Continuation of New Méziriac Knight-Break Pendulum Method (Part IV)

A pendulum

These new Méziriac squares, which I will label PMn* (center cell#) [LD or RU,small or large arc,(n1U,n2R)] where PMn* signifies a small or large arc pendulum move. nxn is a Méziriac square with a certain cell number, breaking in knight fashion down or to the left depending where on the diagonal the opening 1 resides on. n1 or n2 are the variable numbers a or b shown below.

This set of squares consists of six groups of small and large arc pendulum squares according to the following table:

where a = ½(n -3) and b = ½(n -1) and n is the order of the square.

Group C Group D
I (small arc)(aU,bL)I (small arc) (bU,aL)
II (large arc)(bU,aL)II (large arc) (aU,bL)
III (large arc)(aU,bL)III (large arc) (bU,aL)


Construction of 5x5 Knight-Break Méziriac Small Arc Pendulum Magic Squares

Group IC

  1. To generate the semi-magic square, PMK5* 14 [RU,small arc,(1U,2L)] , place a 1 into the center of the first row 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 in knight fashion one cell up and two cells left.
  3. Repeat the process until the square is filled, as shown below in squares 1-5.

       
1
4                
              2
          1    
      3     6
    5            
  ⇒  
2
4     10        
   9        2
7       1    
      3     6
11 5     8    
  ⇒  
3
4 16 10     13
   9    15 2
7   14 1    
   123     6
11 5     8    
  ⇒  
4
4 16 10     13
18921 15 2
7   14 1 20
   123 19 6
11 5 17 8    
  ⇒  
5
4 16 10 22 13
18921 15 2
72314 1 20
25123 19 6
11 5 17 8 24


********************************************************************************************************************************************************
Group ID

  1. To generate the semi-magic square, PMK5* 15 [LD,small arc,(2U,1L)] , 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 in knight fashion two cells up and one cell left.
  3. Repeat the process until the square is filled, as shown below in squares 1-5.

       
1
5                
              3
          1    
      2     6
    4            
  ⇒  
2
5     9        
   10        3
8       1    
      2     6
11 4     7    
  ⇒  
3
5 16 9     12
   10    14 3
8   15 1    
   132     6
11 4     7    
  ⇒  
4
5 16 9     12
171021 14 3
8   15 1 19
   132 20 6
11 4 18 7    
  ⇒  
5
5 16 9 23 12
171021 14 3
82215 1 19
24132 20 6
11 4 18 7 25


********************************************************************************************************************************************************

Construction of 5x5 Knight-Break Méziriac Large Arc Pendulum Magic Squares

Group IIC

  1. To generate the magic square, PMK5* 13 [RU,large arc,(2U,1L)], 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 in knight fashion two cells up and one cell left.
  3. Repeat the process until the square is filled, as shown below in squares 1-5.

       
1
2                
   6        4
          1    
      5        
    3            
  ⇒  
2
2     9     11
   6        4
10       1    
      5     8
    3     7    
  ⇒  
3
2     9     11
   6    15 4
10   13 1    
   125     8
14 3 16 7    
  ⇒  
4
2 20 9     11
186    15 4
10   13 1 17
21125 19 8
14 3 16 7    
  ⇒  
5
2 20 9 23 11
18622 15 4
102413 1 17
21125 19 8
14 3 16 7 25


********************************************************************************************************************************************************
Group IID

  1. To generate the magic square, PMK5* 15 [LD,large arc,(1U,2L)], 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 in knight fashion one cell up and two cells left.
  3. Repeat the process until the square is filled, as shown below in squares 1-5.

       
1
3     6        
              5
          1    
      4        
    2            
  ⇒  
2
3     6        
   9        5
7       1    
   114     8
    2     10    
  ⇒  
3
3     6     12
169    13 5
7   15 1    
   114     8
14 2     10    
  ⇒  
4
3 20 6     12
169    13 5
7   15 1 19
   114 17 8
14 2 18 10 21
  ⇒  
5
3 20 6 24 12
16922 13 5
72315 1 19
25114 17 8
14 2 18 10 21


********************************************************************************************************************************************************
Group IIIC

  1. PMK5* 11 [RU,large arc,(1U,2L)], No magic squares possible.
Group IIID

  1. PMK5* 14 [LD,large arc,(2U,1L)], No magic squares possible.
********************************************************************************************************************************************************

7x7 Méziriac Knight Pendulum Squares Groups IC-IID

Fully constructed 7x7 knight Loubère pendulum squares one each and typical one move break are shown below. Again these squares belong to the set situated on the yellow diagonal.

Group IC and Group ID
PMK7* 27 [RU,small arc,(2U,3L)]        PMK7* 28 [LD,small arc,(3U,2L)]
IC Semi-Magic
6 29 14 37 19 46 24
311336 21 44 26 4
113820 43 28 2 33
401845 27 1 35 9
16 47 25 3 34 8 42
49 23 5 32 10 41 15
22 7 30 12 39 17 48
      
ID Semi-Magic
7 29 13 38 18 47 23
301436 20 45 25 5
123721 43 27 3 32
391944 28 1 34 10
17 46 26 2 35 8 41
48 24 4 33 9 42 15
22 6 31 11 40 1649


********************************************************************************************************************************************************
Group IIC and Group IID
PMK7* 27 [RU,large arc,(3U,2L)]        PMK7* 26 [LD,large arc,(2U,3L)]
IIC
2 32 13 36 21 47 24
34842 19 45 23 4
144017 44 25 6 29
381646 27 1 35 12
18 48 22 7 33 10 37
43 28 5 31 9 39 20
26 3 30 11 41 15 49
      
IID
3 32 8 40 16 48 28
301342 17 46 22 5
113619 44 27 7 31
412145 25 1 33 9
15 47 23 6 35 10 39
49 24 4 29 12 37 20
26 2 34 14 38 1843


********************************************************************************************************************************************************
Group IIIC and Group IIID
PMK7* 24 [RU,large arc,(2U,3L)]        PMK7* 25 [LD,large arc,(3U,2L)]
IIIC
2 35 11 40 20 45 22
331338 15 44 28 4
83721 46 26 6 31
391948 24 1 30 14
17 43 23 7 32 12 41
49 25 5 34 10 36 16
27 3 29 9 42 18 47
      
IIID
3 35 13 37 19 43 25
291138 21 48 23 5
94015 46 24 7 34
422044 26 1 32 10
18 45 28 6 30 12 36
47 22 4 31 14 41 16
27 2 33 8 39 1749


********************************************************************************************************************************************************

9x9 Méziriac Knight Pendulum Squares Groups IC-IIID

Fully constructed 9x9 knight Loubère pendulum squares one each and typical one move break are shown below. Again these squares belong to the set situated on the yellow diagonal. These three groups are semi-magic and only groups I contain at least one magic square with S of 369 as shown in the last table.

Group IC, Group ID, Group IIID

PMK9* 44 [RU,small arc,(3U,4L)]        PMK9* 45 [LD,small arc,(4U,3L)]        PMK9* 42 [LD,large arc,(4U,3L)]
IC Semi-Magic
8 46 18 56 2567 327839
48 17 55 27 6534 76416
15 57 26 64 3674 43450
59 24 66 35 7345 25213
226833 7544 1 54 1161
70 31 77 42 353 106320
29 79 40 5 5112 621972
81 38 7 49 1460 217128
37 9 47 16 5823 693080
      
ID Semi-Magic
9 46 17 57 2468 317938
47 18 55 26 6633 77407
16 56 27 64 3575 42549
58 25 65 36 7344 35114
236734 7445 1 53 1260
69 32 76 43 254 106221
30 78 41 4 5211 631971
80 39 6 50 1361 207228
37 8 48 15 5922 702981
      
IIID Semi-Magic
3 49 17 63 2365 337343
47 15 55 25 6631 80455
13 62 27 68 2978 37748
60 19 70 30 7644 95011
267232 7442 1 52 1258
64 34 75 40 854 145624
36 77 38 6 4616 572271
79 39 4 53 1859 206928
41 2 51 10 6121 673581


9x9 Cell Values and S
Center Value of IC or IDS + ddCenter Value of IIIDS + dd
4540536423789
37333-3641354-15
4439627443756
39351-18393789
42378937354-15
413690383756
40360-9453789
433871840354-15
38342-27433756


This completes this section on Méziriac knight-break pendulum squares (Part IV). To see a new Méziriac regular and variable knight-break full pendulum methods (Part II). To return to homepage.


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