# Time Complexity Analysis | Tower Of Hanoi (Recursion)

• Difficulty Level : Medium
• Last Updated : 27 Jan, 2022

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

```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.

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