Challenge: Interval DP

Start with the idea

Solve the exact task, not just the visible example: Return maximum coins from bursting balloons. On input [3,1,5,8], predict the output before opening the answer reveal. The revealed output is a test oracle for this case, while private checks exercise other legal cases. Start by stating the input grammar and output format in your own words. Parse the complete input, including surrounding whitespace, before relying on any convenient positions in the example.

Build the solution around one maintained claim: Solve increasing interval lengths and choose the last or first action inside each interval. Decide what state represents that claim, how it is initialized, what a single update does, and when the algorithm stops. A multidimensional or structured DP state records the information needed to identify a subproblem: two prefixes, an interval, a bitmask, a tree node, or a resource limit. Trace the visible input by hand and record at least two intermediate states, not only the final answer. If the trace cannot justify your predicted output, there is a gap between the idea and the implementation.

Use the Rust sketch as a guide, not a substitute for a complete solve function. The supplied Rust sketch is a starting point. Identify what it stores, how one iteration changes it, and whether it already solves the exact task or only demonstrates a related operation. Index tables consistently where the state is tabular.