Optiver·Data Scientist·Technical Phone Screen·Intermediate

Intermediate

Jun 2026

Summary

Optiver data scientist interview hit me with a pretty deep algorithmic puzzle that I wasn't expecting to go so far into the math. The whole session was basically one extended problem with a lot of follow-up layers.

Questions Asked (1)

You have an m×n grid of lights where toggling a cell also flips its orthogonal neighbors. Given a starting configuration and a target, find the minimum number of toggles to reach the target, or determine that it's impossible.

Algorithms & Data StructuresTechnical Trade-offsSystem Design

Author's notes

This wrecked me a little because I started thinking BFS and they clearly wanted something more principled.

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AI HintsAI Generated

Suggested Approach

Model the problem as a system of linear equations over GF(2), where each variable represents whether a cell is toggled. Solve for the toggle vector that transforms the start to the target, then minimize the number of 1s in that vector. If the system is inconsistent, it's impossible.

Pro tip: Mention that the toggle matrix is symmetric and often has a unique solution for many grid sizes, but for certain dimensions (e.g., 5x5) there are multiple solutions; use Gaussian elimination to find the solution space and then minimize the Hamming weight.

1. Model as linear system over GF(2)

Define variables x_{i,j} ∈ {0,1} indicating whether to toggle cell (i,j). For each cell, the net toggle it receives is the sum (mod 2) of its own x and the x's of its orthogonal neighbors. This must equal the difference between start and target at that cell.

2. Set up equations and solve

Write the system as A x = b over GF(2), where A is the (mn)×(mn) toggle matrix and b is the vector of required flips. Use Gaussian elimination to determine if a solution exists and to find the solution space.

3. Find minimum weight solution

If solutions exist, the solution space is an affine subspace. Enumerate all solutions (or use linear programming over GF(2) with branch and bound) to find the one with the fewest 1s, i.e., minimum toggles.

4. Handle impossibility and complexity

If the system is inconsistent, return impossible. Discuss time complexity: Gaussian elimination is O((mn)^3) which is fine for small grids; for large grids, exploit structure (e.g., first row determines rest) to reduce to O(2^n * m).

Key Points to Mention

Lights Out puzzle is a classic linear algebra problem over GF(2).
The toggle matrix is symmetric and banded, which can be exploited for efficiency.
Minimum toggles corresponds to minimum Hamming weight solution.
For some grid sizes, the solution is unique; for others, there are multiple solutions.
Impossibility occurs when the target difference is not in the column space of A.
Alternative approach: BFS for small grids, but exponential; linear algebra is more scalable.

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