The Objectives

The purpose of this assignment is to give you practice with:

• recursion
• using command line arguments
• using the timing feature
• and file handling.

The Background

According to The Math Forum @ Drexel University, A Magic Square is an arrangement of the numbers 1 to n² in an n x n matrix with each number occuring exactly once, and such that the sum of the entries of any row, any column, or any main diagonal is the same. It is not hard to show that this sum is n(n² + 1)/2.

The simplest magic square is the 1 x 1 magic square whose only entry is the number 1.

1

The next simplest is the 3 x 3 magic square.

(8 + 1 + 6) or 3(3² + 1)/2
This is also just ONE of many examples for a 3 x 3 Magic Square!!

This 3 x 3 square is definitely magic and satisfies the definition given above. There are 8 different arrangements of entries in a 3 x 3 matrix that fit the definition of a magic square. There are also magic squares of size 4 x 4, 5 x 5, etc.

The Background - Continued

For small n, say n <= 3, you could obviously sit down with pencil and paper and discover a magic square in a fairly short amount of time.
Take a few minutes now and do that, remembering that each of the numbers, 1 through n², appear only once in the square.

Now try it for n = 4 ...

Oh, there has to be a better way. :)

Of course, the computer doesn't have a patience problem. It can solve the problem for us by using the brute force method. This means that the computer will accomplish its task by trying every single possible arrangement of the numbers in a square and check each of the resulting squares to see if it's a magic square.

[6, 2, 7, 5, 4, 8, 3, 9, 1]
[4, 2, 7, 6, 5, 8, 3, 9, 1]
[7, 4, 2, 6, 5, 8, 3, 9, 1]
...

The Task

Your assignment will be to determine all of the magic squares when given an n, display permutations that match the magic squares to the screen AND write it to a file.

• You will only test when n == 3!!! But code it for n of ANY number!!
• It turns out that finding the magic squares for n >= 4 takes too long for us to run it on the shared machines (linux1, 2 & 3)
• Since if n == 1, the square is 1 (too easy!!)
• and since for n == 2 there is no answer!!

Your program must come up with every possible arrangement (permutation) of numbers within the square. These are known as the permutations of that list of numbers. Each permutation needs to be checked to see if it has formed a magic square.

Program Requirements

Your program must:
• use command line arguments. At the command line the user must enter (in this order):
  • the name of the executable file,
  • n (which will ALWAYS be 3, but again could be run with other numbers)
  • and the name of the output file in which to write the squares.
• use a recursive permute to give all of the permutations of the square.
  • In other words, your function called "permute", must be a recursive function.
  • So for each permutation, you should check to see if it is a magic square, display it and write it to the file
• write only the unique magic squares to the SCREEN and FILE. No duplicates allowed.
  • HINT: Let your program produce duplicates first, then see how you can refrain from writing the duplicates.
    Then modify your code.
• use a list of size n².
  • The code for the recursive permute function will be much easier to write if you store the numbers in a list
  • rather than a two-dimensional list (or array) of size n x n. How you print the data is not related.

  [6, 2, 7, 5, 4, 8, 3, 9, 1, ....]   (size to n2)
  [4, 2, 7, 6, 5, 8, 3, 9, 1, ....]
  [7, 4, 2, 6, 5, 8, 3, 9, 1, ....]
  ...
  

• close any files that you have opened as soon as you have finished using them.
• Time how long your program takes to find the magic square and print that time at the end of the SAME file.

Sample Run

Here is a sample of output of this program. Please note that the sample output only shows what is expected from your program if the user is actually entering everything as instructed. This is not a test of the program at all, but just a sample for your clarification.

2   7   6
9   5   1
4   3   8

2   9   4
7   5   3
6   1   8

4   3   8
9   5   1
2   7   6

4   9   2
3   5   7
8   1   6

6   1   8
7   5   3
2   9   4

6   7   2
1   5   9
8   3   4

8   1   6
3   5   7
4   9   2

8   3   4
1   5   9
6   7   2

Total running time: 4.76 seconds.

Submitting your work

You must be logged into your account and in the same directory as the file you are trying to submit.

To submit your design, type the following at the linux prompt:

submit cs201 Proj2 design2.txt

To submit your project, type the following at the linux prompt:

submit cs201 Proj2 proj2.py

To verify that your files were submitted, you can execute the following command at the Unix prompt. It will show all files that you submitted in a format similar to the Unix 'ls' command.

submitls cs201 Proj2