Problem of the Week #23: Monday June 12th, 2023 As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems. Please register for an account if you would like to join the discussion below or share your own problems.
Tower of Hanoi is a one-player game using three vertical rods and a number of disks, all of different diameters. It is set up so that all the disks are on one of the three rods, arranged by size with the largest disk on the bottom, as shown in this picture:
The goal of the game is to move the entire pile of disks onto one of the other two pegs follow these rules:
- Only one disk can be moved per step.
- The disk being moved must be from the top of its pile.
- The disk can only be moved to the top of a pile (can be an empty pile) that doesn’t contain any disk smaller than itself.
a). Try this game using 3 disks. What is the least number of steps it takes?
b). How many steps does it take to finish this game using 4 disks?
c). How many steps does it take to finish this game using n disks, where n is a natural number? Why?
d). If there are four rods instead of three, how many steps does it take to finish this game using 6 disks?
e). If there are four rods instead of three, how many steps does it take to finish the game using n disks, where n is a natural number?
f). If there are five rods, how many steps does it take to finish this game using 10 disks?
g). If there are k rods, how many steps does it take to finish this game using n disks, where k and n are both natural numbers?
h). Share your own problem inspired by this one.
i). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.