Locks Pirates Can't Outsmart

This question describes a treasure safe protected by multiple locks, with keys distributed across a fixed group of pirates, under a strict majority rule. The access control requirement is combinatorial: any sufficiently large coalition must be able to open everything, while any smaller coalition must be blocked by at least one remaining lock. The candidate must reason about how to encode a size-based access structure using discrete objects (locks and keys), and how to minimize the physical resources needed. This style of construction appears in theoretical discussions of secret sharing and in some quant trading interviews that probe abstract combinatorial design.

The solution leans on combinatorial set systems, especially reasoning about subsets just below the threshold and how to treat them as minimal forbidden coalitions. It draws on inclusion–exclusion style thinking, binomial coefficients, and symmetry arguments to keep the construction minimal and elegant. An interviewer is watching for clear identification of the relevant family of subsets, correct mapping between subsets and locks, rigorous justification that both access conditions hold, and an argument that the number of locks and keys cannot be reduced further.

What it tests

This problem class is governed by the combinatorial method of threshold access structures, specifically the idea of designing a system where only groups above a certain size can achieve full access. The core principle is that for any group smaller than the threshold, there must exist at least one resource (here, a lock) inaccessible to them, while any group at or above the threshold always has access to every resource. This is achieved by associating each minimal forbidden group (those just below the threshold) with a unique resource they cannot access, and ensuring all others can. The pattern holds because the intersection properties of subsets guarantee that every larger group will always include at least one member with access to each resource, while smaller groups are always missing at least one key. This approach generalizes to any scenario where access control must be based on group size, not individual identity.