Minimizing Ferry Occupancy After Final Stop

This brain teaser is about tracking a discrete quantity through repeated proportional reductions and fixed additions, under the constraint that everything stays integer at each stage. In a quant prep context, it sits at the intersection of number theory and simple algebra, but wrapped in an accessible, story-like setting that often appears in quant interviews.

It trains your ability to reason about integer structure, modular constraints, and backward propagation of conditions through multiple steps of a process. You practice recognizing when a dynamic system secretly encodes divisibility requirements and when a seemingly forward narrative is better attacked by working from the end back to the start.

This matters for quant interviews because many real quant problems hide algebraic structure inside repetitive rules or flows. Interviewers want to see you detect invariants, handle recursive setups cleanly, and reason precisely about discrete quantities rather than rely on hand-wavy approximations.

What it tests

This class of problems is governed by the principle of integer preservation through repeated affine transformations—specifically, when a quantity is repeatedly scaled by a rational factor and then incremented by an integer. The key is that, after each transformation, the result must remain an integer, which imposes divisibility constraints on the starting value. The structure is recursive: each stage's output becomes the next stage's input, and the divisibility required at the end propagates backward through all steps. The reason this pattern holds is that multiplying by a non-integer factor (like 1/4) introduces denominators, so unless the input at each stage is carefully chosen, the process will yield non-integers. Thus, the problem reduces to finding initial values that, after repeated rational scaling and integer shifting, still yield integers at the end.