3x3 Grid: Bulbs Not Touching

This quant interview question is about counting valid configurations in a small discrete system under a simple local restriction. You are given a tiny grid with binary states, and only certain global patterns are allowed once adjacency rules are enforced. It sits at the intersection of basic combinatorics and graph thinking, a common theme in quant prep and technical interviews for trading and research roles.

It trains your ability to translate a verbal constraint into a combinatorial structure, to recognize an independence condition on a graph, and to organize an exhaustive yet efficient enumeration. It also builds comfort with symmetry, invariants, and clean case partitioning under probability. That combination is central to strong quant interviews.

This matters because many top firms use similar grid, graph, or board-style probability puzzles to test discrete modeling skills. Being fluent with these patterns elevates your overall quant interviews performance and accelerates your quant prep.

What it tests

This problem class is governed by the principle of independent assignment under local adjacency constraints. When each element (such as a bulb) can be independently 'on' or 'off', but certain pairs cannot simultaneously be 'on' due to adjacency, the configuration count is no longer simply $2^n$. Instead, the problem reduces to counting independent sets in a graph, where each node represents a bulb and edges represent adjacency. The independence constraint means that for any valid configuration, no two adjacent nodes are both 'on'. This structure is common in combinatorial problems involving forbidden local patterns, and the count often requires recursive or case-based enumeration, exploiting symmetries or decomposing the grid.