Two Sum - The Equation

You're solving $nums[i] + nums[j] = target$ for indices $i$ and $j$. Rearrange to see what you need:

$nums[j] = target - nums[i]$

This tells you: for each number in the array, check if the complement ($target - nums[i]$) exists elsewhere in the array. This algebraic rearrangement is the key insight.