New Méziriac Pendulum Method (Part III)

Regular and Non-Regular Squares

A pendulum


A Discussion of the New Methods

An important general principle for generating odd magic squares by the Méziriac method is that the center cell must always contain the middle number of the series of numbers used, i.e. a number which is equal to one half the sum of the first and last numbers of the series, or ½(n2 + 1). The properties of these regular or associated Loubère squares are:

  1. That the sum of the horizontal rows, vertical columns and corner diagonals are equal to the magic sum S.
  2. The sum of any two numbers that are diagonally equidistant from the center (DENS) is equal to n2 + 1, i.e., or twice the number in the center cell and are complementary to each other.


The 5x5 and 7x7 regular Méziriac squares are shown below as examples:

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3 16 9 22 15
20821 14 2
72513 1 19
24125 18 6
11 4 17 10 23
                              
4 29 12 37 20 45 28
351136 19 44 27 3
104218 43 26 2 34
411749 25 1 33 9
16 48 24 7 32 8 40
47 23 6 31 14 39 15
22 5 30 13 38 21 46


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Méziriac squares are normally contructed using a stepwise approach where each subsequent number is added consecutively one cell at a time. In this new method each subsequent number is added using the pendulum approach. When a break is encountered this may be a single move (right or down) or a knight move as shown in the construction of the squares.

  1. Fill in the starting number and add the next numbers down the ladder followed by up the ladder until the broken diagonal is filled tracing out an arc of a pendulum or
  2. Fill in the starting number and add the next numbers up the ladder followed by down the ladder until the broken diagonal is filled again tracing out a pendular arc.

These new Méziriac squares, which I will label PMn* (center cell#) [LD or RU,small or large arc] where PMn* signifies a small or large arc pendulum move nxn Méziriac square with a certain center cell number, breaking either down or to the left depending where on the diagonal the opening 1 resides on.

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Construction of the 5x5 Méziriac Pendulum Magic Squares

5x5 Small Arc Squares

Group IA
  1. To generate the square, PM5* 15 [RU, small arc], place a 1 to the right of the center cell of a 5x5 square and fill in empty cells by advancing diagonally first up right, then left down continuously switching until blocked by a previous number.
  2. Move two cells down.
  3. Repeat the process until the square is filled, as shown below in squares 1-5.

       
1
                   
              2
          1    
      3        
                   
  ⇒  
2
4     7        
   6        2
8       1    
      3        
    5            
  ⇒  
3
4     7     11
   6        2
8       1    
      3     10
12 5     9    
  ⇒  
4
4 18 7     11
206    13 2
8   15 1 19
21143 17 10
12 5 16 9    
  ⇒  
5 PM5* 13 [RU,small arc]
4 18 7 25 11
20624 13 2
82215 1 19
21143 17 10
12 5 16 9 23


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Group IB

  1. To generate the magic square, PM5* 13 [LD, small arc], place a 1 to the right of the center cell of a 5x5 square and fill in empty cells by advancing diagonally first up right, then down left continuously switching until blocked by a previous number.
  2. Move two cells right.
  3. Repeat the process until the square is filled, as shown below in squares 1-5.

       
1
                   
              2
          1    
      3        
                   
  ⇒  
2
5     6        
   7        2
          1    
      3        
    4     8    
  ⇒  
3
5     6        
   7        2
9       1    
   113     10
12 4     8    
  ⇒  
4
5 18 6     14
167    15 2
9   13 1 17
   113 19 10
12 4 20 8 21
  ⇒  
5 PM5* 14 [RU,small arc]
5 18 6 22 14
16724 15 2
92513 1 17
23113 19 10
12 4 20 8 21


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5x5 Large Arc Squares

Group IIA

  1. To generate the semi-magic square, PM5* 12 [RU, big arc], place a 1 to the right of the center cell of a 5x5 square and fill in empty cells by advancing diagonally first up right, then down left continuously switching until blocked by a previous number.
  2. Move two cells right.
  3. Repeat the process until the square is filled, as shown below in squares 1-5.

       
1
2                
                 
          1    
                 
    3            
  ⇒  
2
2     8        
   7        4
          1    
      5     6
    3            
  ⇒  
3
2     8        
   7        4
9   12 1    
      5     6
11 3     10    
  ⇒  
4
2 16 8     15
20721 13 4
9   12 1 18
   145 17 6
11 3 19 10    
  ⇒  
5 PM5* 12 [RU,big arc]
2 16 8 24 15
20721 13 4
92512 1 18
23145 17 6
11 3 19 10 22


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Group IIB

  1. To generate the semi-magic square, PM5* 15 [DL, big arc], place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing diagonally first down left, then up right continuously switching until blocked by a previous number.
  2. Move two cells down.
  3. Repeat the process until the square is filled, as shown below in squares 1-5.

       
1
3                
                 
          1    
                 
    2            
  ⇒  
2
3     7        
              5
          1    
      4     6
    2            
  ⇒  
3
3     7        
   8    12 5
10       1    
      4     6
11 2     9    
  ⇒  
4
3 16 7     14
19821 12 5
10   13 1 17
   154 18 6
11 2 20 9    
  ⇒  
5 PM5* 13 [DL,big arc]
3 16 7 25 14
19821 12 5
102413 1 17
22154 18 6
11 2 20 9 23


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7x7 Méziriac Pendulum Squares Groups IA-IIB

Fully constructed 7x7 Méziriac pendulum squares one each and typical two move break are shown below. Again these squares belong to the set situated on the yellow diagonal, with the last two groups belonging to a semi-magic set.

IA
6 32 9 36 17 47 28
30838 19 49 27 4
104021 48 25 2 29
422046 23 1 31 12
18 44 22 3 33 14 41
43 24 5 35 13 39 16
26 7 34 11 37 15 45
      
IB
7 32 8 40 20 44 24
34938 21 46 22 5
113619 48 23 3 35
371749 25 1 33 13
15 47 27 2 31 14 39
45 28 4 29 12 41 16
26 6 30 10 42 18 43
      
IIA Semi-Magic
2 29 10 41 19 46 28
35936 17 48 26 4
114216 43 24 6 33
401849 23 1 31 13
20 47 25 7 30 8 38
45 27 5 32 14 37 15
22 3 34 12 39 21 44
      
IIB Semi-Magic
3 29 9 42 18 47 27
341036 16 49 25 5
124117 43 23 7 32
391948 24 1 30 14
21 46 26 6 31 8 37
44 28 4 33 13 38 15
22 2 35 11 40 20 45


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This completes this section on Méziriac pendulum squares (Part III). To continue this method see new Méziriac pendulum squares method (Part IV). To return to homepage.


Copyright © 2008 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com