Dynamic programming: minimum coins

Implement prototype(sizes, amount) and solve(sizes, amount). Unlimited copies of each positive integer size are available. Return the minimum count totaling amount exactly or -1 if impossible. amount 0 returns 0, including sizes [].

Sizes [1, 3, 4] and amount 6 return 2; [4, 6] and amount 5 return -1. Throw RangeError for negative amount or any size<=0.

Prototype: recursive choices without caching. Final: reuse smaller-amount answers in O(amount * sizes.length) time/O(amount) space. Preserve sizes on success and failure.

All numeric inputs and finite arithmetic results in this lesson are safe-integer Numbers. Use null for documented absence, not undefined or a numeric sentinel. Return actual booleans where the contract requires them. Do not add validation requirements beyond the documented inputs.

Run with Node.js 20+ from this directory:

node main.mjs --stage prototype
node main.mjs --stage final

Edit exercise.mjs; keep main.mjs unchanged. Both stages begin as unfinished learner functions. The website uses the same test functions in a separate browser worker. Tests check deterministic examples, boundaries, documented validation, input preservation, and stage-specific cases. Passing tests does not prove the required time complexity.