Sparse Magic Squares

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In December, 2000 Hercules Lovell created a new type of magic square that contained many blank cells. In other respects, it had the requirements to qualify it as a magic square. i.e. All rows, columns and the two main diagonals summed to the same value. He sent  sample order-8 and order-9 squares to Mutsumi Suzuki who put them on his web page.

 
    4 29     20 13
14 19     30 3    
    5 12     21 28
11 6     27 22    
    26 10     7 23
25 9     8 24    
    31 15     18 2
16 32     1 17    

Hercules Lovell 8x8  Dec. 2000

Uses numbers 1 to 32. Rows, columns and
 main diagonals sum to 66.

 
52 45     36 61    
    62 35     46 51
37 44     53 60    
    43 38     59 54
42 58     55 39    
    41 57     56 40
63 47     50 34    
    48 64     33 49

The numbers in the square on left are complemented, then flipped left to right.
Uses numbers 33 to 64. Rows, columns and
 main diagonals sum to 194.

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

The two previous squares combine to make a simple magic square with S = 260, using numbers from 1 to 64.

 

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

Hercules Lovell 8x8  Dec. 2000

Uses numbers 1 to 54. Rows, columns and main diagonals sum to 165.

 
  70       77 57    
    59 73       72  
61       78       65
  60       64 80    
    66 62       76  
75       71       58
  74       63 67    
    79 69       56  
68       55       81

Mutsumi Suzuki constructed this square to complement Lovell's 9x9.
Rows, columns and main diagonals sum to 204.
The two squares combined form a simple order-9 magic square.

Sometime later, Mutsumi received some similar squares from a Mr. Kobayashi who said he had published them in a local "Puzzle Research" newsletter in October, 1999..

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

The two 4x4 squares use duplicate numbers. Not possible to use distinct numbers? All rows, columns and main diagonals sum to 12 (2nd square 11)

The order-5 squares both use the distinct numbers 1 to 15. All rows, columns and main diagonals sum to 24. These squares cannot be made into regular magic squares by combining with a complementary sparse square, because there are only 5 pairs of numbers from 16 to 25 that sum to 41. Each pair has to supply 41 to a row and a column.

We have found with other types of magic squares, the principle can be applied to higher dimensions.

Who will be first to construct a Sparse Magic Cube?

This page was originally posted February 24, 2011
It was last updated February 24, 2011
Harvey Heinz   harveyheinz@shaw.ca
Copyright 1998-2011 by Harvey D. Heinz