A New Magic Square Method

New Loubère Magic Square Method (Part I)

The Complementary table Variation

A Loubere 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 Loubère and Méziriac methods employ a consecutive number approach. i.e., the first number layed down is a 1 followed by a 2 up to the number n. 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 Loubère 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
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 Loubère 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 Loubère pages. For example see New Loubère Methods and Squares.

The set of Loubère Broken Diagonals
11
11
1 1
11
1 1

The Loubère squares, which I will label Ln* (center cell#)[1D](n1,n2) where (Ln* signifies a nxn Loubère square with the center cell number and a break followed by 1 move Down. 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 Loubère[1D] Method

Example starting with IA

  1. Place the first number of IA at the center of the first row of a 5x5 square (the regular historical 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
1 23
522
421
25 3
2 24
2
1 23 15
522 14
42113
2512 3
11 2 24
3 L5* 13 [1D](1,6)
7 19 1 23 15
18522 14 6
42113 10 17
25129 16 3
11 8 20 2 24

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
6 18
1017
916
20 8
7 19
2
6 18 15
1017 14
91613
2012 8
11 7 19
3 L5* 13 [1D](6,1)
2 24 6 18 15
231017 14 1
91613 5 22
20124 21 8
11 3 25 7 19

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
16 3
202
191
5 18
17 4
2
16 3 15
202 14
19113
512 18
11 17 4
3 L5* 13 [1D](16,21)
22 9 16 3 15
8202 14 21
19113 25 7
51224 6 18
11 23 10 17 4

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
21 8
257
246
10 23
22 9
2
21 8 15
257 14
24613
1012 23
11 22 9
3 L5* 13 [1D](21,16)
17 4 21 8 15
3257 14 16
24613 20 2
101219 1 23
11 18 5 22 9

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 L5* 14 [1D](1,6)
8 20 2 24 11
19123 15 7
52214 6 18
211310 17 4
12 9 16 3 25
 
2 L5* 15 [1D](1,6)
9 16 3 25 12
20224 11 8
12315 7 19
22146 18 5
13 10 17 4 21
 
3 L5* 11 [1D](1,6)
10 17 4 21 13
16325 12 9
22411 8 20
23157 19 1
14 6 18 5 22
 
4 L5* 12 [1D](1,6)
6 18 5 22 14
17421 13 10
32512 9 16
24118 20 2
15 7 19 1 23

This completes this section on New Loubère Magic Square Method (Part I). To continue this series go to New Méziriac Magic Square Method (Part II). To return to homepage.


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