New Bachet de Méziriac Method and Squares (Part I)

Regular and Non-Regular Bachet de Méziriac Squares

A Méziriac square

A Discussion of the New Methods

An important general principle for generating odd magic squares by the Bachet de 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.
  3. the same regular square should be produced when the initial 1 is placed to the top, right, left or bottom of the center cell.

However, for the Bachet de Méziriac methods the four regular squares produce two equal pairs as shown below for 5x5 and 7x7 squares:

3 16 9 22 15
20821 14 2
72513 1 19
24125 18 6
11 4 17 10 23
  
22 9 16 3 15
8202 14 21
19113 25 7
51224 6 18
11 23 10 17 4
 
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
 
45 20 37 12 29 4 28
193611 45 3 27 44
421034 2 26 43 18
9331 25 49 17 41
32 7 24 48 16 40 8
6 23 47 15 39 14 31
22 46 21 38 13 30 5

This web page has been modified from the previous to take into account new findings since the previous Bachet de Méziriac method has been found to be a subset of a general method. It will be shown that the initial numeral 1 can be placed anywhere on two broken diagonals, in which the original Méziriac method is placed on the bottom diagonal and the the second initial 1 is placed on the top diagonal. this page will show that all the squares having an initial 1 on the broken diagonal are related to one another. It will also be shown that both regular and non-regular squares are produced. In fact, more non-regular squares are generated than regular squares. this goes for this and other Méziriac methods that I will be discussing. I will also show that the squares oo the left are produced by one method which give rise to semi-magic squares as well while the squares on the right are generated by a second method which produces magic squares. this page deals with the latter method.

All odd squares having the numerical 1's lying on two broken diagonals symmetrical with the light grey main diagonal behave differently from typical Loubère squares. After a break/2 moves down for the broken yellow diagonal and break/2 moves left for the broken light blue diagonal, one regular square and n - 1 non-regular squares are produced. Squares belonging to one diagonal group are identical to another square on the other diagonal group. Using the 7x7 square as an example shows the two diagonals and typical 1 positions. the second table shows the equation and value of the center cell of each square (starting with the square generated from 1 in the first row) where the values range from ½(n2 - n + 2) to ½(n2 + 7) for squares breaking to the left or down.

The set of Broken Diagonals
1 1
1 1
1 1
1 1
1 1
1 1
1 1
  
Center Value
EquationValue LeftEquationValue Down
½(n2 - 5)22½(n2 + 7)28
½(n2 - 3)23½(n2 - 5)22
½(n2 - 1)24½(n2 - 3)23
½(n2 + 1)25½(n2 - 1)24
½(n2 + 3)26½(n2 + 1)25
½(n2 + 5)27½(n2 + 3)26
½(n2 + 7)28½(n2 + 5)27

These new Méziriac squares, which I will label Mn* (center cell#) (LEFT or DOWN) where (Mn* signifies a nxn Méziriac square with the center cell number of the square and breaking either left or down. this would make the original Méziriac squares depicted in the introduction as M5* 13 RIGHT and M7* 13 RIGHT and which I have said previously are constructed by a different method. the squares exhibit the following properties:

  1. Every number on the main diagonal is represented at least once in this type of square.
  2. Odd squares divisible by 3, i.e., 3(2n + 1) are not magic.

Construction of Regular and Non-regular Bachet de Méziriac squares

A 5x5 Square

  1. Place a 1 to the right of the center cell a 5x5 square and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
  2. Move two cells down, when a block is encountered. (the alternative method is to place a 1 over the center center and move to the left 2 cells when a block is encountered.)
  3. Repeat the process until the square is filled, as shown below in squares 1-2.
1
3 6
10 13 2
912 1
115 8
4 7
2
3 17 6 25 14
161024 13 2
92312 1 20
22 115 19 8
15 4 18 7 21
A five series

Note that the five pairs sum to 29 and seven pairs sum to 24 including the center square when multiplied by 2. In addition none of these pairs are complementary, as in the regular Bachet de Méziriac squares, as shown in the connectivities of the complementary table.

The four other 5x5 Squares

The following four squares complete the set of 5x5 from the yellow broken diagonal :

I
2 16 10 24 13
20923 12 1
82211 5 19
21154 18 7
14 3 17 6 25
 
II
4 18 7 21 15
17625 14 2
102413 2 16
23121 20 9
11 5 19 8 22
 
III
5 19 8 22 11
18721 15 4
62514 3 17
24132 16 10
12 1 20 9 23
 
IV
1 20 9 23 12
19822 11 5
72115 4 18
25143 17 6
13 2 16 10 24

Construction of Regular and Non-regular 7x7Bachet de Méziriac squares

the following 7x7 non-regular Méziriac square (M7* 24 DOWN) is generated using this method:

1
4 14
13 3
1215 2
1 11
7 10
6 9
5 8
2
4 14 17
13 16 3
1215 2
21 1 11
20 7 10
22 6 9 19
5 8 18
3
4 30 14 40 17 43 27
291339 16 49 26 3
123815 48 25 2 35
3721 47 24 1 34 11
20 46 23 7 33 10 36
45 22 6 32 9 42 19
28 5 31 8 41 18 44

Note that the seven pairs sum to 55 and seventeen pairs sum to 48 including the center square when multiplied by 2. In addition none of these pairs are complementary, as in the regular Bachet de Méziriac squares, as shown in the connectivities of the complementary table.

A seven series

The following four 7x7 squares are a sampling of the set from the yellow broken diagonal :

IM7* 22 DOWN
2 35 12 38 45 48 35
341137 21 47 24 1
103620 46 23 7 33
421945 22 6 32 9
18 44 28 5 31 8 41
43 27 4 30 14 40 17
26 3 29 13 39 16 49
 
II M7* 23 DOWN
3 29 13 39 16 49 26
351238 15 48 25 2
113721 47 24 1 34
362046 23 7 33 10
19 45 22 6 32 9 42
44 28 5 31 8 41 18
27 4 30 14 40 17 43
 
III M7* 27 DOWN
7 33 10 36 20 46 23
32942 19 45 22 6
84118 44 28 5 31
401743 27 4 30 14
16 49 26 3 29 13 39
48 25 2 35 12 38 15
24 1 34 11 37 21 47

The Plane of Méziriac Squares

At this point it may be said that alternatively these squares may be constructed using a plane of four squares. For example using the 7x7 square M7* 22 DOWN one can move up the right diagonal on a plane of four M7* 22 DOWN and generate the complete set of 7 squares as is shown in Part IV of the new Bachet de Méziriac method.

This completes this section on regular and non-regular Bachet de Méziriac squares. the next section discusses a new group of regular and non-regular Bachet de Méziriac squares in which all columns and rows have the magic sum, while the left diagonal varies in sum according to S + dn and d varies from 0 to ½(n - 1). To go back to regular and non-regular Loubère squares. To return to homepage.


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