The Sam and Polly Puzzle is a beautiful riddle in which two logicians Sam and Polly must simulate the other’s thought processes several layers deep, deriving information from seemingly unhelpful statements. While many variations have been proposed, I first heard of the puzzle from the excellent list at the XKCD wiki, where it takes the following form:

Sam and Polly are perfectly logical mathematicians. A student walks in and says: “I’m thinking of two numbers `x`

and `y`

, with `3 <= x <= y <= 97`

. I’ll tell their sum to Sam, and their product to Polly.” She does so, then leaves, and the following conversation takes place:

Sam (to Polly): You can’t know what `x`

and `y`

are.

Polly (to Sam): That was true, but now I know.

Sam (to Polly): Now I know too.

Find `x`

and `y`

.

If you haven’t solved this riddle or one of its variations before, I *absolutely* recommend giving it a try before reading on. It took me about three hours to solve, with a whiteboard and the Python programming language.

This post consists of:

- Section 1: A clean Python solution
- Section 2: An investigation of the (somewhat weak) dependence on the bounds 3 and 97
- Section 3: A discussion of unbounded variations of the riddle.

The most common variant on the puzzle (and the original version published by Freudenthal in 1969, available here) uses the following bounds instead: `2 <= x < y`

and `x + y <= 100`

, leading to a different solution. I’m sticking with the above formulation just because that’s the version that I saw and started thinking about before looking around to see what was known. The solution in the following section can be easily modified to solve Freudenthal’s version.

This riddle also known as the Sum and Product Puzzle, or as the Impossible Puzzle (not to be confused with the Hardest Logic Puzzle Ever, another excellent riddle).

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