Problem

Source: Turkey Team Selection Test 2018 P3

Tags: combinatorics, permutation



A Retired Linguist (R.L.) writes in the first move a word consisting of $n$ letters, which are all different. In each move, he determines the maximum $i$, such that the word obtained by reversing the first $i$ letters of the last word hasn't been written before, and writes this new word. Prove that R.L. can make $n!$ moves.