An interesting mathematical puzzle

[AminEE]

I'v chosen two integer number;x and y. We assume 1<x<y and x+y<100. I told x+y to Mr. S and x.y to Mr. P. I want Mr. S and Mr. P to find x and y. The following conversation done between these tow person.
P: I can't determine the two numbers.
S: I knew that.
P: Now I can determine them.
S: So can I.

FIND THE TWO NUMBERS

 

[srieda]

i cant tell what the two numbers are

Do you have an answer for this??

 

[cedance]

1 x y 100 sum product i cannot puzzle

obviously, both r prime.

cedance.

 

[jorgito]

some intersting puzzle

AminEE,

Are you sure about the talk between P and S?

If numbers were 6 and 13, then the sum is 19 and product is 84.
But, 84 is interesting, because its factors 6 and 13 sums 19, and its factors 7 and 12 also sum 19.

(analyzing up to P=165, this case where the two diferent factorization have the same sum is unique)

Best regards!

 

[AminEE]

Re: An intersting puzzle

Do you have an answer for this??

Yes, Of course. I know its solution. It is really nice question. Don't you agree?

Added after 23 minutes:

AminEE,

Are you sure about the talk between P and S?

If numbers were 6 and 13, then the sum is 19 and product is 84.
But, 84 is interesting, because its factors 6 and 13 sums 19, and its factors 7 and 12 also sum 19.

(analyzing up to P=165, this case where the two diferent factorization have the same sum is unique)

Best regards!

The talk between P and S are correct and accurate. the solution is unique.

 

[jorgito]

An intersting puzzle

I guess that the numbers are 3 and 8.

S saw that all additive decomposition of his number give products with two or more diferent factorizations.

The sum 11 can be (2+9), (3+8), (4+7) or (5+6)
The corresponding products would be
18, 24, 28 and 30

This is the first case that I found, so if the solution is unique I bet for 3 and 8.

Best regards!

 

[AminEE]

Re: An intersting puzzle

Dear jorgito
Ok. there are some more numbers like 11, e.g. test 17.
How can Mr. P after the first reply of Mr. S, find the two numbers? And How can Mr. S. find the solution after that? You should use ths last two reply.
Best wishes;

 

[cedance]

Re: An intersting puzzle

AminEE,

Are you sure about the talk between P and S?

If numbers were 6 and 13, then the sum is 19 and product is 84.
But, 84 is interesting, because its factors 6 and 13 sums 19, and its factors 7 and 12 also sum 19.

(analyzing up to P=165, this case where the two diferent factorization have the same sum is unique)

Best regards!

to my knowledge 13*6 = 78!

cedance.

 

[jorgito]

An intersting puzzle

Oooops!

If I have not make another stupid mistake as cedance pointed, the numbers must be 4 and 13.

Best regards!

 

[klug]

Re: An intersting puzzle

Very interesting puzzle. I like it.

At first time it seems simple, but it is not very simple. It has a travel to theory of prime numbers.

I even found here interesting that every even number can be presented as sum of two primes, next I tried to prove it, but it seems for me very difficult.
(Ha-ha, next I found by Google.com that Goldbach formulated this idea near 300 years ago, and it is not proven untill now) ))

P: I can't determine the two numbers.

So, they are not two prime numbers. And x*y is product of more than 2 primes.

S: I knew that.

It means that sum x+y can not be presented as sum of two primes. So, the sum can be one of R = 11, 17, 23, 27, 29, 35, 37 ... (really they all are non prime odd numbers plus 2) Here was the idea that sum can not be even, as every even number can be presented as sum of two primes. (Goldbach has found it 300 years ago, and it is proven now for all few numbers up to 2*10^17 ) )

P: Now I can determine them.

So, it means that x*y can be presented as product of not more than 3 primes (need to be proven) and more than 2, so it has 3 prime numbers: A*B*C , and sum x+y is one of 3 variants :
A+B*C
A*B+C
A*C+B
As we found the sum is odd, this means that at least one of this A, B or C is even and prime, so it is equal for 2.

Now, there 3 variants of sum:
2+B*C
2*B+C
2*C+B
And x*y is 2*B*C

To determine x & y Mr. P must have only two variants (need to be proven). So, two of this 3 primes must be equal then there will be not 3 but 2 variants. Product can be 2*2*C or 2*B*B, and sum can be :
2+2*C even
2*2+C odd *
or
2+B*B odd
2*B+B odd

S: So can I.

Next, I simply checked variants in row R for unique combination of * and found the same numbers, as "Jorgito".

4 & 13.

But seems there must be more elegant solution than checking. Also some of assumptions need to be proven. And I still do not prove that there is only 1 solution.

 

[pareshatlnmiit]

Re: An intersting puzzle

If s know x.y and p knows x+y then it can be possible to found out solution but for this case its really tough

 

[Naveed Alam]

Re: An intersting puzzle

why r all saying the no.s must prime...
if x=3
y=4

then
x.y=12(told to P)
options are 2*6 or 3*4

x+y=7 (told to S)
options are
2+5 or 3+4
P says he don't know the two numbers (P is confused whether its 2*6 or 3*4)
S says he also don't know
P realise that S have also two options as he got. So, he become to know that 2+5 option is impossible b/c it gives 2*5=10; so it can be 3+4 becuase 3*4=12.

S also realise like this..

Kindly correct me.....if i m wrong.


Naveed

 

[klug]

Re: An intersting puzzle

if i m wrong.

I know that.

I can't determine the two numbers.

I knew that.

In (2) S says that he knew (1) before P said (1).

So, there can not be the sum of two prime numbers.

 

[Quibbler]

Re: An intersting puzzle

They seem to be fibbing.
the sum could be 11 as this cannot be made from the sum of two primes so S knows that P cannot determine the two numbers. So the two numbers could be 2 and 9 or 8 and 3 or....So take 2 and 9 say P=18 P's options at the start are 3x6 and 9x2 but when S reveals that the sum is not made up of two primes it can only be 2 and 9. But 8 and 3 will work just as well.

I don't think S can claim to know a unique solution there are just too many combinations

 

[smiles]

An intersting puzzle

x, y is two roots of equation X² - (Mr.S) x X + (Mr.P) = 0
delta = Mr.S² - 4.Mr.P
x = [- sqrt(delta) - Mr.S]/2
y = [- sqrt(delta) + Mr.S]/2
That is the unique result

Exactly, replace Mr.S and Mr.P by S and P in the equations !!!

 

[FvM]

why r all saying the no.s must prime...
if x=3
y=4

Yes. It hasn't been asked to prove that the solution is unique, but I guess it is.

 

[jpanhalt]

This is an old thread and even older puzzle. I remembered it from my days reading Martin Garner's columns in Scientific American. The clue is sums and products, and without giving the actual numeric answer here, you might be interested in these links:

http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/logic_sum_product

Impossible Puzzle - Wikipedia, the free encyclopedia

The first is quite an extensive discussion of the puzzle and variations on it. The Wikipedia article is more abbreviated.

John

 

[vikrant2012]

I'v chosen two integer number;x and y. We assume 1<x<y and x+y<100. I told x+y to Mr. S and x.y to Mr. P. I want Mr. S and Mr. P to find x and y. The following conversation done between these tow person.
P: I can't determine the two numbers.
S: I knew that.
P: Now I can determine them.
S: So can I.

FIND THE TWO NUMBERS

dear AminEE, can you please send answer.