Morgan Stanley·Software Engineer·Technical Phone Screen·Intermediate

Intermediate

Apr 2026

Summary

Morgan Stanley quant engineer interview with a brain-melting geometry/probability puzzle that I was not remotely prepared for. The kind of question where you either know the trick or you're just sitting there drawing circles in your head for 20 minutes.

Questions Asked (1)

A dog runs around the boundary of a unit circle at speed 4. You're somewhere inside the circle and can move at speed 1. The dog always moves toward whichever point on the boundary you're projected to reach first. Can you escape, meaning reach the boundary at a point where the dog isn't? If so, describe a strategy and prove it works.

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

I stared at this for way too long before saying anything useful.

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AI HintsAI Generated

Suggested Approach

First, clarify the problem and identify key variables: dog's speed (4), your speed (1), and the dog's strategy (always moving toward your projected boundary point). Then, propose a strategy: move to the center, then spiral outward while staying opposite the dog, and finally make a dash when close enough. Prove that the strategy works by showing you can maintain angular separation and reach the boundary before the dog.

Pro tip: Draw a diagram and use polar coordinates to explain the strategy; this makes the angular separation and radial movement intuitive and demonstrates strong analytical skills.

1. Understand the problem

Restate the problem in your own words, define the dog's behavior (always moves toward your projected boundary point), and note the speeds: dog 4, you 1.

2. Develop a strategy

Start at the center, then move outward while maintaining an angle opposite the dog. Use the fact that near the center, your angular speed can exceed the dog's, allowing you to get opposite the dog.

3. Prove the strategy works

Show that you can reach a radius where your angular speed equals the dog's, then move radially outward while keeping the dog opposite. Finally, when close enough to the boundary, dash to a point far from the dog.

4. Address edge cases and assumptions

Consider if the dog can change direction instantly, if you start at the center, and if the dog's strategy is optimal. Discuss whether the strategy works if the dog anticipates your move.

5. Conclude and summarize

State clearly that escape is possible, summarize the strategy, and highlight the key mathematical insights (angular speed, radial dash).

Key Points to Mention

The dog's strategy: always moves toward your projected boundary point.
Angular speed: your angular speed at radius r is 1/r, dog's is 4 (since radius=1).
You can outrun the dog angularly when r < 1/4, allowing you to get opposite the dog.
Maintain opposite position while moving radially outward until r = 1/4.
From r = 1/4, dash to the boundary; the dog must travel half the circumference (π) while you travel 3/4, so you arrive first.
The strategy works regardless of the dog's starting position because you can always get opposite.

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