed this question:

A dragon caught two scientists for dinner. It told them that they would be freed if they figure out the (integer) ages of its two children. The dragon then whispers the product of the two numbers to Dr. Jeal and the sum of them to Dr. KyJan . They are not allowed to tell these numbers to each other.

Dr. Jeal: I don't know.
Dr. KyJan: I don't know, either!
Dr. Jeal: Then, I know!
Dr.KyJan : Then I know, too!



How old are the dragon-kids?!

So, did you figure it out?


It's ok if you didn't. I just figured it out yesterday :)

This is actually a pretty tricky problem with a really tricky answer. Hopefully I can explain it to you well.



So let's call the ages of the children x and y. Therefore Dr. Jeal knows xy and Dr. KyJan knows x+y.

Let's try to figure out what each of the "I don't know"s mean:

When Dr. Jeal says, "I don't know" then he is saying that he knows that xy is not a prime. If xy was a prime number, then either x or y would be 1, so the sum would be one more than the product. In fact, the product would be the prime, itself, and Dr. Jeal would know both x and y!

Now, when Dr. KyJan says that he doesn't know what x and y are then the knowledge he is giving Dr. Jeal is much more broad. Since Dr. Jeal knows that the product is not a prime, then when Dr. KyJan say's he doesn't know, this means that x+y is not 4. The only way to sum to 4 is 2+2 or 3+1 (but it can't be 3+1, then Dr. KyJan would have known, since 3 is a prime). Therefore, Dr. KyJan can't know what x & y are.

So we know that x and y aren't prime and don't sum to 4.

So that narrows us down to about.....and infinite number of possibilities.

Except that that's all we really need.

When Dr. Jeal says, "Then I know!" then there is only one possibility. If xy>4, then there would be at least 2 possibilities for every x and y, so there would be no way Dr. Jeal would know the two numbers.

For example, let's look at if x=2 and y=3. If this were the case, then xy=6 and Dr. Jeal would start by saying, "I don't know." This would be true because 6 is not a prime and there are two possibilities to get to 6 (namely, 2*3, and 6*1). Then Dr. KyJan would respond with an, "I don't know" because there are two possibilities to get 5 (namely, 2+3 and 4+1). But Dr. Jeal could not respond saying that he knows the right answer. There would be no way for him to know whether or not the numbers were 6 and 1, or 2 and 3.

However, if the numbers were x=1 and y=4, then there is a different story.

If this was true, then xy=4. To start off, Dr. Jeal would not know the numbers x and y because 4 is not a prime. So he would, in fact, tell Dr. KyJan, "I don't know". Then there are two possibilities for x and y when xy=4, either x=1 and y=4 or x=2 and y=2. So Dr. Jeal needs information from Dr. KyJan.

Dr. KyJan, however, would have to respond with an, "I don't know." If x=1 and y=4, then x+y=5. This would mean that there are 2 possibilities for x and y. Either x=2 and y=3 would work, or x=4 and y=1. So, since Dr. KyJan doesn't have 4, he must say, "I don't know."

But, then Dr. Jeal knows the answer now, since he knows x+y is not equal to 4. He knows that x=1 and y=4 since if y,x=2 then x+y =4, and since this is not the case x and y must be 1 and 4. Because Dr. Jeal knows, Dr. KyJan can infer the values and thus knows as well.

SO, the only way for this to play out is if x=1 and y=4.

That's totally cool, right?!


I really hope this made sense. It seems that it's very difficult for me to explain this solution in words without verbally explaining it to you (with lots of hand motions involved). Here's hoping that you aren't totally confused by my explanation!





P.S. Don't forget to celebrate George Lucas's birthday today!!!!