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

Regular and Non-Regular Méziriac Variable Knight Combo Magic and Semi-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

In continuation from the previous page Part I we have shown one form of a knight move. The second knight move is similar to the first except that after a break one moves in a knight fashion ½(n - 1) moves down followed by ½(n - 3) moves right for the broken yellow diagonal . Similarly, for the broken light blue diagonal we perform ½(n - 1) moves left followed by ½(n - 3) 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 - 3)23
½(n2 - 1)24
½(n2 + 1)25
½(n2 + 3)26
½(n2 + 5)27
½(n2 + 7)28
½(n2 - 5)22

These new Bachet de Méziriac type squares where hte L1 and L2 lengths of the knight move are interchanged and 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 - 1) cell Down,L2= ½(n - 3) cells Right) for the yellow broken diagonal squares or
  2. a knight move L3= (½(n - 1) cell Left,L4= ½(n - 3) 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 5, i.e., 3(2n + 1) are possible by this method, although these squares do have some partial magic properties.
  3. Odd squares divisible by 3 but not 5 are magic or semi-magic.

Construction of Regular and Non-Regular Méziriac-Knight Squares

Examples of 7x7 Squares

For the first square MZK7* 24 K[3L,2U]:

  1. In a 7x7 square place the number 1 at the end of the second row 7x7 and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
  2. Move three 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 8
14 1
13 7
6 12
5 11
4 10
3 9
2
2 8 21 27
14 20 26 1
1319 25 7
18 24 6 12
17 23 5 11
22 4 10 16
28 3 9 15
3
2 33 8 39 21 27
321438 20 26 1
133719 25 7 31
3618 24 6 30 12
17 23 5 29 11 42
22 4 35 10 41 16
28 3 34 9 40 15
4
2 33 8 39 21 45 27
321438 20 44 26 1
133719 43 25 7 31
361849 24 6 30 12
17 48 23 5 29 11 42
47 22 4 35 10 41 16
28 3 34 9 40 15 46

Note that this squares is non-regular. The following complete three 7x7 squares from the 1 on the broken light blue diagonal show 1 regular and 2 non-regular squares.

MZK7* 25 K[3L,2U]
3 34 9 40 15 46 28
33839 21 45 27 2
143820 44 26 1 32
371943 25 7 31 13
18 49 24 6 30 12 36
48 23 5 29 11 42 17
224 35 10 41 16 47
 
MZK7* 26 K[3L,2U]
4 35 10 41 16 47 22
34940 15 46 28 3
83921 45 27 2 33
382044 26 1 32 14
19 43 25 7 31 13 37
49 24 6 30 12 36 18
235 29 11 42 17 48
 
MZK7* 27 K[3L,2U]
5 29 11 42 17 48 23
351041 16 47 22 4
94015 46 28 3 34
392145 27 2 33 8
20 44 26 1 32 14 38
43 25 7 31 13 37 19
246 30 12 36 18 49

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

Examples of 9x9 Squares

The following are examples of regular and non-regular 9x9 square from the broken blue diagonal showing the 8th knight move:

MZK9* 44 K[3L,2U]
2 46 18 62 2569 32 7639
54 17 61 24 6831 75 381
16 60 23 67 3074 37 953
59 22 66 29 7345 8 5215
21 65 28 81 447 51 1458
64 36 80 43 650 13 5720
35 79 42 5 4912 56 1972
78 41 4 48 1155 27 7134
40 3 47 10 6326 70 3377
 
MZK9* 45 K[3L,2U]
3 47 10 63 2670 33 7740
46 18 62 25 6932 76 392
17 61 24 68 3175 38 154
60 23 67 30 7437 9 5316
22 66 29 73 458 52 1559
65 28 81 44 751 14 5821
36 80 43 6 5013 57 2064
79 42 5 49 1256 19 7235
41 4 48 11 5527 71 3478
 
MZK9* 45 LEFT[3,2]
4 48 11 55 2771 34 7841
47 10 63 26 7033 77 403
18 62 25 69 3276 39 246
61 24 68 31 7538 1 5417
23 67 30 74 379 53 1660
66 29 73 45 852 15 5922
28 81 44 7 5114 58 2165
80 43 6 50 1357 20 6436
42 5 49 12 5619 72 3579

The 9x9 and 21x21 Value tables

The following include the 9x9 and 21x21 center cell values and the magic sum S of the left main diagonal. These odd n divisible by 3 but not 5 cycle through the triad of d2 values -n, 0 and n and the yellow entries are magic and the others semi-magic:

9x9 Square Values
center of Square Sleft+d2Value d2
4438413690
453942378n
374043360-n
21x21 Square Values
center of Square Sleft+d2Value d2
227230212215218221224 46410
228231213216219222225 4662n
229211214217220223226 4620-n

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* 25 K[3L,2U] 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 II). To go back to Part I or to return to homepage.


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