# What are the two numbers puzzle?

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Two positive integers < 50 are chosen.
The sum is revealed to A
The sum of the squares is revealed to B.
Both A and B are given the information in this problem.

The conversation between them goes as follows:

• B: I can’t tell what the numbers are
• A: I can’t tell what the numbers are.
• B: I can’t tell what the numbers are.
• A: I can’t tell what the numbers are.
• B: I can’t tell what the numbers are.
• A: I can’t tell what the numbers are.
• B: Now I can tell what the numbers are.

What are the two numbers?

SherlockHolmes Expert Asked on 15th August 2015 in

This problem is intriguing!

Number the sentences in conversation:
<<1>> B: I can’t tell what the numbers are.
<<2>> A: I can’t tell what the numbers are.
<<3>> B: I can’t tell what the numbers are.
<<4>> A: I can’t tell what the numbers are.
<<5>> B: I can’t tell what the numbers are.
<<6>> A: I can’t tell what the numbers are.
<<7>> B: Now I can tell what the numbers are.

Denote the sum S and sum of squares N. S-list will be a list of all possible sums a + b, N-list a list of all possible sums of squares a^2 + b^2. Initially, the S-list contains all positive integers >1, N-list all positive integers expressible as sum of two positive integer squares.

The section <<i>> below describes the information gained after the line <<i>> in the conversation was spoken.

<<1>> N is not expressible as the sum of positive integer squares uniquely. N-list is correspondingly adjusted (numbers expressible as sum of two positive integer squares uniquely are thrown out). S-list is adjusted – sums are thrown out, that don’t allow for summands a, b, such that a^2 + b^2 is on the current N-list.

<<2>> S allows for more values of a, b (a + b = S), such that a^2 + b^2 belongs to the N-list. S-list is adjusted (sums are deleted not allowing for more values of a, b (a + b = the sum), such that a^2 + b^2 belongs to the N-list).

<<3>> N allows for more values of a, b (a^2 + b^2 = N), such that a + b is on the S-list. Those integers that don’t are deleted from the N-list.

<<4>> S allows for more values of a, b, such that a^2 + b^2 belongs to the N-list. Those integers that don’t are deleted from the S-list.

<<5>> N allows for more values of a, b, such that a + b belongs to the S-list. Those integers that don’t are deleted from the N-list.

<<6>> S allows for more values of a, b, such that a^2 + b^2 belongs to the N-list. Those integers that don’t are deleted from the S-list.

<<7>> At this moment there is only one pair of positive integers a, b, such that a + b is on the S-list and a^2 + b^2 = N. If N-list contains more values, they need to be checked to see if they allow for unique solution.

If some bounds are imposed on a,b, I guess the solution could be found by computer or manual deleting from the S- and N-list. I haven’t really thought beyond this outline yet, at this moment solving the problem in full generality seems difficult… I’d love to see the intended solution.

SherlockHolmes Expert Answered on 15th October 2015.

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