Answer / guest

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.

Summerising data for sum 11:

Possible Sum PRODUCT Possible Sum

2+9 18 2+9=11 (possible)

3+6=9

3+8 24 2+12=14

3+8=11 (possible)

4+6=10

4+7 28 2+12=14

3+8=11 (possible)

4+6=10

5+6 30 2+15=17 (possible)

3+10=13

5+6=11 (possible)

Following the same procedure for 17:

Possible Sum PRODUCT Possible Sum

2+15 30 2+15=17 (possible)

3+10= 13

5+6=11 (possible)

3+14 42 2+21=23 (possible)

3+14=17 (possible)

6+7=13

4+13 52 2+26=28

4+13=17 (possible)

5+12 60 2+30=32

3+20=23 (possible)

4+15=19

5+12=17 (possible)

6+10=16

6+11 66 2+33=35 (possible)

3+22=25

6+11=17 (possible)

7+10 70 2+35=37 (possible)

5+14=19

7+10=17 (possible)

8+9 72 2+36=38

3+24=27 (possible)

4+18=22

6+12=18

8+9=17 (possible)

Here, Prashant will be sure of the numbers if the product is
52.

Sachin : "Since Prashant knows the numbers, they must be
(4,13)."

For all other numbers in the Possible Sum List, Prashant
might be sure of the numbers but Sachin won't.

Here is the step by step explaination:

Sachin : "As the sum is 17, two numbers can be either
(2,15), (3,14), (4,13), (5,12), (6,11), (7,10) or (8,9).
Also, as none of them is a prime numbers pair, Prashant
won't be knowing numbers either."

Prashant : "Since Sachin is sure that both of us don't know
the numbers, the sum must be one of the Possible Sum List.
Further, as the product is 52, two numbers can be either
(2,26) or (4,13). But if they were (2,26), Sachin would not
have been sure in advance that I don't know the numbers as
28 (2+26) is not in the Possible Sum List. Hence, two
numbers are 4 and 13."

Sachin : "As Prashant now knows both the numbers, out of all
possible products - 30(2,15), 42(3,14), 52(4,13), 60(5,12),
66(6,11), 70(7,10), 72(8,9) - there is one product for which
list of all possible sum contains ONLY ONE sum from the
Possible Sum List. And also, no such two lists exist. [see
table above for 17] Hence, two numbers are 4 and 13."

Nakul figured out both the numbers just as we did by
observing the conversation between Sachin and Prashant.

It is interesting to note that there are no other such two
numbers. We checked all the possible sums till 500 !!!

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