There are 4 mathematicians - Brahma, Sachin, Prashant and Nakul - having lunch in a hotel. Suddenly, Brahma thinks of 2 integer numbers greater than 1 and says, "The sum of the numbers is..." and he whispers the sum to Sachin. Then he says, "The product of the numbers is..." and he whispers the product to Prashant. After that following conversation takes place :

Sachin : Prashant, I don't think that we know the numbers.
Prashant : Aha!, now I know the numbers.
Sachin : Oh, now I also know the numbers.
Nakul : Now, I also know the numbers.

What are the numbers?

For Solution SCROLL DOWN...

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The numbers are 4 and 13.

As Sachin is initially confident that they (i.e. he and Prashant) don't know the numbers, we can conclude that -
1) The sum must not be expressible as sum of two primes, otherwise Sachin could not have been sure in advance that Prashant did not know the numbers.
2) The product cannot be less than 12, otherwise there would only be one choice and Prashant would have figured that out also.

Such possible sum are - 11, 17, 23, 27, 29, 35, 37, 41, 47, 51, 53, 57, 59, 65, 67, 71, 77, 79, 83, 87, 89, 93, 95, 97, 101, 107, 113, 117, 119, 121, 123, 125, 127, 131, 135, 137, 143, 145, 147, 149, 155, 157, 161, 163, 167, 171, 173, 177, 179, 185, 187, 189, 191, 197, ....

Let's examine them one by one.

If the sum of two numbers is 11, Sachin will think that the numbers would be (2,9), (3,8), (4,7) or (5,6).

Sachin : "As 11 is not expressible as sum of two primes, Prashant can't know the numbers."

Here, the product would be 18(2*9), 24(3*8), 28(4*7) or 30(5*6). In all the cases except for product 30, Prashant would know the numbers.

- if product of two numbers is 18:
Prashant : "Since the product is 18, the sum could be either 11(2,9) or 9(3,6). But if the sum was 9, Sachin would have deduced that I might know the numbers as (2,7) is the possible prime numbers pair. Hence, the numbers must be 2 and 9." (OR in otherwords, 9 is not in the Possible Sum List)

- if product of two numbers is 24:
Prashant : "Since the product is 24, the sum could be either 14(2,12), 11(3,8) or 10(4,6). But 14 and 10 are not in the Possible Sum List. Hence, the numbers must be 3 and 8."

- if product of two numbers is 28:
Prashant : "Since the product is 28, the sum could be either 16(2,14) or 11(4,7). But 16 is not in the Possible Sum List. Hence, the numbers must be 4 and 7."

- if product of two numbers is 30:
Prashant : "Since the product is 30, the sum could be either 17(2,15), 13(3,10) or 11(5,6). But 13 is not in the Possible Sum List. Hence, the numbers must be either (2,15) or (5,6)." Here, Prashant won't be sure of the numbers.

Hence, Prashant will be sure of the numbers if product is either 18, 24 or 28.

Sachin : "Since Prashant knows the numbers, they must be either (3,8), (4,7) or (5,6)." But he won't be sure. Hence, the sum is not 11.


Anonymous said...

4 and 13..................rajesh

Anonymous said...

Clue Less, Would like to know how to achieve the answer

probe said...

Numbers are 6 & 2.
Sachin knows, sum is 8.
Prashant knows, multiple is 12.

Looking from Prashants point of view...
12 can be from 2 x 6 or 3 x 4.
If 3&4 are the numbers sum is 7.
With 7 there are only 2 products,
5x2 = 10 & 3x4 = 12. If 10 is the product
prashant would guess the numbers (5 and 2 being only options)
then sachin would also guess it eventually.
If 12 is the product, sachin can guess 3 & 4 from the fact that
prashant didn't guess it.

Looking from sachins point of view,
He gets 8 as sum, so the numbers can be 5&3 or 6&2 or 4&4.
If 5&3 are the numbers prasanth would guess it,
else neither prashanth nor he can guess it.
So sachin claims that neither can guess it.

This gives prashanth a hint, if sachin knows the number to be 7,
he(sachin) would know that eventually both would guess it.
But he says both cant guess it, which leads him to the conclusion that
sachin knows 8 as the sum so he figures out 2 & 6 are the numbers.

Once prashanth reveals that he knows the number, sachin works
out the same logic and guesses the number.

Once prashanth and sachien guesses the number, Nakul also guesses it
like I did :)

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Anonymous said...

it shuld b 2 & 4 & 13 z possible???

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David Jackman said...

This was really really hard. I thought it was 4 and 3, but that wasn't right. I'm not convinced of the solution. I don't think you did a good enough job explaining it Suman. In fact I know you didn't because you didn't even cover the circumstances covering the solution you posited (4 & 13).

Oh well, on to the next puzzle. I really didn't want to move forward unless I got each and every puzzle, but I'll have to swallow a little defeat to move on.