Classic Brain Teaser: The Two Numbers

**hymie**[$] · #1 03-22-2002, 12:34 AM

I select two numbers between 2 and 99 inclusive. I tell their sum to Sam and their product to Pat.

Sam says to Pat: You don't know what the numbers are, and neither do I.
Pat says to Sam: I know the numbers.
Sam says to Pat: So do I!

What are the numbers?

(Warning - you may need a computer to figure out the answer, or at least a lot of paper and pencil work.)

kurt_messick · #2 03-26-2002, 05:22 PM

And the answer is???

**mjfrombuffalo**[$] · #3 03-26-2002, 05:30 PM

Whole numbers only?

**hymie**[$] · #4 04-07-2002, 11:35 PM

Yes, whole numbers. I don't actually remember the answer - I'll have to work it out myself. Since there don't seem to be many takers, here's how the solution begins. The key is that just from the sum, you know that you can't figure out the numbers from the product. That strongly limits what the sum can be. For example, if the sum was 30, one possible way to form it would be 23 + 7, but then the numbers could be determined from the product because 23 and 7 are prime, so 30 can't be a possible sum.

**hymie**[$] · #5 04-08-2002, 11:46 AM

I scribbled up the program to calculate the answer. The numbers are 4 and 13, but of course, it's how you get there that's interesting!

briwei · #6 04-08-2002, 04:44 PM

Quote:

Originally posted by hymie
I scribbled up the program to calculate the answer. The numbers are 4 and 13, but of course, it's how you get there that's interesting!

Perhaps it's because of daylight savings throwing my brain out of whack, but I'm not sure how this result works. If the numbers are 4 and 13, the product is 52 and the sum is 17.

With a product of 52, there are two possible combos: 2 x 26 and 4 x 13. So, you need to get some information from the sum to know which to pick.

The only information you get from the sum is that the two numbers can't be guessed strictly from the sum. It seems to me that that would hold true whether the sum were 17 (4+13) or 28 (2+26).

So, my head hurts. Anyone got the mathematical equivalent of Excedrin?

Thanks,
Bri

theeye · #7 04-08-2002, 06:09 PM

Quote:

Originally posted by briwei

With a product of 52, there are two possible combos: 2 x 26 and 4 x 13. So, you need to get some information from the sum to know which to pick.

The only information you get from the sum is that the two numbers can't be guessed strictly from the sum. It seems to me that that would hold true whether the sum were 17 (4+13) or 28 (2+26).

Nope. As Pat, you know one additional fact. In addition to knowing that the sum isn't enough information to guess the two numbers, you also know that Sam is convinced that the product isn't enough on its own either.

So here you are knowing that the product is 52. You know there are two possibilities: 2x26 or 4x13.

Let's guess that the answer is 2x26. If that's correct, then Sam knows that the sum is 28. Sam is sure that Pat can't know what the numbers are. But if the sum is 28, then the two numbers might be 23 and 5, as far as Sam knows. And those two numbers are prime. And if the two numbers were both prime, then Pat would instantly know what they were.

So (assuming that Sam is a logical sort of person), the two numbers can't possibly be 2 and 26, because if they were, then Sam couldn't be sure that Pat didn't know what the numbers were.

Once Sam expresses certainty that Pat can't know what the numbers are, Pat knows that (2,26) isn't a possibility. That leaves only 4 and 13, so Pat knows the answer.

Excedrin's right over there in the medicine cabinet if you still need it.

briwei · #8 04-09-2002, 03:59 PM

Quote:

Originally posted by theeye

Nope. As Pat, you know one additional fact. In addition to knowing that the sum isn't enough information to guess the two numbers, you also know that Sam is convinced that the product isn't enough on its own either.

So here you are knowing that the product is 52. You know there are two possibilities: 2x26 or 4x13.

Let's guess that the answer is 2x26. If that's correct, then Sam knows that the sum is 28. Sam is sure that Pat can't know what the numbers are. But if the sum is 28, then the two numbers might be 23 and 5, as far as Sam knows. And those two numbers are prime. And if the two numbers were both prime, then Pat would instantly know what they were.

So (assuming that Sam is a logical sort of person), the two numbers can't possibly be 2 and 26, because if they were, then Sam couldn't be sure that Pat didn't know what the numbers were.

Once Sam expresses certainty that Pat can't know what the numbers are, Pat knows that (2,26) isn't a possibility. That leaves only 4 and 13, so Pat knows the answer.

Excedrin's right over there in the medicine cabinet if you still need it.

Ahhhhhhhhhhhhh. Light dawns on Marble head. :thumbs:

The very existence of a pair of primes as the sum precludes that entire sum from being correct. I got it. I'll definitely take that Excedrin now.

That's some pretty tricky stuff!

Thanks!
Bri

**drmomentum**[$] · #9 04-09-2002, 06:48 PM

Now my question is - why are Sam and Pat so good at math that they can work out all the combinations in their head quickly? Why can't we have a puzzle with 2 dummies trying to solve it?

More my speed.

-JP

**hymie**[$] · #10 04-10-2002, 12:29 AM

Here's another apparent "zero information" puzzle. A village consists of married couples and a celibate priest. Some of the wives have been unfaithful to their husbands, in a way both careless and careful - the only person who does not know whether a woman has been indiscreet is her own husband. All this comes to the attention of the priest, who announces to the village that at least one wife has been unfaithful, that the villagers must not talk about this at all to anyone, even each other, but that any husband who can prove his wife was unfaithful must make an announcement before the entire village at the daily meeting. Days go by, and at each meeting no one says anything. At the tenth meeting, however, a number of husbands stand up to denounce their wives.

How many? How did they know?

theeye · #11 04-11-2002, 03:52 PM

Quote:

Originally posted by hymie
Here's another apparent "zero information" puzzle. A village consists of married couples and a celibate priest. Some of the wives have been unfaithful to their husbands, in a way both careless and careful - the only person who does not know whether a woman has been indiscreet is her own husband. All this comes to the attention of the priest, who announces to the village that at least one wife has been unfaithful, that the villagers must not talk about this at all to anyone, even each other, but that any husband who can prove his wife was unfaithful must make an announcement before the entire village at the daily meeting. Days go by, and at each meeting no one says anything. At the tenth meeting, however, a number of husbands stand up to denounce their wives.

How many? How did they know?

Presumably the priest also reveals to everyone the fact that the unfaithful wives have told everyone except their own husband.

Assuming that is correct, then here is the analysis:

On day one, everyone realizes that any husband who is unaware of any infidelities must realize that his own wife has been unfaithful (as he knows there is at least one unfaithful wife and he knows he'd know about any other one). Such a husband would step forth at the meeting and denounce his own wife. As no such husband stepped forth, at the end of day one everyone knows that there are no husbands in the village who had heard about zero infidelities: every husband knows about at least one. If every husband knows about at least one unfaithful wife (but must not know about his own), then there must be at least two unfaithful wives in the village (and some husbands must know about at least two).

On day two, everyone realizes that any husband who is aware of only one infidelity must realize that his own wife is the second and he would denounce her. So there are no such husbands. There must be at least three unfaithful wives total and some husbands must know about at least three.

By induction, on the tenth day, everyone realizes that any husband who is aware of only ten infidelities must realize that his own wife is the eleventh. Since we know that all husbands are aware of all infidelities other than their own wives and since some men came forward on the tenth day, it must be all eleven men who simultaneously realize that their own wives were unfaithful.

At that point they all turn on the priest, who should have seen this coming and tell him that they are coming to wring his neck, but will only halve the distance between them and him with each step they take. Will they ever get their revenge?

**drmomentum**[$] · #12 04-11-2002, 06:17 PM

And what is the priest's name?

I'm guessing "Zeno."

-JP

**hymie**[$] · #13 04-12-2002, 03:17 PM

Along the same lines, another classic. A prisoner is told that he has been sentenced to die in the next week, but that the exact day will be unexpected by him. He reasons that he can not be executed on Saturday, since not having been executed until then, he would expect to be hung on that day. Once the last day is excluded, there is a new last day, and therefore by induction he reasons that he cannot be executed at all. He is therefore extremely surprised when, unexpectedly, he is hung on Sunday!