Morgan Stanley·Data Scientist·Technical Phone Screen·Intermediate

Intermediate

Jun 2026

Summary

Interviewed for a Data Scientist role at Morgan Stanley and got hit with a pure math/logic puzzle that felt more like a quant interview than anything data science related. Wasn't expecting it at all.

Questions Asked (1)

A dog runs along the boundary of a unit circle at speed 4. You start somewhere strictly inside the circle and can move anywhere inside the disk at speed 1. If you reach the boundary at the same point and time as the dog, you're caught. Can you guarantee an escape, and if so, how?

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

This wrecked me for a few minutes.

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AI HintsAI Generated

Suggested Approach

Model the problem as a pursuit-evasion game on a disk, where the dog's position is constrained to the boundary and yours to the interior. Show that by maintaining a specific geometric relationship (e.g., staying diametrically opposite or at a safe angular distance) you can always reach the boundary at a point different from the dog's, or argue that the dog can always intercept you. Use relative speeds and angular velocities to prove your strategy.

Pro tip: Quantify the angular speed advantage: at radius r, your maximum angular speed is 1/r, while the dog's is 4. This means you can outmaneuver the dog only when r < 1/4. Use this threshold to structure your strategy and demonstrate analytical rigor.

1. Understand the problem and constraints

Restate the problem: you start anywhere strictly inside the unit circle, move at speed 1, and the dog runs along the boundary at speed 4. You are caught only if you reach the boundary at the same point and time as the dog. Determine whether you can guarantee escape.

2. Analyze angular speeds and relative motion

Compute your maximum angular speed at radius r: ω_you = 1/r. The dog's angular speed is ω_dog = 4 (since radius=1). Compare these to find the critical radius r* = 1/4 where you can match the dog's angular speed.

3. Develop a strategy to avoid capture

If you can reach a radius less than 1/4, you can outrun the dog angularly and position yourself diametrically opposite. Then sprint radially outward to the boundary; the dog must cover half the circumference (π) while you cover at most 1 - r. Check if you can reach the boundary before the dog.

4. Prove the strategy works or fails

Show that from the center, you can reach the boundary in time 1, while the dog needs time π/4 ≈ 0.785 to reach any point. Since 1 > 0.785, the dog can intercept you if you go straight. However, by first moving to a smaller radius and aligning opposite, you can increase the dog's required travel distance and reduce yours.

5. Conclude and generalize

Conclude that escape is possible by maintaining a safe angular separation and exploiting the critical radius. Discuss the general principle: the evader can escape if their speed is greater than the pursuer's speed times the ratio of the radius of the safe zone to the boundary radius.

Key Points to Mention

Angular velocity comparison: your angular speed at radius r is 1/r, dog's is 4.
Critical radius r* = 1/4 where angular speeds match.
Strategy: move to radius < 1/4, align diametrically opposite, then sprint to boundary.
Time calculation: dog must cover π radians at speed 4, taking π/4 ≈ 0.785; you cover 1 - r at speed 1, taking 1 - r. Need 1 - r < π/4.
Choose r sufficiently small (e.g., r < 1 - π/4 ≈ 0.215) to ensure escape.
Generalization: escape possible if your speed > dog's speed * (r_safe / R).

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.