## Nov 13, 2009

### Bhaddo, Tawar and KT (Tough)

Mr. Bhaddo choses two different numbers greater than N but less than M & tells their sum to Mr. Tawar and their product to Mr. KT. The following conversation ensues:

Mr. Tawar:   I cannot determine the two numbers.

Mr. KT:   I cannot determine the two numbers either.
Mr. Tawar:   I still cannot determine the two numbers.
Mr. KT:   Now I can determine the two numbers.
Mr. Tawar:   Now I can determine the two numbers also.

Find the greatest value of M for which this puzzle has a unique solution, for N=1, N=2 and N=3.

1. What are the numbers?

From Mr. kt statement "I do not know the numbers", we can deduce that the product is not the product of two primes. If it were, then Product would have been able to factorize the product into two primes, and would then know the two numbers.

From Mr. tawar's statement: "I knew you didn't knew the numbers" we can deduce that the sum must be an odd number, because every even number (at least for small numbers) can be written as the sum of two primes (Goldbach's Conjecture). The only way for S to know that Mr. Product doesn't know the numbers, is for the sum to be an odd number.

From Mr. kt's statement: "Now I know the numbers", we can deduce that the information that the sum is an odd number helps Mr. Product to find the correct two numbers.

From Mr. tawar's statement: "Now I know the numbers, too", we can deduce that the fact that Mr. Product now knows the numbers, is enough information for Mr. Sum to find the correct two numbers.

A summary of what we know:

P1: m and n are not both primes
P2: the sum m + n is an odd number, so one of the numbers is odd, the other number is even.

I am a bit stumped on how to proceed from here. There are still lot of number pairs left to eliminate until a unique solution presents itself. What other deductions can be made from the sentences that Mr. kt and Mr. tawar says?

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