It was first published in 1969 by Hans Freudenthal,[1][2] and the name Impossible Puzzle was coined by Martin Gardner.
The problem is rather easily solved once the concepts and perspectives are made clear.
There is no need for any advanced knowledge like Goldbach's conjecture or the fact that for the product B·C of such a 2-split to be unique (i.e. there are no other two numbers that also when multiplied yield the same result).
The reader can then deduce the only possible solution based on the fact that S was able to determine it.
But by the exclusion of products with factors that sum to numbers between these boundaries, there are no longer multiple ways of factoring all non-solutions, leading to the information yielding no solution at all to the problem.
Then 35 is eliminated when S declares that P cannot know the factors of the product, which it would not have been if the sum of 64 was allowed.
The first solution with no prime number is the fourth which appears at X + Y ≤ 2522 or higher with values X = 16 = 2·2·2·2 and Y = 111 = 3·37.
Intuitively, the problem space got "sparser" when the prime number 2 is no longer available as the factor X, creating fewer possible products X·Y from a given sum A.
If the condition X + Y ≤ t for some threshold t is exchanged for X·Y ≤ u instead, the problem changes appearance.
A reasonable value for u could be u = t·t/4 for the corresponding t based on the largest product of two factors whose sum are t being (t/2)·(t/2).