How do I approach Coding & Algorithms interview questions?

Coding & Algorithms questions require understanding of core concepts and practice. PracHub provides solutions with explanations to help you master coding & algorithms interviews.

What difficulty level is this interview question?

This is a medium difficulty Coding & Algorithms question, commonly asked during Onsite rounds at Jane Street.

What role is this question designed for?

This question is commonly asked for Data Scientist candidates at Jane Street during technical interviews.

Optimal Stopping for Die Rolls That Bust on Perfect Squares

Q: Optimal Stopping for Die Rolls That Bust on Perfect Squares

This is a Coding & Algorithms interview question from Jane Street for Data Scientist roles. View the full question and solution on PracHub.

Last updated: Jul 2, 2026

|Home/Coding & Algorithms/Jane Street

Optimal Stopping for Die Rolls That Bust on Perfect Squares

Jane Street

Aug 3, 2025, 12:00 AM

mediumData ScientistOnsiteCoding & Algorithms

You repeatedly roll a fair six-sided die and keep a running cumulative sum $S$ of all the values rolled so far (starting from $S = 0$ before the first roll).

After any roll, you may choose to stop and walk away with a payoff equal to the current sum $S$ .
However, if at any point the cumulative sum $S$ lands exactly on a positive perfect square ( $1, 4, 9, 16, 25, \dots$ ), you immediately lose everything : the game ends with payoff $0$ .
If you keep rolling and have not busted, the game continues indefinitely — there is no limit on the number of rolls.

Determine the optimal strategy: for which values of the running sum should you keep rolling, and at which values should you stop and take the money? Justify why the strategy is optimal, and compute the expected payoff of the game under optimal play, starting from a sum of $0$ .