New De Méziriac Knight-step Method and Squares (Part I)

A Méziriac square

A Discussion of the New Methods

An important general principle for generating odd magic squares by the 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 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.

The 5x5 and 7x7 regular Méziriac squares are shown below as examples:

3 16 9 22 15
20821 14 2
72513 1 19
24125 18 6
11 4 17 10 23
 
4 29 12 37 20 45 28
351136 19 44 27 3
104218 43 26 2 34
411749 25 1 33 9
16 48 24 7 32 8 40
47 23 6 31 14 39 15
22 5 30 13 38 21 46

Méziriac squares are normally contructed using a stepwise approach where each subsequent number is added consecutively one cell at a time. In this new method each subsequent number is added in a stepwise knight step manner, followed by a right or an up break. In addition, n odd squares may be constructed with the initial number 1 on either of two broken diagonals shown below in light blue or yellow separated by symmetry by a light grey diagonal for a 5x5 square.

The set of 5x5 Méziriac Broken Diagonals
1 1
1 1
1 1
11
1 1
 
The set of 7x7 Méziriac Broken Diagonals
1 1
1 1
1 1
1 1
1 1
1 1
1 1

These new Méziriac type squares, which have a one cell break as opposed to a double cell break for a regular Méziriac square, will be labeled, as in previous web pages, as follows:

  1. KMn* (center cell#) [(2D,1L),1U] where KMn* signifies a Knight-step nxn Méziriac square with the center cell number of the square and breaking up.
  2. KMn* (center cell#) [(1D,2L),1R] where KMn* signifies a Knight-step nxn Méziriac square with the center cell number of the square and breaking to the right.
  3. KMn* (center cell#) [(2U,1L),1D] where KMn* signifies a Knight-step nxn Méziriac square with the center cell number of the square and breaking down. These squares are identical to entry 1 above by 180° deg rotation along the horizontal axis.
  4. KMn* (center cell#) [(1D,2R),1L] where KMn* signifies a Knight-step nxn Méziriac square with the center cell number of the square and breaking to the left. These squares are identical to entry 2 above by 180° deg rotation along the vertical axis.

depending whether the 1's lie on the yellow or blue diagonal. In the next part the Méziriac square method employing the two cell break will also be shown.

  1. Every number on the main diagonal is represented at least once in this type of square.
  2. For Méziriac Knight-step where n is divisible by 3, i.e., 3(2n + 1) no squares are magic.

Construction Knight-step Méziriac Magic Squares

The set of 5x5 Squares

  1. To generate the regular square, KM5* 13 [(2D,1L),1U], place a 1 into the center of the first row of a 5x5 square and fill in cells by advancing in a knight fashion to the down to the left until blocked by a previous number.
  2. Move one cell up.
  3. Repeat the process until the square is filled, as shown below in squares 1-5.
1
5
3
1
4
2 6
2
9 5
113 7
10 1
48
2 6
3
9 13 5
113 7
1014 1
4168 12
15 2 6
4
17 9 21 13 5
11320 7
1014 1 18
4168 12
15 2 19 6
5
17 9 21 13 5
11320 7 24
102214 1 18
4168 25 12
23 15 2 19 6

The Other Four 5x5 Méziriac Knight-step Squares

A KM5* 7 [(2D,1L),1U]
15 2 19 6 23
92113 5 17
3207 24 11
22141 18 10
16 8 25 12 4
 
B KM5* 16 [(2D,1L),1U]
24 11 3 20 7
181022 14 1
12416 8 25
62315 2 19
5 17 9 21 13
 
C KM5* 23 [(2D,1L),1U]
1 18 10 22 14
25124 16 8
19623 15 2
13517 9 21
7 24 11 3 20
 
D KM5* 5 [(2D,1L),1U]
8 25 12 4 16
2196 23 15
21135 17 9
20724 11 3
14 1 18 10 22

A 7x7 Méziriac Knight-step Square

The construction of a 7x7 Méziriac Knight-step square KM7* 18 [(2D,1L),1U] from the broken yellow diagonal is shown below using the knight-step approach.

1
3 8
12 7
4 9
13 1
5 10 15
2 14
11 6
2
20 3 25 8
2212 17 7
214 26 9
13 18 1 23
5 27 10 15
2919 2 24 14
28 11 16 6
3
30 20 3 42 25 8 47
221244 34 17 7 39
21436 26 9 48 31
134535 18 1 40 23
5 37 27 10 49 32 15
46 2919 2 41 24 14
38 28 11 43 33 16 6

The Plane of Méziriac 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 KM7* 39 [(2D,1L),1U] one can move up the right diagonal on a plane of four KM7* 39 [(2D,1L),1U] and generate the complete set of 7 squares as is shown in Part IV of the new Bachet de Méziriac method.

This completes this section on regular and non-regular De La Méziriac two step squares (Part I). The next section deals with a new Méziriac Knight-step square method (Part II). To return to homepage.


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