New Generalized Procedure of Méziriac Method (Part II)

A stairs

A Discussion of the New Method

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 Méziriac 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 7x7 regular Méziriac squares is shown below as an example:

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

Normally the Méziriac method involves a stepwise approach of consecutive numbers, i.e., 1,2,3... In previous methods I showed that these numbers do not have to be consecutive but may be placed semi-consecutively both frontwards or backwards as in Pendulum method I and Pendulum method II.

If the numbers are added non-consecutively,using a 7x7 example i.e. partially or fully random heptad 1,7,4,3,2,5,6 then it gets patricularly tricky keeping tracks of how the next subsequent numbers are added to complete the rest of the broken diagonals. I will show that there are two ways to accomplish this, both of which give different squares using the same heptad of opening numbers. Using the heptad above, the next number 8 can come after the 7 (the last number of the heptad) or after the 6 (the last number of the set). This method in fact may be used for any group of numbers (random or consecutive) to produce magic or semi-magic squares and may, therefore, be classified as general.

In all our examples we will be using 7x7 tables, so therefore, the heptad table is first produced by inserting in the first 7 numbers in the first row. To acquire the next row of heptad n (in this case 7) is added to all the previous numbers. This is continued until the heptad table is filled. See the first example below. Both translational and knight breaks are shown. Knight break moves are variable where the length of the moves are ½(n - 1) or ½(n - 1) depending on the group.

Construction of Generallized Procedure of (Random or Symmetrical) Méziriac Square

Heptad Table I
1 5 6 74 3 2
81213 14 11 109
151920 21 18 1716
22 26 27 28 25 2423
293334 35 32 3130
36 40 41 42 39 3837
43 47 48 49 46 4544

7x7 Squares

Method I: Group IA
  1. Generate the heptad table.
  2. Take the first row of the heptad table I and add it to the square being built (Square 1) in a stepwise manner, until blocked by a previous number.
  3. Move two cells right from the number 7 (the last number of the heptad).
  4. Repeat the process until the square is filled, always taking the next heptad set from the heptad table 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 21 226
1020 23 5
19 24 1 11
1525 2 14
28 3 29 13 16
27 4 12 17
3
7 31 8 4118 26
32940 21 226
103620 23 535
37 19 24 1 3411
1525 233 1438
28 3 29 13 3916
27 4 30 12 42 1743
4
7 31 8 4118 44 26
32940 21 45 226
103620 46 23 535
37 19 49 24 1 3411
154825 233 1438
47 28 3 29 13 3916
27 4 30 12 42 1743
Method I: Group IB
  1. Take the first row of the heptad table I and add it to the square being built (Square 1) in a stepwise manner, until blocked by a previous number.
  2. Move two cells down from the number 7 (the last number of the heptad).
  3. Repeat the process until the square is filled, always taking the next heptad set from the heptad table as shown below in squares 1-4.
1
7 13
12 15 6
8 5
1 9
2 10
3 11
4 14
2
7 13 19 22
12 15 236
816 24 5
17 25 1 9
1828 2 10
27 3 11 21
26 4 29 14 20
3
7 30 13 3819 22
311239 15 236
84216 24 532
41 17 25 1 359
184328 234 1040
27 3 33 11 3621
26 4 29 14 37 20
4
7 30 13 3819 46 22
311239 15 49 236
84216 48 24 532
41 17 47 25 1 359
184328 234 1040
44 27 3 33 11 3621
26 4 29 14 37 2045
Method II: Group IA (Semi-Magic)
  1. Take the first row of the heptad table I and add it to the square being built (Square 1) in a stepwise manner, until blocked by a previous number.
  2. Move two cells right from the number 2 (the last number of the set of the heptad).
  3. Repeat the process until the square is filled, always taking the next heptad set from the heptad table 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
1927 2 8
26 3 9 15
22 4 10 16
3
7 29 11 4017 23
301436 18 246
133721 43 25 531
38 20 28 1 3212
1927 235 839
26 3 34 9 4215
22 4 33 10 41 16
4 Semi-Magic
7 29 11 4017 48 23
301436 18 47 246
133721 43 25 531
38 20 44 28 1 3212
194527 235 839
46 26 3 34 9 4215
22 4 33 10 41 1649
Method II: Group IB
  1. Take the first row of the heptad table I and add it to the square being built (Square 1) in a stepwise manner, until blocked by a previous number.
  2. Move two cells down from the number 2 (the last number of the set of the heptad).
  3. Repeat the process until the square is filled, always taking the next heptad set from the heptad table as shown below in squares 1-4.
1
7 9
10 6
1115 5
1 14
2 13
3 12
4 8
2
7 9 20 24
2910 19 256
1115 28 5
16 27 1 14
1726 2 13
22 3 12 18
23 4 8 21
3
7 33 9 3920 43 24
291042 19 256
114115 28 530
40 16 27 1 3114
1726 232 1336
22 3 35 12 3718
23 4 34 8 38 21
4
7 33 9 3920 43 24
291042 19 44 256
114115 45 28 530
40 16 46 27 1 3114
174926 232 1336
48 22 3 35 12 3718
23 4 34 8 38 2147

Construction of Generallized Procedure of (Random or Symmetrical) Knight-Break Loubère Square

Below is again listed the heptad table I and is shown here for easier viewing. These are the only two groups that are either magic or semi-magic.

Heptad Table I
1 5 6 74 3 2
81213 14 11 109
151920 21 18 1716
22 26 27 28 25 2423
293334 35 32 3130
36 40 41 42 39 3837
43 47 48 49 46 4544
Method I: Group IC
  1. This group of squares is semi-magic.
  2. Take the first row of the heptad table I and add it to the square being built (Square 1) in a stepwise manner, until blocked by a previous number.
  3. Using a knight break, move three cells up and two cells left from the number 3 (the last number of the heptad).
  4. Repeat the process until the square is filled, always taking the next heptad set from the heptad table 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
1927 2 8
26 3 9 15
22 4 10 16
3
7 29 11 4017 23
301436 18 246
133721 43 25 531
38 20 28 1 3212
1927 235 839
26 3 34 9 4215
22 4 33 10 41 16
4 Semi-Magic
7 29 11 4017 &48 23
301436 18 47 246
133721 43 25 531
38 20 44 28 1 3212
194527 235 839
46 26 3 34 9 4215
22 4 33 10 41 1649
Method I: Group ID
  1. None are magic
Method II: Group IC
  1. Take the first row of the heptad table I and add it to the square being built (Square 1) in a stepwise manner, until blocked by a previous number.
  2. Using a knight break, move three cells up and two cells left from the number 2 (the last number of the set of the heptad).
  3. Repeat the process until the square is filled, always taking the next heptad set from the heptad table as shown below in squares 1-4.
1
7 12
8 6
9 5
1 10
2 11
3 14
4 13 15
2
7 12 16 25
8 17 286
918 27 529
21 261 10
2022 2 11
23 3 14 19
24 4 13 15
3
7 34 12 3616 25
33837 17 286
93818 27 529
39 21 261 3010
2022 231 1142
43 23 3 32 14 4119
24 4 35 13 40 15
4
7 34 12 3616 45 25
33837 17 46 286
93818 49 27 529
39 21 48 261 3010
204722 231 1142
43 23 3 32 14 4119
24 4 35 13 40 1544
Method II: Group ID
  1. Take the first row of the heptad table I and add it to the square being built (Square 1) in a stepwise manner, until blocked by a previous number.
  2. Using a knight break, move two cells up and three cells left from the number 2 (the last number of the set of the heptad).
  3. Repeat the process until the square is filled, always taking the next heptad set from the heptad table as shown below in squares 1-4.
1
7 13
12 15 6
8 5
1 9
2 10
3 11
4 14
2
7 13 19 22
12 15 236
816 24 5
17 251 9
1828 2 10
27 3 11 21
26 4 29 14 20
3
7 30 13 3819 22
311239 15 236
84216 24 532
41 17 251 359
184328 234 1040
27 3 33 11 3621
26 4 29 14 37 20
4
7 30 13 3819 46 22
311239 15 49 236
84216 48 24 532
41 17 47 251 359
184328 234 1040
44 27 3 33 11 3621
26 4 29 14 37 2045

Results of a typical Méziriac Odd n Square (n divible by 3)

For odd magic squares n divided by three no magic squares for either Method I and II.

This completes this section on Generallized Procedure for Méziriac squares (Part II). To return to homepage.


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