Tower of Hanoi

The game The Tower of Hanoi is basically a mathematical game or puzzle that starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape. Our team tried the game for several times but always takes more steps to complete it. At the end of this game we find an equation show the pattern of this game and can we complete the game perfectly. The equation we prove out is Tn=2^n -1.  The two steps are to show true for n=1 and assume the statement is true for n=k, and prove true for k+1. After we followed these steps to show that k+1 is true and it follows from mathematical induction that the statement is true for every positive integer. For the recursion and mathematical induction, I learned that it is a form of direct proof, and it is done in two steps. The first step, known as the base case, is to prove the given statement for the first natural number. The second step, known as the inductive step, is to prove that the given statement for any one natural number implies the given statement for the next natural number.