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

Regular and Non-Regular Méziriac-Knight Combo Magic Squares

A Break Followed by a Knight Move

Two Knights

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 one moves in a knight fashion 2 moves to the right followed by 1 move down for the broken yellow diagonal and 2 moves up and 1 move to the 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 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 + knight) Diagonals
1 1
1 1
1 1
1 1
1 1
1 1
1 1
  
Center Value
EquationValue Right
½(n2 - 5)22
½(n2 - 3)23
½(n2 - 1)24
½(n2 + 1)25
½(n2 + 3)26
½(n2 + 5)27
½(n2 + 7)28

these new Bachet de Méziriac type squares, which I will label MZ7* (center cell#) (K[L1,L2] or K[L3,L4]) where (MZ7* signifies a 7x7 Méziriac-Knight square with the center cell number of the square and a break followed by a L1= 1D and L2= 2R for the yellow broken diagonal squares or a knight move L3= 1L and L4= 2U for the light blue broken diagonal. 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.

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-Knight Squares

5x5 and 7x7 Squares

For the first square MZ5* 13 [1D,2R]:

  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 one cells to the down and two cell right.
  3. Repeat the process until the square is filled, as shown below in squares 1-3.
 
1
8 1
75
4 6
3
2
2
8 1 15
75 14
413 6
312 10
11 9 2
3
24 8 17 1 15
7165 14 23
20413 22 6
31221 10 19
11 25 9 18 2

Note that this squares is regular and complementary. the squares that follow are all non-regular and not complementary and just the final squares without their construction are given.

A
25 9 18 2 11
8171 15 24
16514 23 7
41322 6 20
12 21 10 9 3
 
B
21 10 19 3 12
9182 11 25
17115 24 8
51423 7 16
13 22 6 20 4
 
C
22 6 20 4 13
10193 12 21
18211 25 9
11524 8 17
14 23 7 16 5
 
D
23 7 16 5 14
6204 13 22
19312 21 10
21125 9 18
15 24 8 17 1

The complementarty tables for the last four 5x5 squares may be seen on new Bachet de Méziriac squares, new Loubère squares and new Bachet de Méziriac squares. Note that both de Méziriac and Loubère squares can give rise to the same complementary table.

For the first square MZ7* 25 [1D,2R]:

1
10 1
7 9
6 8
5
4
3
2
2
10 19 1 28
2918 7 27 9
176 26 8
165 25 14
4 24 13 15
23 12 21 3
22 11 20 2
3
1048 30 19 1 39 28
472918 7 38 27 9
35176 37 26 8 46
16536 25 14 45 34
4 42 24 13 44 33 15
41 23 12 43 32 21 3
22 11 49 31 20 2 40
4
10 48 30 19 1 39 28
472918 7 38 27 9
35176 37 26 8 46
16536 25 14 45 34
4 42 24 13 44 33 15
41 23 12 43 32 21 3
22 11 49 31 20 2 40

Note that this squares is regular and complementary. the following three 7x7 squares (out of a possible 7) are all non-regular and not complementary and just the final squares are given.

MZ7* 26 [1D,2R]
11 49 31 20 2 40 22
483019 1 39 28 10
29187 38 27 9 47
17637 26 8 46 35
5 36 25 14 45 34 16
42 24 13 44 33 15 4
2312 43 32 21 3 41
 
MZ7* 28 [1D,2R]
13 44 33 15 4 42 24
433221 3 41 23 12
31202 40 22 11 49
19139 28 10 48 30
7 38 27 9 47 29 18
37 26 8 46 35 17 6
2514 45 34 16 5 36
 
MZ7* 24 [1D,2R]
9 47 29 18 7 38 27
463517 6 37 26 8
34165 36 25 14 45
15442 24 13 44 33
3 41 23 12 43 32 21
40 22 11 49 31 20 2
2810 48 30 19 1 39

The complementary tables for the last three 7x7 squares may be seen on new bachet squares (Part II) and new bachet squares (Part III).

The Plane of Loubère-Knight 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 MZ7* 25 [1D,2R] one can move up the right diagonal on a plane and generate the complete set of 7 squares as is shown in Part IV of the Bachet de Méziriac series.

this completes this section on the new Bachet de Méziriac method and squares (Part V). Click to view the new variable knight Bachet de Méziriac method and squares or to return to homepage.


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