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 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.

******************************************************************************************************************************************************** For the second MT7^* 28 DOWN 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 down.
3. Repeat the process until the square is filled, as shown below in squares 1-3.

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.

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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.

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.

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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.

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.

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.

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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.

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.

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