Three ants sit on the corners of an equilateral triangle. Each ant simultaneously picks one of the two adjacent corners uniformly at random and walks to it. What is the probability that no two ants collide?
That is the simplest version of a question family that Goldman Sachs, Jane Street, Citadel, and most other quant interviewers have been asking continuously for at least 30 years. The clockface version puts an ant on the 12 of a clock and asks for the probability that, after a random walk of N steps, the ant is back at 12. Or never returns to 12. Or visits every position. The question has a hundred variants and they are still asked in 2026, because unlike the brainteasers in tech, the underlying skill — probability fluency under time pressure — is genuinely job-relevant for a quant trader.
The triangle version
The cleanest setup is three ants on a triangle. Each ant has two neighbors. Each picks uniformly. We want P(no collisions).
For there to be no collisions, all three ants must move in the same direction around the triangle — all clockwise, or all counterclockwise. Any other configuration has at least one pair of ants walking toward each other.
The probability of all three picking clockwise is (1/2)³ = 1/8. Same for all counterclockwise. So:
P(no collision) = 2 × (1/8) = 1/4.
The interview signal is whether the candidate sees the symmetry argument quickly. A candidate who tries to enumerate all 2³ = 8 configurations and check each one will get the right answer but burn three minutes; a candidate who says “they all need to go the same way around” gets the answer in 15 seconds.
The clockface version
Place an ant at the 12 of a clock. At each tick, the ant moves to an adjacent number (11 or 1) with probability 1/2 each. After 12 steps, what is the probability that the ant is back at 12?
This is a random walk on a cycle of length 12. The cleanest approach uses linearity and parity. After N steps, the ant’s net displacement is the difference between rightward and leftward steps, taken modulo 12. Returning to 12 requires the net displacement to be a multiple of 12.
For 12 steps with each step ±1, the net displacement is in {-12, -10, -8, ..., 0, ..., 10, 12}. Net displacement k happens when you take (12+k)/2 rightward steps and (12-k)/2 leftward steps. Net 0 has multiplicity C(12,6) = 924; net ±12 has multiplicity 1 each. Total configurations: 212 = 4096.
Returns to 12 require net displacement in {-12, 0, 12} (since 12 ≡ 0 mod 12). Total favorable configurations: 1 + 924 + 1 = 926.
P(return to 12 in exactly 12 steps) = 926 / 4096 ≈ 22.6%.
The clean way to extract this in an interview is to recognize that the problem is “random walk on a cycle with mod-12 wraparound” and compute the central binomial coefficient and the two endpoint cases.
The “never returns” variant
Now make the walk infinite. What is the probability the ant ever returns to 12?
This is the recurrence question for a 1D random walk, and the celebrated answer is: probability 1. A simple symmetric random walk on the integers (or on a cycle) is recurrent. The ant returns to its origin with probability 1, eventually.
The proof requires martingales or generating functions in full, but the interview-friendly version is: the expected number of visits to the origin in N steps grows like √N (from the Stirling approximation of central binomials), which goes to infinity, which means the walk is recurrent. A polished candidate sketches this without doing the full algebra.
The expected-time-to-return variant
Asked at quant interviews far more often: “what is the expected number of steps until the first return to 12?”
For 1D random walks the answer is infinite. The walk returns with probability 1 but the expected return time is unbounded — the walk is null recurrent. For random walks on a finite cycle of length n, the expected return time is exactly n (the stationary distribution is uniform, and by Kac’s lemma the expected return time is the reciprocal of the stationary probability).
So for the 12-position clockface, the expected return time to 12 is exactly 12 steps. That is a clean answer with a clean derivation, and a polished candidate can pull it out without writing anything.
Why this question family survives
Goldman Sachs has been asking the ant-on-clockface family for decades. So have Jane Street, Citadel, HRT, Two Sigma, DRW, Optiver, IMC, and SIG. The reason the question never went out of style is that the skill it tests — manipulating expectations, recognizing recurrence/transience, computing combinatorial probabilities under time pressure — is exactly the skill a quant trader uses every day. Pricing an option requires computing expectations on random walks. Hedging a portfolio requires reasoning about path dependencies. The ant problem is a toy version of the actual job.
The brainteaser-era critique that landed in tech (“this question does not predict performance”) never landed in finance, because in finance the question does predict performance. A trader who cannot solve a 1D random walk on a clockface in two minutes is not going to be able to size positions in real time when an event print breaks the market.
Variations that interviewers ask in 2026
- Two ants chasing. Two ants start at opposite ends of an n-cycle. At each step, each picks a direction. Expected time until they meet?
- Asymmetric walk. Probability p of clockwise, 1-p of counterclockwise. How does the answer change?
- Walk on a graph. The ant is on a cube, walking along edges. Expected time to visit all 8 vertices?
- 2D ant. The ant is on a 2D grid. Is the walk recurrent? (Yes — Pólya’s theorem for d ≤ 2.)
- 3D ant. Same question on a 3D lattice. (No — Pólya’s theorem says d ≥ 3 is transient. Approximately 34% recurrence in 3D.)
The 2D-vs-3D recurrence result (Pólya’s theorem) is one of the most-asked extension questions, and the polished answer is “Pólya proved in 1921 that simple random walks on Zd are recurrent iff d ≤ 2″, followed by an intuition about why dimensions ≥ 3 give the walk too much room to escape.
What a good answer looks like
The signal Goldman Sachs and the others are looking for has three layers:
- Setup speed. Can the candidate translate the words (“ant on clockface”) into a mathematical structure (random walk on Z/12Z) within 30 seconds?
- Clean computation. Can they identify the right tool (symmetry, generating function, martingale, Markov chain stationary distribution) and execute it without algebra errors?
- Generalization. When the interviewer adds a twist (“now the steps are biased”, “now the ant has to visit two specific positions”), can they adapt their tool or reach for a new one?
What makes the question hard is that it has to be done in real time, conversationally, often without paper. The interviewer is not just testing whether the candidate knows the answer — they are testing whether the candidate can think probabilistically while talking out loud, which is exactly the skill a junior trader needs on the desk.
Is it asked in 2026?
Yes, constantly, at every tier-1 quant firm and many tier-2 ones. The ant-on-clockface family is core to the Wall Street probability tradition and shows no signs of fading. If anything, the questions have gotten harder as candidates prepare more — the modern version often skips the simple ant and goes directly to “expected time to absorption on a graph with one trapping state” or “compute the partition function of this random walk in closed form”.
Frequently Asked Questions
Where do these questions come from?
Recreational mathematics, going back to Pólya in the early 20th century and earlier. The application to quant interviews stabilized in the 1980s and 1990s as quantitative trading firms started hiring math and physics PhDs and needed a fast probability filter.
Is the symmetric walk on a cycle always recurrent?
Yes, on any finite cycle. Recurrence on infinite Z is true for 1D and 2D walks but not for 3D and higher (Pólya’s theorem).
How do quant interviewees prepare?
The standard references are Mark Joshi’s Quant Job Interview Questions and Answers and the older Heard on the Street. Most candidates also drill probability problems on quantnet.com forums, glassdoor leaked questions, and direct practice with random walk and generating function problems.
Do tech interviews ask this?
Almost never. The tech industry retreated from probability puzzles after Google’s 2013 brainteaser mea culpa, and they were never as central to tech hiring as they are to finance. A senior ML engineer might be asked something probability-flavored, but it will be framed as a model-related question, not as an ant on a clockface.