CSCE 235
Handout
19: Induction and Recurrence for Tower of Hanoi
Assigned April 7, 2002
Induction
For the Tower of Hanoi problem with 3 pegs and n disks, we know
that the number of moves to solve the problem is 
. Now, using Mathematical
Induction, show that 
for all 
, 
, where 
is the number of
moves to solve a Tower of Hanoi problem with 3 pegs and n disks.
First, we want to show that the basis is true. When n = 1, 
. Thus, we have shown
that the basis is true by inspection.
Then, we want to show that the induction step is true. Given that the kth proposition, 
, is true, we want to show that the (k+1)th
proposition 
is true. We know that 
. So, if we can show
that 
is true, then we are
done. Here is how:
.
We have shown that induction step is true.
Thus, by the Principle of Mathematical Induction, we have shown that 
is true for all 
.
Recurrence
Solve the recurrence relation 
+1 by computing the explicit formula, given that 
, and 
. The explicit
formula is 
.
Here is the trick that we use
to do this. If we include the initial setup
of the puzzle as one move, that means, 
and 
. We also have 
. Now, we can try to
solve for the explicit formula. We now
have 
where a = 2
and b = 0, and 
and 
. Solving the
characteristic equation, we have 
or 
. Thus, 
and 
. Since we have two
solutions, we have (see Handout 18):
.
Since 
, 
. Thus, 
. Or, if we ignore 
and treat the problem
as if we have one distinct solution, then we have (see Handout 18):
Since we are now
including the initial position as one move, we know that 
and 
. So, we have:
Solving the above
equations for the constants, we have 
= 1, and 
= 0. Thus, we have 
!
Now, we need to finish
the trick. Remember that 
is the result of
counting the initial setup of the puzzle as one move. So, now we need to take that move out to obtain 
:
.