Lozenge method of John Horton Conway

 

With the Lozenge method of John Horton Conway you get a magic square of odd order and you find all odd numbers in the (white) 'diamond' and all even numbers outside the diamond (in the dark area). The Lozenge method looks like the diagonal method.

 

Put the odd numbers in sequence cross upwards in the magic square. Start with number 1 exactly in the middle of the left column.

 

 

        5      
      3        
    1          
               
               
               
               
        5      
      3 9      
    1 7        
               
               
               
               
        5      
      3 9 15    
    1 7 13      
      11        
               
               
               
        5      
      3 9 15    
    1 7 13 19    
      11 17      
               
               
               
    1 21 65 57 25  
  39           39
5       5      
27     3 9 15    
65   1 7 13 19 25  
51     11 17 23    
21       21      

 

 

The remaining (even) numbers are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22 and 24.

 

In the second column (from left) you need 44 and in the fourth column you need 8. With 20+24 you get 44 and with 2+6 you get 8. To get a symmetric magic square, you must put 20 and 6 respectively 24 and 2 crosswards in the magic square. There are two possibilities and we choose:

 

 

    1 65 65 65 25  
  39           39
35     24 5 6    
27     3 9 15    
65   1 7 13 19 25  
51     11 17 23    
43     20 21 2    

 

 

Now puzzle the second and the fourth row in a similar way.

 

 

    33 65 65 65 45  
  39           39
35     24 5 6    
65   22 3 9 15 16  
65   1 7 13 19 25  
65   10 11 17 23 4  
43     20 21 2    

 

 

And now we put the remaining four numbers in the magic square:

 

 

    65 65 65 65 65  
  65           65
65   18 24 5 6 12  
65   22 3 9 15 16  
65   1 7 13 19 25  
65   10 11 17 23 4  
65   14 20 21 2 8  

 

 

Is all this puzzling really a method to construct a magic square?

 

Analyzing the magic square, I discovered that it is possible to construct a Lozenge magic square by using a specific row grid and column grid. Put in both grids the numbers 0 up to n-1 (that is number 0 up to 4 in the 5x5 Lozenge magic square) in the middle row,. Shift the row one by left respectively one by right in the row grid and reverse it in the column grid (to get all numbers in the magic square).

 

 

Take 1x number from row grid +1

 

    10 10 10 10 10  
  10           10
10   2 3 4 0 1  <--
10   1 2 3 4 0  <--
10   0 1 2 3 4  
10   4 0 1 2 3  -->
10   3 4 0 1 2  -->

 

 

+ 5x number from column grid

 

    10 10 10 10 10  
  10           10
10   3 4 0 1 2  -->
10   4 0 1 2 3  -->
10   0 1 2 3 4  
10   1 2 3 4 0  <--
10   2 3 4 0 1  <--

 

 

= 5x5 Lozenge magic square

 

    65 65 65 65 65  
  65           65
65   18 24 5 6 12  
65   22 3 9 15 16  
65   1 7 13 19 25  
65   10 11 17 23 4  
65   14 20 21 2 8  

 

 

Use this method to construct magic squares of odd order (= 3x3, 5x5, 7x7, ... magic square).

 

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5x5, Lozenge method.xls
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