Maximum Sum Circular Subarray

Problem

Given a circular integer array nums of length n, return the maximum possible sum of a non-empty subarray of nums.

Examples

The brute-force approach

Try all subarrays including wrapping ones. For wrapping: concatenate nums with nums and try all subarrays of length at most n.

max_sum = -inf
doubled = nums + nums
for i in range(n):
    cur = 0
    for j in range(i, i+n):
        cur += doubled[j]
        max_sum = max(max_sum, cur)
return max_sum

O(n²) — nested loops.

Spotting the pattern

This is a Kadane's Algorithm problem. The key question to ask yourself:

A wrapping subarray takes elements from the end and the start, skipping a middle portion. If you removed the middle portion (the minimum subarray), what's left?

Answering that is where it clicks, and it's exactly what the guided walkthrough below builds with you: the pattern reasoning, a progressive hint ladder that never spoils the answer, a row-by-row dry run, the optimized solution, and an in-browser editor to run your code against real test cases.