Lessons from Range Sum

What did you learn?

$1.$ Precomputation trades space for query speed. You spend $O(n)$ space to make queries $O(1)$.

$2.$ Cumulative sums let you compute any range sum by subtraction. Here's the core prefix sum trick.

$3.$ The $+1$ offset in prefix arrays simplifies edge cases. $prefix[0] = 0$ means you never special-case the start of the array. This pattern generalizes. Can you extend it to two dimensions?