Recursion program

1. 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.
2. The base case in this method is when the second argument is 0:
   n^0 = 1, by definition.
3. Hint: 3^5 = 3*(3^4)
4. Why is recursion not the best solution for this problem? If you do not know, ask a TA.
   1. Create a similar method named
      power2 that is functionally identical but does not use recursion. It should still NOT use
      Math.pow().

1. 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 sub-directories
   1. Directories count as files, too.
2. Your method should look through all files and sub-directories 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.
3. Hint:
   1. What is the base case?
   2. What is the recursive step?
   3. What is the combination?
4. You should consult the
   java.io.File API for more information.
5. 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

1. Write a
   public static function named
   hanoi that recursively solves the Tower of Hanoi puzzle (see the description below).
2. Your function should take 1 integer and 3 chars as input (in this order):
   n,
   src,
   dest,
   aux.
   1. n is the number of disks.
   2. src (source),
      dest (destination), and
      aux (auxiliary) are tower letters.
   3. For example, to solve the 3 disk puzzle, you would call
      hanoi(3,'A','C','B');
      1. hanoi(3,'A','C','B'); means “move 3 disks from tower A to tower C using tower B”
3. Your function should print the solution to the puzzle to standard output in the following format:
   1. “move [disk] from [tower1] to [tower2]”
      1. [disk] is the number of the disk being moved. Disks are numbered 1 to
         n, 1 being the smallest and
         n being the largest.
      2. [tower1] is the letter of the tower
         from which the disk is being moved.
      3. [tower2] is the letter of the tower
         to which the disk is being moved.
   2. For example: “move 1 from A to C” is the first step of solving the 3 disk problem.
   3. Print each step on its own line.
4. The base case is the 0-disk problem, for which nothing must be done (just return).
   1. What is the recursive step?
   2. What is the combination?
5. Hints:
   1. Solve the puzzle by hand (
      one of many online versions or
      another).
   2. Divide and Conquer.
      1. How do you use the solution to the 1-disk problem to help you solve the 2-disk problem?
      2. How about using the 2-disk solution to solve the 3-disk problem?
      3. And so on.
   3. The whole function is just 5 lines of code.
      1. 2 of these are the base case
      2. The other 3 are the steps to use the (n-1)-disk solution to solve the n-disk problem.

There is a temple in Kashi Vishwanath which contains a large room with three time-worn 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:

1. Only one disk may 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.
3. No disk may be placed on top of a smaller disk.

1. Figure out the pattern from the table below.
2. 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
   i-th row and
   j-th column).
   1. mysterySeries(6,2) should return 15
3. Base Cases:
   1. If
      i < 0 or
      j < 0 or
      i <
      j, then A(
      i,
      j) = 0
   2. The above 3 base cases are
      not sufficient, what other base case(s) do you need?
4. Hint: Each element of the
   i-th row can be computed from elements in the (
   i-1)-th row.