New Méziriac Magic Square Method (Part II)

The Complementary table Variation

A Méziriac square

A Discussion of the New Method

An important general principle for generating odd magic squares by the De La Loubère 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.
  3. The same regular square is produced when the initial 1 is placed on the center of the first row or the center of the last column, however, this is not the case for the first column or last row.

As examples, the 5x5 regular Loubère and Méziriac squares are shown below:

17 24 1 8 15
2357 14 16
4613 20 22
101219 21 3
11 18 25 2 9
 
3 16 9 22 15
20821 14 2
72513 1 19
24125 18 6
11 4 17 10 23

Setting up the Complementary tables

The typical Méziriac methods employ a consecutive number approach. i.e., the first number layed down is a 1 followed by a 2 up to the numbern. The first non-consecutive approach used a number shift method where the main right diagonal is always the same, but the rest of the numbers are consecutive but shifted. The new method employs groups of numbers, which while not consecutive as a whole are consecutive within their own group, for example (1,2,3,4,5) and (21,22,23,24,25). On this site we use this approach to manipulate the complementary table of an nxn square, generate two modified tables which are then used to construct magic squares.

As an example we begin with the complementary 5x5 table shown below:

1 2 3 4 5 6 7 8 9 10 11 12
13
25 24 23 22 21 20 19 18 17 16 15 14

Perform the next steps for a 5x5 square:

  1. Break up the table into three groups.
  2. Flip the 1st, 2nd, 6th and 7th groups.
  3. Move the 7th group to the first position and each of the others down one, not including lines 3,4 and 5. These last moves are needed since the middle table does not give rise to magic squares.
  4. Label each of the groups (not including lines 3,4 and 5) composing each complementary table. For example for a 5x5 square these labelings are IA, IIA, IB and IIB.
  5. Construct the magic squares for each table using either the A or B configuration. The right main diagonal always uses the numbers from 11-15. For the 5x5 Méziriac squares there a a total of four squares.
IA
 
 
 
 
IB
 
1 2 3 4 5
25 24 23 22 21
11 12
13
15 14
6 7 8 9 10
20 19 18 17 16
Flip
5 4 3 2 1
21 22 23 24 25
11 12
13
15 14
10 9 8 7 6
16 17 18 19 20
Move up
IIA
 
 
 
 
IIB
 
16 17 18 19 20
5 4 3 2 1
11 12
13
15 14
21 22 23 24 25
10 9 8 7 6

The Set of Méziriac Broken Diagonals

The table below correponds to the positions where the initial numerical 1's are placed. Those one's in the broken yellow diagonal correspond to by symmetry to those in the the broken light blue diagonal. This follows the rules employed in previous Méziriac pages. For example see New Loubère Methods and Squares.

The set of Méziriac Broken Diagonals
1 1
1 1
1 1
1 1
1 1

The Méziriac squares, which I will label Mn* center cell#)[2R](n1,n2) where (Mn* signifies a nxn Méziriac square with the center cell numbers and a break followed by 2 moves Right. Since the complementary table is composed of two groups A and B a total of four squares are possible with the numbers n1 and n2 in any of four combinations.

Examples of Magic 5x5 Squares Using the Méziriac[2R] Method

Example starting with IA

  1. Place the first number of IA at the center of the first row of a 5x5 square and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
  2. Move down one cell, by taking the last number from the next line on the complementary table and placing this into the cell.
  3. Fill in the main right diagonal.
  4. Repeat the process with IB until the square is filled, as shown below in squares 1-3.
IA
 
 
 
 
IB
 
1 2 3 4 5
25 24 23 22 21
11 12
13
15 14
6 7 8 9 10
20 19 18 17 16
 
1
3 24
23 2
22 1
5 21
4 25
2
3 24 15
23 14 2
2213 1
125 21
11 4 25
3 M5* 13 [2R](1,6)
36 24 17 15
102316 14 2
222013 1 9
19125 8 21
11 4 7 25 18

Example starting with IB

  1. Place the first number of IB at the center of the first row of a 5x5 square and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
  2. Move down one cell, by taking the last number from the next line on the complementary table and placing this into the cell.
  3. Fill in the main right diagonal.
  4. Repeat the process with IA until the square is filled, as shown below in squares 1-3.
IA
 
 
 
 
IB
 
1 2 3 4 5
25 24 23 22 21
11 12
13
15 14
6 7 8 9 10
20 19 18 17 16
 
1
8 19
18 7
17 6
10 16
9 20
2
8 19 15
18 14 7
1713 6
1210 16
11 9 20
3 M5* 13 [2R](6,1)
81 19 22 15
51821 14 7
172513 6 4
241210 3 16
11 92 20 23

Example starting with IIA

  1. Place the first number of IIA at the center of the first row of a 5x5 square and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
  2. Move down one cell, by taking the last number from the next line on the complementary table and placing this into the cell.
  3. Fill in the main right diagonal.
  4. Repeat the process with IIB until the square is filled, as shown below in squares 1-3.
IIA
 
 
 
 
IIB
 
16 17 18 19 20
5 4 3 2 1
11 12
13
15 14
21 22 23 24 25
10 9 8 7 6
 
1
18 4
3 17
2 16
20 1
19 5
2
18 4 15
3 14 17
213 16
1220 1
11 19 5
3 M5* 13 [2R](16,21)
1821 4 7 15
2536 14 17
21013 16 24
91220 23 1
11 19 22 5 8

Example starting with IIB

  1. Place the first number of IIB at the center of the first row of a 5x5 square and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
  2. Move down one cell, by taking the last number from the next line on the complementary table and placing this into the cell.
  3. Fill in the main right diagonal.
  4. Repeat the process with IIA until the square is filled, as shown below in squares 1-3.
IIA
 
 
 
 
IIB
 
16 17 18 19 20
5 4 3 2 1
11 12
13
15 14
21 22 23 24 25
10 9 8 7 6
 
1
23 9
8 22
7 21
25 6
24 10
2
23 9 15
8 14 22
713 21
1225 6
11 24 10
3 M5* 13 [2R](21,16)
2316 9 2 15
2081 14 22
7513 21 19
41225 18 6
11 24 17 10 3

The other four 5x5 Examples Starting at 1

Below are the four examples that complete the 5x5 series starting at 1. These squares may also be generated using a plane of four squares described previously in Part IV of the new Bachet de Méziriac :

1 M5* 14 [2R](1,6)
4 7 25 18 11
62417 15 3
231614 2 10
20131 9 22
12 5 8 21 19
 
2 M5* 15 [2R](1,6)
5 8 21 19 12
72518 11 4
241715 3 6
16142 10 23
13 1 9 22 20
 
3 M5* 11 [2R](1,6)
1 9 22 20 13
82119 12 5
251811 4 7
17153 6 24
14 2 10 23 16
 
4 M5* 12 [2R](1,6)
2 10 23 16 14
92220 13 1
211912 5 8
18114 7 25
15 3 6 24 17

This completes this section on New Méziriac Magic Square Method (Part I). To continue this series go to New Loubère Magic Square Method-7x7 Squares (Part III). To return to homepage.


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