Ok, so a couple of days ago our analysis (maths) teacher gave us an interesting math problem. I actually solved (without any help from any source) it a day ago and he confirmed that my result was correct. I am among the extremely small amount of people on earth who are capable of solving, or even understanding the problem.
I am now wish to find out how many people over here can solve it. Using the internet to find out the solution might be very easy, for I have never tried, but I advise that you not do so for it makes no sense to lie to people on the internet, because you are only lying to yourself.

Here is the enigma:
There is a person who comes up with 2 numbers that range from 2 to 999. (Numbers can be the same). He then tells the PRODUCT of those two numbers to a person called "Philip". He also tells the SUM of those two numbers to a person called "Sam".
Philip and Sam have a conversation:
Philip: "I do not know the two numbers".
Sam: "I knew you wouldn't know them".
Philip: "I know know the two numbers."
Sam: "Now I too know the numbers"

Note: Their conversation was truthful.
Note: They did not speak before the conversation and they did not know what numbers the other one was given, before the conversation.

You know have to find those two numbers, and their products and sums. That is all the information you will get/need.

Philip: "I know know the two numbers."

(Corrected)
Philip: "I now know the two numbers."

Philip doesn't know any of them, so it's not a product of primes.
Sam knew this, so the sum can't be written as the sum of 2 primes.

Maybe I'm still missing something.

hmm... time to get really stoned and re-read this, laugh at it for a few mins apon realisation that they repeat the number "two" twice, chew on some dry 2 min noodles, then make a flash about my whole adventure...

whos in?

16 and 13

4 and 13?

numbers can be the same

2?

At 9/18/08 07:53 AM, AnalogStick wrote: You know have to find those two numbers, and their products and sums. That is all the information you will get/need.

I've only considered it for a few minutes but I'm stumped. Math-wise the furthest I've gotten is that the product can be anywhere from 4 to 998,001 and the sum can be anywhere from 4 to 1,998. It seems like the clue is supposed to be found in Sam and Philip's conversation, but given the circumstances I'm having trouble seeing how either of them could have deduced the numbers from just that little talk.

What I also don't understand is, if "they did not speak before the conversation and they did not know what numbers the other one was given before the conversation," then how could Sam tell Philip "I knew you wouldn't know them" and their conversation still be truthful? So... Philip knows the product but can't figure out the two numbers for himself, and Sam knows the sum and ALSO knows that Philip wouldn't have be able to figure out the product. So that means the sum number indicates that the product number has more than two divisors between 2 and 999...? And then by knowing this is the situation, they both suddenly know what those two numbers are?

Yeah, I'm clueless here.

At 9/18/08 09:23 AM, ghostface619 wrote:

numbers can be the same

2?

I started at the bottom too... but I think if both numbers were 2 then Sam and Philip would have been able to figure it out even before their conversation. 4 is the lowest possible sum/product so that would've made the two numbers obvious. Or something like that.

At 9/18/08 09:32 AM, StephanosGnomon wrote: the furthest I've gotten is that the product can be anywhere from 4 to 998,001

The product can't be a prime number, though. It has to be divisible by at least 2.

I don't know if pointing this out helps anyone, I'm just as clueless as you.

At 9/18/08 09:39 AM, AapoJoki wrote:

At 9/18/08 09:32 AM, StephanosGnomon wrote: the furthest I've gotten is that the product can be anywhere from 4 to 998,001

The product can't be a prime number, though. It has to be divisible by at least 2.

I don't know if pointing this out helps anyone, I'm just as clueless as you.

It seems like that would definately be a part of the solution, I think. I'm just not really up on my math properties anymore... it's been a looong time. Maybe I should just stick to logic puzzles. :P

I look forward to hearing the solution though.

Well all the extra remarks were made so the result should come out of their conversation.

So suppose the numbers are x and y and the product equals P and sum S.
P=xy
S=x+y

When does P give away the 2 numbers right away. -> P is the product of primes or P is the 3d power of a prime: x^3=x(x^2)

Him not knowing that elimenates the fact that both x and y are prime or y=x^2 with x a prime.

S knows this, so the sum can't be x+y with x and y both prime or y=x^2

Than you can eliminate a whole range of values for S. (e.g. 9=2+7, cause 2 and 7 are prime)

Then you can look at the rest of the values for S and split them into the sum of 2 numbers, looking at their products.

Philippe has the same reasoning as above (we may assume) so he will look at his product and it's divisors and checking their sums. If he knows the numbers, then their is only one sum which corresponds to the facts above. So he'll know the numbers.

This way however gives a sollution by process of elimination and a lot of basic sums and product calculations. So one can pray
1. the solution is unique
2. it's in the lower region of the numbers (so we don't have products that split into 16 primes, cause there would be a shitload of calculating
Or of course you can always let the PC do your calculations for you. Programming skills come in handy here.

I tried it by hand.

Phillip says two.
Sam says two.

2*2=4
2+2=4

The answer is 4.

At 9/18/08 09:57 AM, videogamer0810 wrote: Phillip says two.
Sam says two.

2*2=4
2+2=4

The answer is 4.

Provided philip isn't a retard, he should be able to work out that the only way to get 4 by multiplying two integers between 2 and 999 is 2*2,

At 9/18/08 07:53 AM, AnalogStick wrote: I am among the extremely small amount of people on earth who are capable of solving, or even understanding the problem.

Lol, no.

That's just unwarranted self importance. Give me proof that only " an extremely small amount of people on Earth" can solve a problem given to you in school. Anyone can figure it out, some people find it boring and give up, some people take longer than others etc.

At 9/18/08 10:11 AM, BananaBreadMuffin wrote: Provided philip isn't a retard, he should be able to work out that the only way to get 4 by multiplying two integers between 2 and 999 is 2*2,

Yes, but before the conversation, he was not told the two numbers or the sum/product.
Therefore, he either has to be psychic, or the two numbers have to be mentioned somewhere in the conversation.

Let me congratulate you and your massive ego on being able to solve a brainteaser.

At 9/18/08 10:13 AM, videogamer0810 wrote:

At 9/18/08 10:11 AM, BananaBreadMuffin wrote: Provided philip isn't a retard, he should be able to work out that the only way to get 4 by multiplying two integers between 2 and 999 is 2*2,

Yes, but before the conversation, he was not told the two numbers or the sum/product.

I mean, we were not told. He was told, but the problem doesn't tell us.

Well, he COULD be a retard.

I don't know anymore.

At 9/18/08 10:12 AM, xXMajuniorXx wrote:

At 9/18/08 07:53 AM, AnalogStick wrote: I am among the extremely small amount of people on earth who are capable of solving, or even understanding the problem.

Lol, no.

That's just unwarranted self importance. Give me proof that only " an extremely small amount of people on Earth" can solve a problem given to you in school. Anyone can figure it out, some people find it boring and give up, some people take longer than others etc.

You need logical reasoning to figure this out. MANY MANY people lack logical reasoning, hence the average IQ is 100. Some things just cannot be done with time, for those people can have all the time in the world and still not solve it. In all the people that have given what they think is the solution only one of them has a reasonable explanation as to what the numbers are (unless he didn't google).

At 9/18/08 10:20 AM, AnalogStick wrote: You need logical reasoning to figure this out. MANY MANY people lack logical reasoning, hence the average IQ is 100. Some things just cannot be done with time, for those people can have all the time in the world and still not solve it. In all the people that have given what they think is the solution only one of them has a reasonable explanation as to what the numbers are (unless he didn't google).

That's becuase it's not fun or interesting. It takes minimal logic and then some working out.

That's becuase it's not fun or interesting. It takes minimal logic and then some working out.

Than why don't you work it out if it takes MINIMAL logic and SOME working out. Why, you should do it in 5 minutes!

At 9/18/08 10:25 AM, AnalogStick wrote: Than why don't you work it out if it takes MINIMAL logic and SOME working out. Why, you should do it in 5 minutes!

Because it's boring. Brainteasers should be at least entertaining. To a logical mind anyway. This is just work, I could write out the theory of relativity and I'd get about as much out of the experience.

Because it's boring. Brainteasers should be at least entertaining. To a logical mind anyway. This is just work, I could write out the theory of relativity and I'd get about as much out of the experience.

If it's boring, why the hell are you even posting on this thread? Not entertaining? Any puzzle that you can't solve is fun to try to solve. And is does minutes of your life count as "work"?

At 9/18/08 10:34 AM, AnalogStick wrote: If it's boring, why the hell are you even posting on this thread? Not entertaining? Any puzzle that you can't solve is fun to try to solve. And is does minutes of your life count as "work"?

Work is defined by what your doing, not the amount of time you spent doing it. The main reason I posted in this thread is because of the ridiculous claim that this is the most difficult enigma in the world. Maybe your teacher told you it was to make you feel more special, but there are alot more difficult things to work out than this.
If you can prove there are infinitely many twin primes, then I'll admit you're within the top minds on Earth.

OH! I love these! hold on a sec while I go to work to figure this out!

oh boy! something to do now!

The numbers were written on each other heads, but they are both emo so they were looking at the ground, then something happened and they both looked up and now they know the numbers.
Duh.

I usually spend 5 mins attempting to work out problems like this and then after that I just give up.
Can't be bothered attempting to work out the problem when i'll learn the answer later.

Yeh, i'm lazy like that.

Work is defined by what your doing, not the amount of time you spent doing it. The main reason I posted in this thread is because of the ridiculous claim that this is the most difficult enigma in the world. Maybe your teacher told you it was to make you feel more special, but there are alot more difficult things to work out than this.
If you can prove there are infinitely many twin primes, then I'll admit you're within the top minds on Earth.

You mean the Goldbach conjecture? The title was just an exaggeration. Of course I don't literally think that this is the most difficult problem.

At 9/18/08 11:31 AM, AnalogStick wrote: You mean the Goldbach conjecture?

That's something quite different.

The title was just an exaggeration. Of course I don't literally think that this is the most difficult problem.

How about the bull about being one of the few people who even understands the problem? There's seems plenty of people in this restricted pool of intellect that are quite capable of understanding the problem.

At 9/18/08 11:43 AM, HeartbreakHoldout wrote:

At 9/18/08 11:31 AM, AnalogStick wrote: You mean the Goldbach conjecture?

That's something quite different.

Yeah forgot myself, the goldbach conjecture is that you can create any even number with primes. It's actually a useful thing to know to solve this.

The title was just an exaggeration. Of course I don't literally think that this is the most difficult problem.

How about the bull about being one of the few people who even understands the problem? There's seems plenty of people in this restricted pool of intellect that are quite capable of understanding the problem.

I see only 2 people to have understood this problem.