Min Coins to Cover 1 to 100

This brain teaser is about designing an efficient "toolkit" of bills so that every dollar amount in a fixed range can be built from some subset of them. It lives at the intersection of discrete math and combinatorial design: you are trading off denomination size against flexibility. Top trading firms like this style of quant prep question because it looks simple but encodes a careful global coverage requirement that candidates must internalize from a short statement.

It trains your ability to reason about worst-case constructions, subset coverage, and how local denomination choices affect a full interval of targets. You practice spotting critical gaps, reasoning about bottleneck values, and building compact, globally consistent structures under a tight resource budget.

This matters for quant interviews because market-making, pricing, and hedging all involve constructing minimal, robust portfolios that span many scenarios. Interviewers want to see if you can turn a small, combinatorial specification into a clean, globally valid design without brute force.

What it tests

This problem class is governed by the concept of covering all target values within a range using the smallest possible set of elements, where each element can be used at most once per construction. The underlying structure is a variant of the subset sum problem, but with the additional constraint of minimizing the number of elements used in the hardest-to-represent case. The key is to identify the 'bottleneck' amounts—those that require the largest number of bills due to the denominations' gaps. The principle holds because the largest denominations efficiently cover high values, but the inability to combine them exactly for every target means smaller denominations must fill the gaps, often leading to a peak in the number of bills needed at amounts just below the largest denomination. The pattern is about balancing coverage (reach all targets) and efficiency (minimize bill count in the worst case), which is why the solution is dictated by the hardest-to-represent amounts, not the average case.