Pumpkin Weight Squared Sum

This quant interview question is a clean linear algebra puzzle wrapped in a simple story. You are given partial aggregate information and asked to infer something structural about the underlying variables. It sits at the intersection of systems of equations and vector norms, which makes it ideal quant prep material for candidates reviewing linear algebra for interviews.

It trains your ability to encode a word problem as a linear system, reason about uniqueness of solutions, and connect pairwise relationships to global quantities like squared norms. It also exercises algebraic manipulation, comfort with symmetry, and checking consistency of given data, all of which are central to quantitative problem solving.

This matters for quant interviews because many trading, research, and data problems look exactly like this: reconstructing hidden parameters from incomplete aggregated observations. Top trading firms use such questions to see if you can translate intuition into precise linear structure quickly and reliably, a core skill in quant interviews.

What it tests

Whenever you are given the sums of all pairs of elements from a set of unknowns, you can reconstruct the individual values by treating the pairwise sums as a system of linear equations. For three unknowns, each pair sum gives a linear equation, and the system is always solvable (if consistent) because you have as many equations as unknowns. The key is that each unknown appears in exactly two of the three pairwise sums, so by combining and manipulating these equations (adding, subtracting), you can isolate each variable. This pattern generalizes: for $n$ unknowns, knowing all $n(n-1)/2$ pairwise sums gives you enough information to reconstruct the original set, provided the system is consistent. The reason this works is that the mapping from individual values to pairwise sums is linear and invertible (except for pathological cases like repeated values or inconsistent data).