1. Computing

Doubly-Even Magic Squares

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Level: Beginner

Focus: Loops, Arrays, Methods

Doubly-Even Magic Squares Question

This programming exercise is a follow-on from Odd Magic Squares. You will need the program you made from that exercise or alternatively you could use the version that can be found on the Odd Magic Squares solution page.

You could be forgiven for thinking that I'm going to ask you to add a new method to the odd magic squares program so that it creates even magic squares (e.g., 2x2, 4x4, 6x6). However, when it comes to even numbers and magic squares things get a little bit tricky. Therefore the goal of this programming exercise is to create doubly-even magic squares. Yes that's right, just when you think numbers are either odd or even, someone comes up with the idea of doubly-even. These are numbers that are divisible by four. In terms, of our magic squares these are squares like 4x4, 8x8, 12x12, 16x16 etc..

To make a doubly-even magic square consider the sequence of numbers that make up the square. For example, a 4x4 square is made up of the numbers 1 to 16. If we fill up the square sequentially from the top left corner to the bottom right we would get:

  1  2  3  4 
  5  6  7  8 
  9 10 11 12
 13 14 15 16
 

The trick is to fill the squares according to a generic pattern. Split the magic square so that there is a mini-square at each corner and a large square in the center. The size of a mini-square is the length of the magic square divided by 4 (i.e., for a 4x4 it would be 4/4=1, therefore each mini-square is 1x1) and the size of the large square is double the size of the mini-square (i.e., 2x2):

For a 4x4 square the pattern is a 1x1 mini-square in each corner and a 2x2 square in the center:

 #      #
   #  #   
   #  #   
 #      # 
 

Only fill in the sequential numbers where there are squares in the pattern:

  1       4 
     6  7    
    10 11 
 13      16
 

With the remaining numbers in the sequence start at the top row and going from left to right count downwards filling in the squares (i.e., 15,14 fit into the first row, 12,9 into the second row and so on). What you should end up with is:

  1 15 14  4 
 12  6  7  9 
  8 10 11  5
 13  3  2 16
 

Program Requirements:

  • a user must be able to enter in the size of the magic square
  • they must only be allowed to enter in an odd or doubly-even number
  • add a new method to the odd magic squares program to create the doubly-even magic square
  • the program should display the square
  • the program should display the magic constant (i.e., the sum of all the numbers in a row or column or diagonal)

Can your program create the 8 x 8 magic square below?

  1  2 62 61 60 59  7  8
  9 10 54 53 52 51 15 16 
 48 47 19 20 21 22 42 41 
 40 39 27 28 29 30 34 33 
 32 31 35 36 37 38 26 25 
 24 23 43 44 45 46 18 17 
 49 50 14 13 12 11 55 56 
 57 58  6  5  4  3 63 64 
 

To get the most out of this question try and figure out the answer before coming back to read the solution on the next page.

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