Five Rings Interview Questions
87 quant-trader and quant-researcher questions from Five Rings: fast expectation/EV, dice-and-coin probability, mental-math brain-teasers, and a few stats and stochastic-process pieces.
Inside the Five Rings interview
Five Rings is a young, fast-growing proprietary trading and market-making firm whose interviews are speed-first: rapid-fire expected-value and probability puzzles, mental-math estimation, and brain-teasers meant to be reasoned through out loud, not ground out on paper.
What they test
The core is expectation and probability — together more than half the set — built on dice, coins, and uniform random variables: expected maxima/minima of rolls, fair prices for flip games, waiting times until a pattern or first success. Around it sit mental-math brain-teasers (estimating a tenth root, log2(50!), the share of primes ending in 7) and lighter combinatorics, with a thin layer of statistics, stochastic processes, and game theory for the researcher track.
The recurring shapes
Expect dice and coins everywhere: expected value of the max/min/2nd-highest of several rolls, fair price to enter a coin-flip game, and waiting-time questions like flips to two consecutive heads or the longest heads streak across many players. A second cluster is geometric probability — breaking a stick into a triangle, random points on a disk or circle. The trader track leans on instant EV and estimation; the researcher track adds estimators, regression slopes, and random walks.
How to approach
Train for speed and clean reasoning under pressure. Reach first for linearity of expectation and indicator variables to skip casework, use symmetry and first-step / conditioning arguments on the waiting-time and random-walk problems, and keep order-of-magnitude estimation sharp for the mental-math teasers. Say your assumptions aloud and sanity-check the number — the interviewer is watching how you think, not just the final answer.
Difficulty leans easy-to-medium (43 easy, 34 medium) with 10 hard problems to stretch on — a set built for building fast, reliable EV and probability instincts.
Five Rings expected value questions (29)
- Maximize the Product of Two Dice
- Expected Maximum of Two Dice
- Expected Area of a Random Triangle on a Unit Circle
- Expected Flips for Two Consecutive Heads
- Expected Maximum and Minimum of Three Dice Rolls
- Expected Value of the Second-Highest of Several Dice Rolls
- Generating Arbitrary Probabilities from a Fair Coin
- Expected Flips to Reach a Head-Tail Boundary
- Expected Longest Heads Streak Across 30 People
- Expected Value of the Maximum of Two Dice
- Expected Number of Connected Components in a Random Coloring
- Expected Number of All-Black 2x2 Squares in a Random Grid
- Cover Time of a Complete Graph
- Expected Product of Two Dice
- Bus Waiting Paradox
- Expected Draws Until Uniform Sum Exceeds One
- Expected Number of Uniforms to Exceed One
- Expected Length and Ratio of the Smaller Stick Piece
- Expectation of Sine of a Uniform Angle
- Expectation of the Ratio of Two Dice
- Expected Draws to First Ace
- Doubling Account With Fixed Withdrawals
- Expected Pulls to Collect All Five Characters
- Expected Number of Fixed Points in a Random Permutation
- Expected Radial Distance on a Circular Disk
- Conditional Expectation Given an Even Function
- Expected Twos Before the First One
- Doubling Coin Game: Expected Balance
- Fair Price to Enter a Coin Flip Game
Five Rings probability questions (23)
- Three Six-Sided Dice vs One Twenty-Sided Die
- Coin Flip Probabilities: One, Two, and n Coins
- Pattern Race: HHT vs THH
- Longest Head Streak in Fifteen Flips
- Two-Stage Conditional Dice Roll
- Hidden Face of a Painted Cube
- Triangle From Breaking the Longer Piece
- Equal Heads in Split Coin Flips
- Random Walk to the Boundary of a Square
- Probability of a Linear Combination of Normals Being Positive
- Probability That a Stick Broken in Three Forms a Triangle
- Probability One Player Gets More Heads With an Extra Coin
- At Least One Low Dice Sum in Five Rolls
- Biased Random Walk Reaching Position 1
- Absolute Uniform Difference
- Conditional Probability for a Sum of Two Uniforms
- Probability of No Decay in One Second
- Even Heads with Mixed Fair and Unfair Coins
- Probability That a Dice Sum Beats the Dice Product
- Mean Versus Median for a Decreasing Density
- Number of Draws to the First Ace: Mean Versus Median
- Momentum-Switching Tennis Tournament
- Beating a Loaded Die
Five Rings combinatorics questions (8)
Five Rings brain teasers questions (6)
Five Rings random variables questions (5)
Five Rings stochastic processes questions (5)
Five Rings statistics questions (4)
Five Rings finance questions (2)
Five Rings game theory questions (1)
Five Rings machine learning questions (1)
Five Rings linear algebra questions (1)
Five Rings regression questions (1)
Five Rings optimization questions (1)
Five Rings interview FAQ
What kind of questions does Five Rings ask in quant interviews?
Candidates most often report expected value, probability and combinatorics questions. This page collects 87 of them, 70 stamped with the month they were last reported — each with a full worked solution.
How hard are Five Rings interview questions?
The set spans 43 easy, 34 medium and 10 hard problems. Most sit at medium difficulty — solvable in a few minutes with clean reasoning — with a harder tail that rewards knowing the canonical tricks.
How do I prepare for the Five Rings quant interview?
Work through this set by topic (use the sidebar), starting from your weakest area. 12 problems are free to open with their full solution, so you can judge the quality before anything else. Then walk the full Five Rings interview guide for the round-by-round funnel and the online assessment.
Are these the actual Five Rings interview questions?
They are built from candidate-reported Five Rings questions. We rewrite each prompt for clarity and author the worked solutions ourselves — we don't claim the wording is verbatim, and we never invent questions or recycle generic lists. 70 of 87 carry the month they were last reported.