A New True Bachet de Méziriac Variable Knight Method and Squares (Part I)

Regular and Non-Regular Méziriac Variable Knight Combo Magic Squares

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 based on the true, original Bachet de Méziriac square is now possible. The difference with this knight method and previous ones is that the knight moves take on a dynamic or variable character, meaning that as the order of the square n increases as the length of the knight moves increases. In addition two possible knight moves are possible both going in the same direction. The next page (Part II) will show the second type of knight move. Previous Loubère and Méziriac type knight methods employed a regular knight move (2 moves followed by a 1 move). Because the left main diagonal and the broken diagonals of a true Méziriac square are composed of arithmetic progressions only the 5x5 squares can be constructed using the knight equation from above, i.e., (2,1) moves. Construction of higher order squares, it appears, require a dynamic knight square move to produce magic squares. Although the new broken diagonals generated are not geometrical they are very close to it.

So to continue 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 ½(n - 3) moves down followed by ½(n - 1) moves right for the broken yellow diagonal . Similarly, for the broken light blue diagonal we perform ½(n - 3) moves left followed by ½(n - 1) moves up. This produces one regular square and n - 1 non-regular squares. 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 for the light blue diagonal. Starting with the square generated from 1 in the first row the values range from ½(n2 - n + 2) to ½(n2 + 7) for squares breaking to the left.

The set of (Break + knight) Diagonals
1 1
1 1
1 1
1 1
1 1
1 1
1 1
 
center Value
EquationValues for light blue diagonal
½(n2 + 5)27
½(n2 + 7)28
½(n2 - 5)22
½(n2 - 3)23
½(n2 - 1)24
½(n2 + 1)25
½(n2 + 3)26

These new Bachet de Méziriac type squares, which I will label MZK7* (center cell#) (K[L1,L2] or K[L3,L4]) where (MZK7* signifies a true 7x7 Méziriac-Knight square with the center cell number of the square and a break followed by:

  1. a knight move L1= (½(n - 3) cells Down, L2= ½(n - 1) cells Right) for the yellow broken diagonal squares or
  2. a knight move L3= (½(n - 3) cell Left,L4= ½(n - 1) cells Up) 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, although these squaresdo have some partial magic properties.

Construction of Regular and Non-Regular Méziriac-Knight Squares

5x5 and 7x7 Squares

For the first square MZK5* 13 K[1L,2U]:

  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 cell to the left and two cells up.
  3. Repeat the process until the square is filled, as shown below in squares 1-3.
 
1
2 6
10 1
9 5
4 8
3 7
2
2 19 6 15
1810 14 1
913 5 17
124 16 8
11 3 7
3
2 19 6 23 15
181022 14 1
92113 5 17
25124 16 8
11 3 20 7 24

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
3 20 7 24 11
19623 15 2
102214 1 18
21135 17 9
12 4 16 8 25
 
B
4 16 8 25 12
20724 14 3
62315 2 19
22141 18 10
13 5 17 9 21
 
C
5 17 9 21 13
16825 12 4
72411 3 20
23152 19 6
14 1 18 10 22
 
D
1 18 10 22 14
17921 13 5
82512 4 16
24113 20 7
15 2 19 6 23

The complementary 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 MZK7* 25 K[2L,3U]:

 
1
6 11
10 5
9 4
3 8
2 14
1 13
7 12
2
6 11 16 28
10 15 27 5
921 26 4
20 25 3 8
19 24 2 14
23 1 13 18
22 7 12 17
3
6 33 11 38 16 28
321037 15 27 5
93621 26 4 31
4220 25 3 30 8
19 24 2 29 14 41
23 1 35 13 40 18
22 7 34 12 39 17
4
6 33 11 38 16 43 28
321037 15 49 27 5
93621 48 26 4 31
422047 25 3 30 8
19 46 24 2 29 14 41
45 23 1 35 13 40 18
22 7 34 12 39 17 44

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.

MZK7* 28 LEFT[2L,3U]
2 29 14 41 19 46 24
351340 18 45 23 1
123917 44 22 7 34
381643 28 6 33 11
15 49 27 5 32 10 37
48 26 4 31 9 36 21
253 30 8 42 20 47
 
MZK7* 24 LEFT[2L,3U]
5 32 10 37 15 49 27
31936 21 48 26 4
84220 47 25 3 30
411946 24 2 29 14
18 45 23 1 35 13 40
44 22 7 34 12 39 17
286 33 11 38 16 43
 
MZK7* 27 LEFT[2L,3U]
1 35 13 40 18 45 23
341239 17 44 22 7
113816 43 28 6 33
371549 27 5 32 10
21 48 26 4 31 9 36
47 25 3 30 8 42 20
242 29 14 41 19 46

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

An Example of an 11x11 Square

The following is an example of a non-regular 11x11 square from the broken blue diagonal and shows the[4,5] knight moves at the 1st and 7th breaks:

MZK11* 58 K[4L,5U]
6 70 13 88 3195 38102 45 12063
69 12 87 30 9437 10155 119 625
22 86 29 93 36100 54118 61 468
85 28 92 35 11053 11760 3 6721
27 91 34 109 52116 592 77 2084
90 44 108 51 11558 176 19 8326
43 107 50 114 5711 7518 82 2589
106 49 113 56 1074 1781 24 9942
48 112 66 9 7316 8023 98 41105
111 65 8 72 1579 3397 40 10447
64 7 71 14 7832 9639 103 46121

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 MZK7* 28 LEFT[2L,3U] 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 Knight method and squares (Part I). To continue Méziriac method (Part II). To return to homepage.


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