Given two lowercase words, find the fewest single-character edits — inserting, deleting, or swapping one character for another — needed to turn the first word into the second. Every partial answer only depends on smaller versions of the same question: how many edits does it take to turn some prefix of word1 into some prefix of word2? Once the last pair of letters is decided — either they already agree and cost nothing, or one of the three edits has to cover the mismatch — what's left is exactly the same problem on shorter prefixes. Solving every prefix pair once, smallest first, and reading off three already-known neighbors for each new cell turns an otherwise exponential search into a single pass over a grid.