Max Plane Cuts by 10 Lines

This quant interview question is a classic combinatorics puzzle about how arrangements grow when you keep adding more structure. It asks you to reason about how a configuration in the plane evolves under a constraint and to recognize that there is an underlying counting pattern, not just ad-hoc casework. In the context of quant prep, it introduces a clean, self-contained setting where you must see beyond the surface description to the abstract combinatorial object.

It trains your ability to recognize and manipulate combinatorial growth patterns and to translate a geometric description into a discrete counting framework. You practice generalizing from small cases to a fully general expression, spotting regularities, and articulating why they persist. It also reinforces rigor in arguing about maximal configurations and edge cases.

This matters for quant interviews because many trading, research, and systematic roles require fast, reliable reasoning about structure, scaling, and constraints. Top trading firms use questions like this to see whether candidates can discover patterns, justify them clearly, and move from examples to formulas under time pressure. It is excellent quant interview prep for sharpening combinatorial intuition and disciplined problem framing.

What it tests

This class of problems is governed by the principle of maximal incremental partitioning: each new object (here, a line) can create the most new regions when it intersects all previous objects in the maximal way allowed by the problem's constraints. The underlying structure is combinatorial: each new line can intersect all previous lines at most once, and each intersection divides a region, so the total number of regions is the sum of all possible ways to choose 0, 1, 2, ..., n lines (i.e., the sum of binomial coefficients up to n). This is why the formula $L_n = \frac{n(n+1)}{2} + 1$ emerges: it counts the initial region, plus all new regions created by each intersection. The pattern holds because every new intersection point corresponds to a new region, and the arrangement is always constructed to maximize these intersections.