New Méziriac Full Pendulum Method (Part II)

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:

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

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 full pendulum approach. When a break is encountered this may be a single move (right or down) or a variable 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.

The new Méziriac squares, which I will label PMn* (center cell#) [LD or RU,full arc] where PMn* signifies a full arc pendulum move nxn Méziriac square with a certain center cell number, breaking either down or to the right. All squares in this group are magic except for those where n is divisible by three which are neither magic or semi-magic.

The new Méziriac Knight break squares, which I will label PMn* (center cell#) [LD or RU,full arc,(n1U,n2L)] where PMn* signifies a full arc pendulum move nxn Méziriac square with a certain center cell number, with a variable knight move breaking n1 up then n2 left. All squares in this group even where n is divisible by three are semi-magic.

Construction of the 5x5 Méziriac Full Pendulum Magic Squares

5x5 Full Arc Squares

Group IA
  1. To generate the square, PM5* 15 [RU, full arc], place a 1 to the right of the center cell of a 5x5 square and fill in empty cells by advancingcontinuously diagonally up right, then continuously down left 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
6 2
1
4
5
2
3 7 11
6 2
9 1
4 10
5 8
3
3 7 11
6 14 2
915 1
134 10
12 5 16 8
4
3 19 7 11
206 14 2
915 1 18
21134 17 10
12 5 16 8
5 PM5* 14 [RU,full arc]
3 19 7 25 11
20623 14 2
92215 1 18
21134 17 10
12 5 16 8 4
Group IB
  1. To generate the magic square, PM5* 14 [LD, full arc], place a 1 to the right of the center cell of a 5x5 square and fill in empty cells by advancing continuously diagonally down left, then continuously up right 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
5 6
4
6 1
2
3
2
5 6
7 4
8 1
112 10
3 9
3
5 6 13
167 15 4
814 1
112 10
12 3 9
4
5 19 6 13
167 15 4
814 1 17
112 18 10
12 3 20 9 21
5 PM5* 14 [RU,full arc]
5 19 6 22 13
16723 15 4
82514 1 17
24112 18 10
12 3 20 9 21

7x7 Full Arc Squares

Group IA
  1. To generate the semi-magic square, PM23* 12 [RU, full 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 down.
  3. Repeat the process until the square is filled, as shown below in squares 1-4.
1
4 9
8 3
12 2
1 13
5 14
6 11 16
7 10 15
2
4 9 19 28
8 20 25 3
1221 24 2 29
18 231 13
17 22 5 14
266 11 16
27 7 10 15
3
4 31 9 36 19 28
30840 20 25 3
124121 24 2 29
4218 231 33 13
17 22 5 34 14 39
43266 35 11 38 16
27 7 32 10 37 15
5 PM5* 23 [RU,full arc]
4 31 9 36 19 48 28
30840 20 49 25 3
124121 46 24 2 29
421845 231 33 13
17 44 22 5 34 14 39
43266 35 11 38 16
27 7 32 10 37 15 47
Group IB
  1. To generate the semi-magic square, PM7* 24 [DL, full arc], place a 1 into the center of the first row of a 5x5 square and fill in empty cells by advancing continouosly diagonally down left, then continuously up right 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 1-4.
1
7 8
9 6
10 5
1 11
15 2 14
3 13
4 12
2
7 8 18 26
9 21 22 6
1020 23 5
19 241 11
15 25 2 14
283 29 13 16
27 4 12 17
3
7 31 8 41 18 26
32940 21 22 6
103620 23 5 35
3719 241 34 11
15 25 2 33 14 38
283 29 13 39 16
27 4 30 12 42 17 43
4 PM7* 24 [DL,full arc]
7 31 8 41 18 44 26
32940 21 45 22 6
103620 46 23 5 35
371949 241 34 11
15 48 25 2 33 14 38
47283 29 13 39 16
27 4 30 12 42 17 43

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

Group IC
  1. To generate the square, PMK5* 15 [RU, full arc,(1U,2L)], place a 1 to the right of the center cell of a 5x5 square and fill in empty cells by advancing continuously diagonally up right, then continuously down left until blocked by a previous number.
  2. For the knight break move one cell up two cells left.
  3. Repeat the process until the square is filled, as shown below in squares 1-5.
1
3
2
1
4 6
5
2
3 10
8 2
7 1
4 6
11 5 9
3
3 16 10 14
8 15 2
713 1
124 6
11 5 9
4
3 16 10 14
19821 15 2
713 1 20
124 18 6
11 5 17 9
5 Semi-Magic
3 16 10 22 14
19821 15 2
72413 1 20
25124 18 6
11 5 17 9 23
Group ID
  1. To generate the square, PMK5* 15 [RU, full arc,(2U,1L)], place a 1 to the right of the center cell of a 5x5 square and fill in empty cells by advancing continuously diagonally up right, then continuously down left until blocked by a previous number.
  2. For the knight break move two cells up one cell left.
  3. Repeat the process until the square is filled, as shown below in squares 1-5.
1
5
4
1
2 6
3
2
5 8
10 4
9 1
2 6
11 3 7
3
5 16 8 12
10 13 4
915 1
142 6
11 3 7
4
5 16 8 12
171021 13 4
915 1 18
142 20 6
11 3 19 7
5 Semi-Magic
5 16 8 24 12
171021 13 4
92215 1 18
23142 20 6
11 3 19 7 25

Construction of 7x7 Knight-Break Méziriac Full Arc Pendulum Magic Squares

Group IC
  1. To generate the square, PMK7* 25 [RU, full arc,(2U,3L)], place a 1 to the right of the center cell of a 5x5 square and fill in empty cells by advancing continuously diagonally up right, then continuously down left until blocked by a previous number.
  2. For the knight break move two cells up three cells left.
  3. Repeat the process until the square is filled, as shown below in squares 1-4.
1
4 14
11 3
10 2
1 9
5 8
6 12 15
7 13
2
4 29 14 20 26
11 21 273
1018 28 2
17 251 9
16 24 5 8
23 6 12 15
22 7 13 19
3
4 29 14 37 20 26
331136 21 273
104018 43 28 234
4117 251 359
16 24 5 32 842
23 6 31 12 3915
22 7 30 13 38 19
4 Semi-Magic
4 29 14 37 20 45 26
331136 21 44 273
104018 43 28 234
411747 251 359
16 48 24 5 32 842
49 23 6 31 12 3915
22 7 30 13 38 1946
Group ID
  1. To generate the square, PMK7* 28 [RU, full arc,(3U,2L)], place a 1 to the right of the center cell of a 5x5 square and fill in empty cells by advancing continuously diagonally up right, then continuously down left until blocked by a previous number.
  2. For the knight break move three cells up two cells left.
  3. Repeat the process until the square is filled, as shown below in squares 1-4.
1
7 11
14 6
13 5
1 12
2 8
3 9 15
4 10
2
7 29 11 17 23
14 18 246
1321 25 5
20 28 1 12
19 27 2 8
26 3 9 15
22 4 10 16
3
7 29 11 40 17 23
301436 18 246
133721 43 25 531
3820 28 1 3212
19 27 2 35 839
26 3 34 9 4215
22 4 33 10 41 16
4 Semi-Magic
7 29 11 40 17 48 23
301436 18 47 246
133721 43 25 531
382044 28 1 3212
19 45 27 2 35 839
46 26 3 34 9 4215
22 4 33 10 41 1649

This completes this section on Méziriac full pendulum squares (Part II). A General staircase procedure for both Méziriac and Loubère methods may be found at New Meziriac Method and New Loubere Method. To return to homepage.


Copyright © 2009 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com