Voleon·Software Engineer·Technical Phone Screen·Senior

SeniorPrefer not to say

Jun 2026

Summary

Voleon gave me a coding problem about simulating a Kac ring, which sounds like a physics toy but turns into a pretty tricky math/CS puzzle once you hit the O(1) constraint on kstep. The question was well-designed and clearly not off-the-shelf leetcode, which I respect even if it wrecked me a little.

Questions Asked (1)

Implement a Kac ring simulation with N positions on a circle, where balls move clockwise each step and flip color when leaving a marked position. The class needs a step() method, a kstep(k) method that runs in O(1) time even for k up to 10^18, and a color() method that returns (W - B) / N in O(1).

Algorithms & Data StructuresSystem DesignTechnical Trade-offs

Author's notes

The step() part was fine, just bookkeeping.

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AI HintsAI Generated

Suggested Approach

First, clarify the problem and define the state representation, then derive the O(1) k-step formula using the ring's periodicity, and finally implement the class with efficient updates. Emphasize the mathematical insight that the color difference evolves linearly and can be computed in constant time.

Pro tip: Mention that the Kac ring's behavior is periodic with period dividing 2N, so k can be reduced modulo 2N, making k-step O(1). This shows deep understanding and avoids unnecessary complexity.

1. Clarify the problem and define state

Restate the problem to ensure understanding: N positions, M marked, balls move clockwise, flip when leaving marked. Define state as array of ball colors and marked positions.

2. Derive the O(1) k-step formula

Observe that after each step, the number of white balls changes by a predictable amount based on marked positions. Derive that W - B evolves linearly, and the system is periodic with period 2N, so k can be reduced modulo 2N.

3. Implement step() and kstep(k)

For step(), update the state in O(N) by moving balls and flipping colors. For kstep(k), reduce k modulo 2N and then apply the precomputed transformation or directly compute the new W-B.

4. Implement color() in O(1)

Maintain the current W-B value as a variable updated during steps, so color() simply returns (W-B)/N.

5. Analyze complexity and trade-offs

Discuss time and space complexity: step() O(N), kstep() O(1) after O(N) preprocessing, color() O(1). Mention that storing the full state allows O(N) step but kstep can be optimized.

Key Points to Mention

Periodicity of the Kac ring with period 2N, allowing k modulo 2N reduction.
Linear evolution of the difference W - B, which can be computed in O(1) per step.
Maintaining a running sum of W - B to answer color() in O(1).
Handling large k up to 10^18 by using modulo arithmetic to avoid overflow.
Trade-off between storing full state for O(N) step and using mathematical shortcuts for kstep.
Edge cases: N=0, M=0, M=N, and ensuring correct modulo behavior for negative values.

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