New De La Loubère Method and Squares (Part IIIB Continued)

Regular and Non-Regular Loubère-Variable Knight Combo Magic and Semi-Magic Squares

the Full Monty III

A Loubere square

A Discussion of the New Methods

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

the 5x5 and 7x7 regular Loubère squares are shown below as examples:

17 24 1 8 15
2357 14 16
4613 20 22
101219 21 3
11 18 25 2 9
  
30 39 48 1 10 19 28
38477 9 18 27 29
4668 17 26 35 37
51416 25 34 36 45
13 15 24 33 42 44 4
21 23 32 41 43 3 12
22 31 40 49 2 11 20

this page which is a continuation of new Loubère Knight methods Part IIIA. To start we place the number 1 on either of 2 broken diagonals, filling the square in a Loubère fashion until a block is encountered, then proceeding with a knight move results in the generation of regular and non-regular squares.

All odd squares having the numerical 1's lying on two broken diagonals symmetrical with the light grey main diagonal behave differently from typical Loubère squares. After a break/ knight[2,1] (2 down,1 left) for the broken yellow diagonal and break/knight[1,2] (1 down, 2 left) for the broken light blue diagonal, one regular square and n - 1 non-regular squares are produced. Squares belonging to one diagonal group are identical to another square on the other diagonal group. Using the 5x5 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 (starting with the square generated from 1 in the first row) where the values range from ½(n2 - n + 2) to ½(n2 + 5).

the set of Broken Diagonals
11
11
11
1 1
11
11
11
 
center Value
EquationValue K[3U,4R])
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 Loubère squares, which I will label Ln* (center cell#) (K[L1,L2] or K[L3,L4]) where (Ln* signifies a nxn Loubère square with the center cell number of the square and having a break followed by a Knight move L1= number1(Up) and L2=number2(Right) Or a Knight move L3= number3(Down) and L4=number4(Left). this is opposed to the original Loubère squares depicted in the introduction as L5* 13 [1D] and L7* 25 [1D]. the squares on this page exhibit the following properties:

  1. Every number on the main diagonal is represented at least once in this type of square.
  2. For the Knight[3U,4R] method no odd squares divisible by 3 are magic, except for those also divisible by 5 which are semi-magic.
  3. For the Knight[3D,2L] method no odd squares divisible by 3 are magic, however those divisible by 7 are semi-magic.

Construction of Regular and Non-regular Loubère-Knight squares

Examples of Magic 7x7 Squares Using the Knight[3U,4R] Method

The result of construction of these squares using the Knight[3U,4R] method is : All 7x7 Knight[3U,4R] squares identical to 7x7 knight[4D,3L] squares.

See bottom of page. Since no Knight[3U,4R] 9x9 square is magic, the 11x11 will be constructed:

An 11x11 Knight[3U,4R] Example

  1. Place the number 1 at the center of the first row of a 7x7 square and fill in cells by advancing diagonally upwards to the right until blocked by a previous number.
  2. Using a knight move 3 up and 4 right.
  3. Repeat the process until the square is filled, as shown below in squares 1-4 (Show only first knight move per square).
  4. Compare to the knight move 4 down and 3 left from New De La Loubère Method and Squares (Part IIIA) which this is meant to complement.
1
1 213039
1120 2938
10 1928 37
9 18 2736
817 26 35
71625 34
152444 6
2343 514
42 4 1333
312 3241
22231 40
2
1 213039 4857
1120 293847 56
10 1928 374666
9 18 2736 4565
817 26 3555 64
71625 34 5463
152444 53 62 6
234352 61 514
425160 4 1333
5059 312 3241
5867 22231 4049
3 L11* 63 K[3U,4R]
778695 104 1131 213039 4857
8594103 112 1120 293847 5676
93102111 10 1928 374666 7584
1011219 18 2736 456574 83 92
120817 26 3555 647382 91100
71625 34 5463 728190 110119
152444 53 6271 8089109 1186
234352 61 7079 99108117 514
425160 69 7898 1071164 1333
505968 88 97106 115312 3241
586787 96 105114 22231 4049
 
A L11* 65 K[4D,3L]
758096 101 1171 173338 5459
7995100 116 1116 323753 5874
94110115 10 1531 365257 7378
1091149 14 3035 515672 8893
113813 29 3450 667187 92108
71228 44 4965 708691 107112
222743 48 6469 8590106 1116
264247 636884 89105121 521
814662 67 8399 1041204 2025
496177 82 98103 119319 2440
607681 97 102118 21823 3955
A eleven series   A eleven series

Note that both squares are non-regular, i.e, not complementary as shown in the connectivities of the complementary table. Also to decrease crowding only the first and middle eleven entries of the connectivity table are shown, since the second to fifth groups have identical connectivities as the first.

The Knight[3D,2L] Method: A Semi-Magic 7x7 Square and 11x11 Magic Square

A 7x7 and 11x11 Knight[3D,2L] Examples which complement the Knight[2U,3R] squares. the 7x7 is the first instance of a semi-magic, wherein the left main diagonal sums to 168. the sums (S+d) of the set of 7x7 squares are also shown, where only one square is fully magic:

A L7* 24 K[3D,2L]]
31 42 46 1 12 16 27
41457 11 15 26 30
44610 21 25 29 40
5920 24 35 39 43
8 19 23 34 38 49 4
18 22 33 37 48 3 14
28 3236 47 2 13 17
 
7x7 Square Values of K[3D,2L]
center ValueS + dd
22154-21
23161-14
24168-7
251750
261827
2718914
2819621
 
B L11* 60 K[3D,2L]]
698499 103 1181 163135 5065
8398102 117 1115 303449 6468
97101116 10 1429 444863 6782
1001159 13 2843 476277 8196
114812 27 4246 617680 95110
72226 41 4560 757994 109113
212540 55 5974 7893108 1126
243954 58 7388 92107111 520
385357 72 8791 1061214 1923
525671 86 90105 120318 3337
667085 89104119 21732 3651

Equality of Knight[(number1*U),(number2*R)] and Knight[(number2*D,number1*L)]

It was mentioned above that All 7x7 Knight[3U,4R] squares identical to 7x7 knight[4D,3L] squares. At this point this applies to all squares which come under the general formula Knight[(number1*U),(number2*R)] and Knight[(number2*D,number1*L)]. thus for example when n = 11, K[5U,6R] is identical to K[6D,5L] where number1 = ½(n - 1) and number2 = ½(n + 1) and where number1 + number2 = n. thus the same result is obtained by moving in any of two knight directions. To go back up.

L7* 28 K[3U,4R] or K[4D,3L]
29 41 46 2 14 19 24
40451 13 18 23 35
44712 17 22 34 39
61116 28 33 38 43
10 15 27 32 37 49 5
21 26 31 36 48 4 9
25 3042 47 3 8 20

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 L11* 63 K[3U,4R] one can move up the right diagonal on a plane of four L11* 63 K[3U,4R] 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 Loubère-Knight squares. To go back to previous page New De La Loubère Method and Squares (Part IIIA). To return to homepage.


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