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

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, have a two cell break like the regular Méziriac square and is labeled, as in previous web pages, as follows:

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

depending whether the 1's lie on the yellow or blue diagonal.

  1. Every number on the main diagonal is represented at least once in this type of square.,/li>
  2. For Méziriac Knight-step where n is divisible by 3, i.e., 3(2n + 1) only semi-magic squares are possible for the (2D,1L),2L or 2D series.

Construction Knight-step Méziriac Magic Squares

The set of 5x5 Squares (Method II)

  1. To generate the regular square, KM5* 19 [(2D,1L),2L], place a 1 left of center on 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 two cells left.
  3. Repeat the process until the square is filled, as shown below in squares 1-5.
1
6 5
3
1
4
2
2
6 5
3 9
7 1
411 10
8 2
3
12 6 5
16315 9
7 1 13
411 10
8 2 14
4
12 6 18 5
16315 9
719 1 13
411 10 17
8 20 2 14 21
5
12 24 6 18 5
16315 22 9
25719 1 13
41123 10 17
8 20 2 14 21

The Other Four 5x5 Méziriac Knight-step Squares

 
A KM5* 22 [(2D,1L),2L]
20 2 14 21 8
24618 5 12
31522 9 16
7191 13 25
11 23 10 17 4
  
B KM5* 11 [(2D,1L),2L]
9 16 3 15 22
13257 19 1
17411 23 10
21820 2 14
5 12 24 6 18
  
C KM5* 8 [(2D,1L),2L]
1 13 25 7 19
10174 11 23
14218 20 2
18512 24 6
22 9 16 3 15
  
D KM5* 5 [(2D,1L),2L]
23 10 17 4 11
21421 8 20
6185 12 24
15229 16 3
19 1 13 25 7

A 7x7 Méziriac Knight-step Square

The construction of a 7x7 Méziriac Knight-step square KM7* 33 [(2D,1L),2L] from the broken yellow diagonal is shown below using the knight-step approach.

1
11 3
8 7
4 12
9 1
5 13
10 2
15 14 6
2
11 3 27 19
16 8 7 24
29428 20 12
179 1 25
5 22 21 13
10 2 26 18
23 15 14 6
3
11 35 3 27 44 19 36
481640 8 32 7 24
29428 45 20 37 12
17419 33 1 25 49
5 22 46 21 38 13 30
42 1034 2 26 43 18
23 47 15 39 14 31 6

Three 9x9 Méziriac Knight-step Squares

Three 9x9 Méziriac Knight-step square from the broken yellow diagonal are shown along with the accompanying table of 9 squares showing the triad sums of the left diagonal. Square A shows a typical knight (2D,1L) move and left break move.

A KL9* 51 [(2D,1L),2L]
792969 195918 49839
14544 447534 652455
307020 601050 94080
46545 763566 255615
712161 11511 418131
63777 366726 571647
226212 52242 733272
387828 682758 17487
631353 34374 336423
 
B KL9* 66 [(2D,1L),2L]
13533 437433 642363
296919 591849 83979
54444 753465 245514
702060 10509 408030
54576 356625 561546
216111 51141 813171
377736 672657 16476
621252 24273 327222
782868 275817 48738
 
C KL9* 9 [(2D,1L),2L]
286827 581748 73878
53343 743364 236313
691959 18498 397929
44475 346524 551454
206010 50940 803070
457635 662556 15465
611151 14181 317121
773667 265716 47637
12522 427332 722262
9x9 Cell Values and S of K[(2D,1L),1R]
center ValueS + dd
2345081
683690
32288-81
7745081
413690
5288-81
5045081
143690
59288-81

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* 33 [(2D,1L),2L] one can move up the right diagonal on a plane of four KM7* 33 [(2D,1L),2L] 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). To return to homepage.


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