The strategy: to move n disks from peg A to peg C using peg B as a spare, you move the top n−1 disks to B, move the single biggest disk to C, then move those n−1 disks from B onto C. That same trick nests inside itself at every smaller size, which is why the whole thing unwinds into a clean pattern.

Minimum moves: 2⁷ − 1 = 127 (unavoidable — this is the mathematically optimal solution, no shorter one exists).

(https://xoxohth.com/thread.php?thread_id=5901055&forum_id=2#50118812)