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

Regular and Non-Regular Bachet de 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.

This web page is a continuation of the previous Méziriac method Part I. Note that MT7* (center cell#) (RIGHT or LEFT) signifies a 7x7 Méziriac type square with the center cell number of the square and breaking either to the right or to the left. Construction of all the squares on the non-main diagonals will continue on the next two pages: Méziriac method Part III and Méziriac method Part IV.

Construction of Méziriac type Squares

Second Set of Four 7x7 Squares

For the first MT7* 24 RIGHT square:

  1. Place the number 1 in cell 3 of row 3 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.
1nr
11 19 3
18 2 10
171 9
167 8
6 14 15
13 5
12 4
2nr
11 19 3 27
18 2 26 10
171 25 9
167 24 8
6 23 14 15
22 13 21 5
28 12 29 20 4
3nr
11 44 35 19 3 36 27
433418 2 42 26 10
33171 41 25 9 49
16740 24 8 48 32
6 39 23 14 47 31 15
38 22 13 46 30 21 5
28 12 45 29 20 4 37
A seven series

For the second MT7* 26 DOWN square:

  1. Place the number 1 in cell 3 of row 3 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.
  4. To keep the colored square simple and uncluttered only the three pairs of colors in the beginning of each group are shown.
1nr
14 183
17 2 13
161 12
157 11
6 10
9 5
8 19 4
1nr
14 183 22
17 2 28 13
161 27 12
157 2611
6 25 10 29 21
24 9 205
23 8 19 4
2nr
14 48 33 183 3722
473217 2 36 28 13
31161 4227 1246
15741 2611 4530
640 25 1044 29 21
3924 9 43 35 205
23 8 49 34 19 438
A seven series

Note that fourteen pairs sum to 47 and ten pairs sum to 54 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 complementary table is identical to the 7x7 semi-square in non-regular Loubère squares.

For the third MT7* 25 RIGHT square:

  1. Place the number 1 in cell 2 of row 4 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.
1
12 4
19 3 11
182 10
171 9
7 8 16
14 15 6
13 5
2
12 29 20 4 28
19 3 27 11
182 26 10
171 25 9
7 24 8 16
23 14 15 6
22 13 21 5
3
12 45 29 20 4 37 28
443519 3 36 27 11
34182 42 26 10 43
17141 25 9 49 33
7 40 24 8 48 32 16
39 23 14 47 31 15 6
22 13 46 30 21 5 38

Note that three pairs sum to 56 including the center square when multiplied by 2 and twenty-one pairs sum to 49. In addition none of these pairs are complementary, as in the regular Méziriac squares, as shown in the connectivities of the complementary table. Also his complementary table is identical to the one in non-regular Loubère squares

For the fourth MT7* 27 DOWN square:

  1. Place the number 1 in cell 2 of row 4 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.
  4. To avoid cluttering only the first color in the group is shown.
1
8 194
18 3 14
172 13
161 12
7 11 15
10 6
9 5
2
8 194 23
18 3 22 14
172 28 13
161 2712
7 26 11 15
25 10 29 216
24 9 20 5
3
8 49 34 194 3823
483318 3 37 22 14
32172 3628 1347
16142 2712 4631
741 26 1145 30 15
4025 10 44 29 216
24 9 43 35 20 539
A seven series

Note that seven pairs sum to 45 and seventeen pairs sum to 52 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 complementary table is identical to the one in non-regular Loubère squares.

This completes this section on new Bachet de Méziriac method and squares. To proceed to Part III or to return to homepage.


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