# Time Complexity Analysis | Tower Of Hanoi (Recursion)

Tower of Hanoi is a mathematical puzzle where we have three rods and n disks. The objective of the puzzle is to move the entire stack to another rod, obeying the following simple rules:

1) Only one disk can be moved at a time.

2) Each move consists of taking the upper disk from one of the stacks and placing it on top of another stack i.e. a disk can only be moved if it is the uppermost disk on a stack.

3) No disk may be placed on top of a smaller disk.

Pseudo Code

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TOH(n, x, y, z) { if (n >= 1) { // put (n-1) disk to z by using y TOH((n-1), x, z, y) // move larger disk to right place move:x-->y // put (n-1) disk to right place TOH((n-1), z, y, x) } }

Analysis of Recursion

Recursive Equation : ——-equation-1

Solving it by Backsubstitution :

———–equation-2

———–equation-3

Put the value of T(n-2) in the equation–2 with help of equation-3

——equation-4

Put the value of T(n-1) in equation-1 with help of equation-4

After Generalization :

Base condition T(1) =1

n – k = 1

k = n-1

put, k = n-1

It is a GP series, and the sum is

, or you can say which is exponential

for 5 disks i.e. n=5 It will take 2^5-1=31 moves.