The Bands Are a Ladder

A partial-scoring problem is really a stack of problems with a shared statement. The scoring bands are not consolation prizes — they are a design document:

• "n ≤ 20" → the full 2^n enumeration is intended for that band.
• "n ≤ 2000" → O(n²) is invited.
• "n ≤ 200000, no updates" → static prefix structures.
• "values ≤ 10^9" → coordinate compression or 64-bit.
• "all a_i equal" / "a_i ≤ 2" → degenerate-case specialization.

Reading order

1. Read the full statement once.
2. Read the bands top-down from the easiest: what algorithm does each band's constraint permit? Name it. That is the band's intended solution.
3. Estimate implementation time per band, in order. Implement band 1 completely before thinking about band 2.
4. The bands often teach the full solution: the n² band's DP frequently becomes the full solution's recurrence with a data structure grafted on.

The scoring arithmetic

If band 1 is worth 20 and costs 10 minutes, and the full solution is uncertain and costs 90, the expected-value arithmetic is not close. Bank the certain band first; you can always upgrade later — you cannot submit an empty file at the end.