Derivation of the formula of Hanoi pass term

Source: Internet
Author: User

Hanoi pass Formula Proof:

Set three towers for a, B, C, respectively. When a tower has n plates initially, it needs T (n) steps to transfer to Tower C.

First, there are the following rules:

T (0) = 0 (of course 0 when there is no plate)

T (1) = 1 

T (2) = 3

T (3) = 7

... /p>

T (n) = t (n-1) + 1 + t (n-1) = (n-1) + 1

Why t (n) = 2 * t (n-1) + 1?

It is easy to think that when n = n-1,

(1) The number of steps required to move all plates from tower A to Tower C is T (n-1).

  (2) If the number of steps moved from a to C is T (n-1), then moving from A to B also requires T (n-1)

So when n = N:

(1) First move all trays of a tower to Tower B, need t (n  -1 Step

(2) moves the last plate of tower A to tower C, which requires 1 steps

(3) to move all trays of Tower B to tower C, requiring t (n  -1) steps. The

Final result requires a T (n  -1) + 1 step.

So T (n) = 2 * t (n  -1) + 1 

What is the general formula? How do you prove it?

is simple ~

To add 1 to both ends of the equation:

T (n) + 1 = 2 * (t (n-1) + 1)

Set T (n) + 1 = S (n)

Then: s (n) = 2 *s (n-1)

And when n = 1, S (1) = 1;

Then s (n) = 2 ^ n

So: T (n) + 1 = S (n) = 2 ^ n

That is   T (n) = 2 ^ n-1


is completed.

Hanoi formula derivation

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