Recursion program
Problems
power

Write a
public static
function named
power
that takes in two integers,
a and
b, and returns
a^
b, the first argument to the power of the second argument. You may
NOT use
Math.pow()
, because that would be boring. 
The base case in this method is when the second argument is 0:
n^0 = 1, by definition. 
Hint: 3^5 = 3*(3^4)

Why is recursion not the best solution for this problem? If you do not know, ask a TA.

Create a similar method named
power2
that is functionally identical but does not use recursion. It should still NOT use
Math.pow()
.

fileCount

Write a
public static
function named
fileCount
that takes as input a directory (as a File object) and returns the total number of files in all subdirectories
Directories count as files, too.


Your method should look through all files and subdirectories in the directory that was passed in. While searching through the directory, use the
java.io.File.isDirectory()
method to check if the File you are looking at is a file or directory. If it is a normal file, count it. If it is a directory, count it and use a recursive call to count its contents. 
Hint:

What is the base case?

What is the recursive step?

What is the combination?


You should consult the
java.io.File API for more information. 
The following directory structure has 14 files:
root_dir 1 +dir 2  +file 3 +dir 4  +dir 5   +dir 6    +file 7   +file 8  +file 9 +dir 10  +dir 11   +file 12  +file 13 +file 14
hanoi

Write a
public static
function named
hanoi
that recursively solves the Tower of Hanoi puzzle (see the description below). 
Your function should take 1 integer and 3 chars as input (in this order):
n,
src,
dest,
aux.
n is the number of disks.

src (source),
dest (destination), and
aux (auxiliary) are tower letters. 
For example, to solve the 3 disk puzzle, you would call
hanoi(3,'A','C','B');

hanoi(3,'A','C','B');
means “move 3 disks from tower A to tower C using tower B”



Your function should print the solution to the puzzle to standard output in the following format:

“move [disk] from [tower1] to [tower2]”

[disk] is the number of the disk being moved. Disks are numbered 1 to
n, 1 being the smallest and
n being the largest. 
[tower1] is the letter of the tower
from which the disk is being moved. 
[tower2] is the letter of the tower
to which the disk is being moved.


For example: “move 1 from A to C” is the first step of solving the 3 disk problem.

Print each step on its own line.


The base case is the 0disk problem, for which nothing must be done (just return).

What is the recursive step?

What is the combination?


Hints:

Divide and Conquer.

How do you use the solution to the 1disk problem to help you solve the 2disk problem?

How about using the 2disk solution to solve the 3disk problem?

And so on.


The whole function is just 5 lines of code.

2 of these are the base case

The other 3 are the steps to use the (n1)disk solution to solve the ndisk problem.

About the Tower of Hanoi
There is a temple in Kashi Vishwanath which contains a large room with three timeworn posts, on which are stacked 64 golden disks. Brahmin priests, acting out the command of an ancient prophecy, have been moving these disks, in accordance with the immutable rules of the Brahma, since that time. When the last move of the puzzle is completed, the world will end. Despite the location of the legendary temple, the puzzle is most often called the Tower of Hanoi.
The objective of the puzzle is to move the entire stack of disks from the first post to the third post, obeying the following simple rules:

Only one disk may be moved at a time.

Each move consists of taking the upper disk from one of the stacks and placing it on top of another stack.

No disk may be placed on top of a smaller disk.
mysterySeries

Figure out the pattern from the table below.

Write a
public static
function named
mysterySeries
that take integers
i and
j as input and returns the (
i,
j)th mystery number (the number in the
ith row and
jth column).
mysterySeries(6,2)
should return 15


Base Cases:

If
i < 0 or
j < 0 or
i <
j, then A(
i,
j) = 0 
The above 3 base cases are
not sufficient, what other base case(s) do you need?


Hint: Each element of the
ith row can be computed from elements in the (
i1)th row.
A(i,j)  0  1  2  3  4  5  6  … 

0  1  
1  1  1  
2  1  2  1  
3  1  3  3  1  
4  1  4  6  4  1  
5  1  5  10  10  5  1  
6  1  6  15  20  15  6  1  
…  …  …  …  …  …  …  …  … 