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

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.

this web page is a continuation of the previous Méziriac method Part I and Méziriac method Part II. 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 Non-regular Méziriac type Squares

The Last Six 7x7 Squares

For the first MT7* 26 RIGHT square:

  1. Place the number 1 five cells down on column 1 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
13 21 5
20 4 12
193 11
182 10
1 9 17
8 16 7
14 15 6
2nr
13 30 21 5 22
2920 4 28 12
193 27 11
182 26 10
1 25 9 17
24 8 16 7
23 14 15 6
3nr
13 46 30 21 5 38 22
452920 4 37 28 12
35193 36 27 11 44
18242 26 10 43 34
1 41 25 9 49 33 17
40 24 8 48 32 16 7
23 14 47 31 15 6 39
A seven series

For the second MT7* 28 DOWN square:

  • Place the number 1 five cells down on column 1 of a 7x7 square and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
  • Move three cells down.
  • Repeat the process until the square is filled, as shown below in squares 1-3.
  • 1nr
    9 205
    19 4 8
    183 14
    172 13
    1 12 16
    11 157
    10 21 6
    2nr
    9 205 24
    19 4 23 8
    183 22 14
    172 2813
    1 27 12 16
    26 11 157
    25 10 29 21 6
    3nr
    9 43 35 205 3924
    493419 4 38 23 8
    33183 3722 1448
    17236 2813 4732
    142 27 1246 31 16
    4126 11 45 30 157
    25 10 44 29 21 640
    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.

    Only the results of the last four squares will be shown, since it is now an easy matter to construct these squares. For the third MT7* 27 RIGHT square:

    1. Place the number 1 six cells down on column 7 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 square I.
    4. To keep the colored square simple and uncluttered only the three pairs of colors in the beginning of each group are shown.
    I
    14 47 31 156 3923
    463021 5 38 22 13
    29204 3728 1245
    19336 2711 4435
    242 26 1043 34 18
    4125 9 49 33 171
    24 8 48 32 16 740
    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 fourth MT7* 22 DOWN square:

    1. Place the number 1 six cells down on column 7 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 square II.
    II
    10 44 29 216 4025
    433520 5 39 24 9
    34194 3823 849
    18337 2214 4833
    236 28 1347 32 17
    4227 12 46 31 161
    26 11 45 30 15 741
    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 fifth MT7* 28 RIGHT square:

    1. Place the number 1 seven cells down on column 6 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 square III.
    III
    8 48 32 167 4024
    473115 6 39 23 14
    30215 3822 1346
    20437 2812 4529
    336 27 1144 35 19
    4226 10 43 34 182
    25 9 49 33 17 141
    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 sixth MT7* 23 DOWN square:

    1. Place the number 1 seven cells down on column 6 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 square IV.
    4. To keep the colored square simple and uncluttered only the three pairs of colors in the beginning of each group are shown.
    IV
    11 45 30 157 4126
    442921 6 40 25 10
    35205 3924 943
    19438 238 4934
    337 22 1448 33 18
    3628 13 47 32 172
    27 12 46 31 16 142
    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.

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


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