The goal of this assignment is to write five short Java programs so that you get used to writing code and debugging it. Do five of the following seven exercises. All questions have equal weight. Feel free to do more than five, but we will choose which ones to grade if you submit more than the required amount.

1. Average value. Write a program Average.java that takes an integer N as a command line argument and uses Math.random() to generate N random numbers betwen 0.0 and 1.0, and then prints their average value. Do not print all the numbers. Only print the average. What do you expect the average to approach when N is large?

2. Distinct values. Write a program Distinct.java that takes three integer command line parameters a, b, and c and prints out the number of distinct values (1, 2, or 3) among a, b, and c.

3. What's your sign? Write a program Zodiac.java that takes two command line integers M and D and prints the Zodiac sign corresponding to month M (1 = January, 12 = December) and day D. Use the following table

   SIGN	FROM	TO
   Capricorn	December 22	January 19
   Aquarius	January 20	February 17
   Pisces	February 18	March 19
   Aries	March 20	April 19
   Taurus	April 20	May 20
   Gemini	May 21	June 20
   Cancer	June 21	July 22
   Leo	July 23	August 22
   Virgo	August 23	September 22
   Libra	September 23	October 22
   Scorpio	October 23	November 21
   Sagittarius	November 22	December 21

4. Checkerboard. Write a program Checkerboard.java that reads an integer, N from the command line, and prints out a two dimensional N-by-N checkerboard pattern with alternating spaces and asterisks, like the following 4-by-4 pattern.

   * * * * * * * * * * * * * * * *

5. Ordinals. Write a program Ordinals.java that takes a command line argument N and prints out the first N ordinals, followed by Hello.

   % java Ordinals 22
   1st Hello
   2nd Hello
   3rd Hello
   4th Hello
   5th Hello
   ...
   10th Hello
   11th Hello
   12th Hello
   13th Hello
   ...
   20th Hello
   21st Hello
   22nd Hello
   
   

   Hint: consider using (i % 10) and (i % 100) to determine when to use "st", "nd", "rd", or "th" for printing the ith Hello.

6. Prime counting function. The prime counting function π(N) plays an important role in number theory. It is defined to be the number of prime numbers less than or equal to N. Write a program PrimeCounter.java that reads an integer N from the command line and prints out π(N). For enrichment purposes only, here is a link with more information about the prime counting function.

7. Bohr radius. Write a program BohrRadius.java that uses Newton's method to find the radii where the probability of finding the electron in the 4s excited state of hydrogen is zero. The probability is given by: (1 - 3r/4 + r²/8 - r³/192)² e^-r/2, where r is the radius in units of the Bohr radius. Start Newton's method by reading a real number from the command line for the initial value of r. Hint: use the initial values r = 0, 5, and 13. Challenge: explain what happens if you use an initial value of r = 4, 10, and 12 (behavior may depend on your implementation).