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Let f( n ) be the minimal number of steps to solve the Tower of Hanoi game ...

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Let f( n ) be the minimal number of steps to solve the Tower of Hanoi game with n disks. f( n ) can be defined recursively as follows:

The basis step: f(1)=1
The recursive step: f( n )=2f(n-1)+1, when n > 1

Use mathematical induction to prove that, for any positive integer n, f( n ) = 2^n-1.

Using the direct proof technique, we assume that $P( k ) \equiv T$ P( k ) \equiv T so f( k ) = 2^k-1.

Inductive step: to prove that $\forall k ( P( k ) \to P( k+1 ) ) \equiv T$ \forall k ( P( k ) \to P( k+1 ) ) \equiv T.

For P( 1 ), since f(1)=1=2^1-1, $P( 1 ) \equiv T$ P( 1 ) \equiv T.

Define predicate P( n ) as the equation f( n ) = 2^n-1, where the domain of n is Z^+ and the smallest element in the domain is 1.

proof

For P( k+1 ), using the recursive step, we have $f(k+1) = 2f(k)+1=2\times 2^k -2 + 1 = 2^{k+1} - 1$ f(k+1) = 2f(k)+1=2\times 2^k -2 + 1 = 2^{k+1} - 1, so $P( k+1 ) \equiv T$ P( k+1 ) \equiv T.

To use the UG rule, we consider an arbitrary positive integer k.

Basis step: to prove that $P( 1 ) \equiv T$ P( 1 ) \equiv T.

QED

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