Exhaustive Enumeration and Approximate Solutions

The first genuinely useful algorithm this vault writes is also its simplest: try every candidate answer in turn until one works, or the search space runs out. This "guess and check by brute force" strategy — exhaustive enumeration — looks naive, but it is often the correct first algorithm to reach for: easy to implement, easy to verify, and, for many problems of modest size, fast enough. This note builds it twice — first as a while loop finding a perfect cube root, then as an equivalent for loop — and then extends it to an approximate root-finding problem, where the goal shifts from "find the exact answer" to "find an answer within a tolerance $\epsilon$ of the true one". That shift in goal — from exact to approximate, with a named tolerance — is the pattern behind essentially every iterative numerical routine in this vault.

Learning objectives. By the end of this note you should be able to: