New Bachet de Méziriac Three Step Staircase and Knight Methods and Squares (Part II)

Regular and Non-Regular Squares

A Loubere square

A Discussion of the New Methods

An important general principle for generating odd magic squares by the 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 3-step manner. 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 or 7x7 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 squares, which I will label Mn* (center cell#) [3S,2R or 2U] where (Mn* signifies a three step nxn Méziriac square with the center cell number of the square and breaking either 2 cells right or 2 cells up.

In the second method, a Knight, the squares are labeled MKn* (center cell#) [3S,(3D,1L)] where (Mn* signifies a three step nxn Méziriac square with the center cell number of the square and breaking using either a 3 down, 1 left knight move or a MKn* (center cell#) [3S,(3L,1D)] using a 3 left, 1 down knight move.

  1. Every number on the main diagonal is represented at least once in this type of square.
  2. For Méziriac odd squares where n is divisible by 3, i.e., 3(2n + 1) are non-magic, while odd squares where n is divisible by 5 are semi-magic. All others are magic.
  3. For Méziriac Knight squares odd squares n is divisible by 3 are non-magic and those where n is divisible by 5 are semi-magic. All others are magic.

Construction of a Regular 5x5 Three Step Méziriac Magic Square

  1. To generate the regular square, M5* 13 [3S,2R], place a 1 to the right of the center of the third row of a 5x5 square and fill in cells by advancing diagonally upwards to the right in steps of 3 until blocked by a previous number.
  2. Move two cells right.
  3. Repeat the process until the square is filled, as shown below in squares 1-4.
1
 
3
1
 
2
2
5 6
9 3
7 1
114 10
2 8
3
5 6 12
169 15 3
713 1
114 10
14 2 8
4 M5* 13 [3S,2R]
5 18 6 24 12
16922 15 3
72513 1 19
23114 17 10
14 2 20 8 21

The Four other Regular 5x5 Three Step Méziriac Magic Squares

The fully constructed non-regular squares shown below all have left diagonal S values of 65.

A M5* 14 [3S,2R]
1 19 7 25 13
171023 11 4
82114 2 20
24125 18 6
15 3 16 9 22
 
B M5* 11 [3S,2R]
3 16 9 22 15
19725 13 1
102311 4 17
21142 20 8
12 5 18 6 24
 
C M5* 15 [3S,2R]
2 20 8 21 14
18624 12 5
92215 3 16
25131 19 7
11 4 17 10 23
 
D M5* 12 [3S,2R]
4 17 10 23 11
20821 14 2
62412 5 18
22153 16 9
13 1 19 7 25

The Regular 7x7 Three Step Méziriac Magic Square

The construction of the 7x7 regular Méziriac square M7* 25 [3S,2R] from the broken yellow diagonal is shown below again in steps of 3.

1
2 10
12 4
1415 6
1 9
3 11
5 13
7 8
2
2 10 18 26
2912 20 28 4
1415 23 6
17 25 1 9
19 27 3 11
22 5 13 21
24 7 8 16
3 M7* 25 [3S,2R]
2 34 10 42 18 43 26
291237 20 45 28 4
143915 47 23 6 31
411749 25 1 33 9
19 44 27 3 35 11 36
46 22 5 30 13 38 21
24 7 32 8 40 16 48

Construction of a Non-regular 5x5 Three step Méziriac Knight Magic Square

5x5 Squares

  1. To generate the non-regular square, MK5* 15 [3S,(3D,1L)], place a 1 to the right of the center of the third row of a 5x5 square and fill in cells by advancing diagonally upwards to the right in steps of 3 until blocked by a previous number.
  2. Move three cells down one cell left (the knight break).
  3. Repeat the process until the square is filled, as shown below in squares 1-5.
1
5
3
1
4 6
2
2
5 7 14
10 123
815 1
134 6
112 9
3
5 167 14
1910 123
815 117
134 206
11218 9
4 MK5* 15 [3S,(3D,1L)]
5 167 2314
191021 123
82415 117
22134 206
11218 925

The Other Four Regular and Non-regular 5x5 Three Step Méziriac Knight Magic Squares

 
A MK5* 11 [3S,(3D,1L)]
1 17 8 24 15
20622 13 4
92511 2 18
23145 16 7
12 3 19 10 21
 
B MK5* 13 [3S,(3D,1L)]
3 19 10 21 12
17824 15 1
62213 420
25112 189
14516 723
 
C MK5* 12 [3S,(3D,1L)]
218 9 2511
16723 145
102112 319
24151 178
13420 622
 
D MK5* 14 [3S,(3D,1L)]
420 6 2213
18925 112
72314 516
21123 1910
15117 824

A Non-Regular 7x7 Three Step Méziriac Knight Square

The construction of a 7x7 non-regular 3 Step Méziriac Knight square MK7* 28 [3S,(3D,1L)] from the broken yellow diagonal similar to the method above is shown below again in steps of 3, followed by a 3 down,1 left knight break. The last table shows also the sums (S) of the left main diagonal for all the 7x7 regular and non-regular squares in this set.

1
2 11 20
13 15 4
817 6
19 1 10
21 3 12
5 14 16
7 9 18
2
2 31 11 20 22
3313 15 244
817 26 635
19 281 3010
2123 332 12
255 3414 36 16
27729 9 18
3
2 31 11 4020 4922
331342 1544 244
83717 4626 635
391948 281 3010
214323 332 12 41
45255 3414 36 16
27729 938 18 47
 
7x7 Cell Values and S of K[3S,(3D,1L)]
Center ValueS + dd
231750
271750
241750
281750
251750
221750
261750

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 MK7* 28 [3S,(3D,1L)] one can move up the right diagonal on a plane of four MK7* 28 [3S,(3D,1L)] 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 3 step regular and non-regular Méziriac squares (Part II). To return to homepage.


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