The Sum-and-Product Puzzle

Two numbers $x$ and $y$ are chosen with $2 \le x \le y$ and $x + y \le 100$. Mr Sum is told only $x+y$; Mr Product is told only $x \times y$. Neither can see the other's clue. In front of a room, this exchange happens:

• Mr Product says: "I don't know the numbers."
• Mr Sum says: "I knew you didn't know. I don't know them either."
• Mr Product says: "Now I know the numbers."
• Mr Sum says: "Now I know them too."

What are $x$ and $y$? The puzzle looks like it needs more information than it's given — the trick is realizing that each statement is itself a piece of information, and everyone in the room, including you, can use it.

The core idea: statements are data

Every sentence spoken rules out possibilities, and — crucially — everyone hears every statement, so each person can re-run the other's reasoning. "I don't know" isn't a shrug, it's a precise claim: "my clue is consistent with more than one pair." "I knew you didn't know" is an even stronger claim: "every possible pair matching my clue would have made your clue ambiguous too." Solving the puzzle means translating each sentence into a filter on the set of candidate pairs, applying the filters one at a time, and watching the candidate set shrink.

Working the logic step by step

Statement 1 — Product doesn't know. This means $x \times y$ factors into at least two valid pairs $(x,y)$ with $2 \le x \le y$, $x+y \le 100$. This immediately rules out any product with a unique factorization into such a pair — for example if the product were $2 \times 3 = 6$, no other valid pair gives 6, so Product would've known instantly. So the true pair's product must be "ambiguous."

Statement 2 — Sum knew Product didn't know, and doesn't know either. "Sum knew Product didn't know" means: for every way to split Sum's total $x+y$ into a valid pair, the resulting product is one of the ambiguous ones from Statement 1. This is a strong constraint on the sum — it rules out sums that have even one "clean" (uniquely-factoring) split. Then "Sum still doesn't know" removes sums that, after this filtering, are left with only one candidate pair (those would have been solved already).

Statement 3 — Product now knows. After Statement 2 has trimmed the surviving pairs, Product's number now matches only one of them.

Statement 4 — Sum now knows. After Statement 3's trimming, Sum's number also matches only one survivor.

Carried through by hand (or by a short program that just filters a table of all $(x,y)$ pairs against each statement in turn), this converges on the unique answer $x=4$, $y=13$. The exact numbers matter less than the method: a table of candidates, filtered four times, each filter derived from what a statement logically implies rather than what it literally states.

What this means in practice

Interviewers use this family of puzzle to see whether a candidate can hold a set of possibilities in mind and update it correctly on each new piece of information — precisely the skill needed to interpret a counterparty's quote, a trade's fill pattern, or a market's reaction as evidence rather than noise. The habit of writing out the candidate set explicitly, rather than trying to reason about it purely in your head, is the transferable skill: it's the same discipline as building a small table of scenarios in a probability question and crossing off the ones each new fact rules out.