The tower of Hanoi was a popular puzzle of the late nineteenth century. The puzzle includes three pegs and eight rings of different sizes placed in order of size, with the largest on thebottom, on one of the pegs. The goal of the puzzle is to move all of the rings, one at a time, without ever placing a larger ring on top of a smaller ring, from the first peg to the second, using the third as an auxiliary peg. a$)$ Use mathematical induction to show that the minimum number of moves to transfer n rings from one peg to another, with the rules we have described, is $2$n $1.$b$)$ An ancient legend tells of the monks in a tower with $64$ gold rings and $3$ diamond pegs. They started moving the rings, one move per second, when the world was created. When they finish transferring the rings to the second peg, the world will end. How long will the world last?