The puzzle: three discs - Small, Medium, Large - are stacked on the Start peg, largest at the bottom. Move the whole tower to the Target peg. Rules: move one disc at a time, and never place a larger disc on a smaller one. The Spare peg may be used. Simplify first - solve the 2-disc version (Small and Large only, Start -> Target): 1) Small -> Spare. 2) Large -> Target. 3) Small -> Target, on top of Large. Three moves, and the tower stands on Target. Now the 3-disc version: work backward from the finish. The LAST move puts Large onto Target, and Large can only move when nothing sits on it and Target is clear for it - so just before that move, Small and Medium must both be stacked on the single remaining peg (Spare). Getting that 2-disc tower onto Spare is exactly the smaller problem, solved the same way. The full optimal solution: 1) Small -> Target. 2) Medium -> Spare. 3) Small -> Spare (onto Medium). 4) Large -> Target. 5) Small -> Start. 6) Medium -> Target (onto Large). 7) Small -> Target (onto Medium). Seven moves. Look for the pattern: minimum moves: 1 disc -> 1; 2 discs -> 3; 3 discs -> 7. Each step: double the previous count and add 1 (1 x 2 + 1 = 3; 3 x 2 + 1 = 7), so 4 discs need 7 x 2 + 1 = 15 moves. Say it plainly: the winning move in hard puzzles is usually made BEFORE the puzzle - shrink it, solve the small one, and let the pattern carry you up.

In the 7-move solution above, which disc moves on move 6, and where does it land? -> Medium - onto Large on the Target peg