A New Set of Bachet de Méziriac Methods and Squares (Part I)

Regular and Non-Regular Magic and Semi-Magic Méziriac Squares

A Break Followed by three Moves

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 Bachet de 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.
  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.

However, for the Bachet de Méziriac methods two regular squares are produced for 5x5 squares, while one regular square is produced for a Loubère square as shown below:

 
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
 
17 24 1 8 15
2357 14 16
4613 20 22
101219 21 3
11 18 25 2 9

A new set of Bachet de Méziriac squares are now possible. All odd squares having the numerical 1's lying on two broken diagonals symmetrical with the light grey main diagonal behave differently from typical Méziriac squares. After a break 3 moves to the right or to the bottom for the broken yellow diagonal and 3 moves to the left or to the top 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 one in the other diagonal group. So that for example a square starting with the number 1 in the first column is identical to that with the number 1 in the last row. 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 right or to the bottom. As one can see the center values are displaced by two units in the table for square starting with a 1 in the same position:

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

These new Bachet de Méziriac type squares, which I will label MT7* (center cell#) (RIGHT or LEFT) where (MT7* signifies a 7x7 Méziriac type square with the center cell number of the square and breaking either to the right or to the left. the squares exhibit the following properties:

  1. Every number on the main diagonal is represented at least once in this type of square.
  2. No odd squares divisible by 3, i.e., 3(2n + 1) are possible by this method.
  3. Odd squares divisible by 5 and not by 3, produce magic and semi-magic squares.

Examples of the odd squares divisible by 5 and not 3 are shown in Méziriac type Squares Part IV.

Construction of Regular and Non-Regular Méziriac type Squares

7x7 Squares 1-4

For the first square MT7* 22 RIGHT:

  1. In the first row place the number 1 one cell to the right of the center column of a 7x7 square and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
  2. Move three cells to the right.
  3. Repeat the process until the square is filled, as shown below in squares 1-3.
1
9 17 1
16 7 8
156 14
5 13
4 12
11 3
10 2
2
9 17 1 25
16 7 24 8
156 23 14
215 22 13
4 28 12 29 20
27 11 19 3
26 10 18 2
3
9 49 33 17 1 41 25
483216 7 40 24 8
31156 39 23 14 47
21538 22 13 46 30
4 37 28 12 45 29 20
36 27 11 44 35 19 3
26 10 43 34 18 2 42
A seven series

Note that three pairs sum to 44 and including the center square when multiplied by 2 and twenty-one pairs sum to 51. In addition none of these pairs are complementary, as in the regular Méziriac squares, as shown in the connectivities of the complementary table.Also this complementary table is identical to the one in non-regular Loubère squares.

For the second square MT7* 24 DOWN:

  1. In the first row place the number 1 one cell to the right of the center column of a 7x7 square and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
  2. Move three cells down.
  3. Repeat the process until the square is filled, as shown below in squares 1-3.
 
1
12 161
15 7 11
6 10
5 9
4 8 19
14 183
13 17 2
2
12 161 27
15 7 26 11
29216 25 10
205 249
4 23 8 19
22 14 183
28 13 17 2
3
12 46 31 161 4227
453015 7 41 26 11
29216 4025 1044
20539 249 4335
438 23 849 34 19
3722 14 48 33 183
28 13 47 32 17 236
A seven series

Note that 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 Méziriac squares, as shown in the connectivities of the complementary table. Also this complentary table is identical to the one in non-regular Loubère squares

For the third MT7* 23 RIGHT:

  1. Place the number 1 in the center of row 2 of a 7x7 square and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
  2. Move three cells right.
  3. Repeat the process until the square is filled, as shown below in squares 1-3.
  4. To keep the colored square simple and uncluttered only the three pairs of colors in the beginning of each group are shown.
1nr
10 182
17 1 9
167 8
156 14
5 22 13 21
12 204
11 19 3
2nr
10 34 182 26
3317 1 25 9
32167 24 8
156 2314 31
5 22 13 30 21
28 12 29 204
27 11 35 19 336
3nr
10 43 34 182 4226
493317 1 41 25 9
32167 4024 848
15639 2314 4731
538 22 1346 30 21
3728 12 45 29 204
27 11 44 35 19 336
A seven series

Note that ten pairs sum to 46 including the center square when multiplied by 2 and fourteen pairs sum to 53. In addition none of these pairs are complementary, as in the regular Méziriac squares, as shown in the connectivities of the complementary table.

For the fourth MT7* 25 DOWN:

  1. Place the number 1 in the center of row 2 of a 7x7 square and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
  2. Move three cells down.
  3. Repeat the process until the square is filled, as shown below in squares 1-3.
1r
13 17 2
16 1 12
157 11
6 10
5 9
8 4
14 3
2r
13 17 2 28
16 1 27 12
157 26 11
216 25 10 29
5 24 9 20
23 8 19 4
22 14 18 3
3r
13 47 32 17 2 36 28
463116 1 42 27 12
30157 41 26 11 45
21640 25 10 44 29
5 39 24 9 43 35 20
38 23 8 49 34 19 4
22 14 48 33 18 3 37

this completes this section on the new Bachet de Méziriac method and squares. To continue to Part II or to return to homepage.


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