A New Procedure for Magic Squares (Part VII) Continuation

Loubère Knight Block trans-Modified 7x7 Squares

A Loubere knight square

A Discussion of the New Method

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.

The 7x7 regular Loubère square is shown below:

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

Normally the Loubère and Méziriac methods involve a stepwise approach of consecutive numbers, i.e., 1,2,3...n2. For example, A new generalized Loubère procedure where the initial number is placed anywhere on any n - 1 cells of the first row. However, in this new method although the numbers are added consecutively starting with 1 and ending with n2, which are then used to generate modified Loubère having numbers greater than n2.

The non-magic square may use a variety of methods. For example, adding a constant to a non-magic diagonal and converting that diagonal into one that is readily amenable to a transposition (trans) is one method.

Accordingly, the numbers are added consecutively starting out with 1 and ending with numbers greater than n2. Whenever a break occurs a knight move is performed instead of a normal translational move (up, down or sideways). The knight move is either a (2,1) or a variable knight move , e.g (3,2), for Méziriac squares squares or Loubère squares. Both these square types were covered in Loubère Knight and Méziriac Knight square methods. Dual modification is then performed as discussed above.

Moreover, it must be stated here that the magic sum has been modified from the known equation S = ½(n3 + n) to the general equation:

S = ½(n3 ± an)

which takes into account these new squares. The variable a, an odd number, is equal to 1,3,5,7 or ... and may take on + or - values. For example when a = 1 the normal magic sum S is implied. When a takes on different odd values S gives the magic sum of a modified magic square. It will be shown that the addition or subtraction of n2 to some of the cells in the square gives rise to a new magic square.

trans-Conversion of Loubère non-magic Squares to Magic Squares

Method I: Start on the first row center (1 ⇒ (2,1) up knight break)
  1. Begin generating the non-magic Loubère square by placing 1 in the center cell of the first row and add six non-consecutive using heptad table I and method of New generalized Loubère procedure.
  2. Perform a knight break two cells right one cell up as shown in color (this is done in order to produce a non-magic square).
  3. Repeat the process until the square is filled as shown below in squares 1-2.
  4. As shown square 2 is non-magic and the last column corresponds to how much must be added or subtracted to the previous column numbers to get to 175.

  5. 1
    1 8
    3
    5
    2
    6
    7
    4 11
    2
    175
    29 36 431 81522 154+21
    38 45 310 172431168+7
    47 5 1219 263340182-7
    2 9 1623 303744161+14
    13 20 2734 41486189-14
    21 28 3542 49714196-21
    25323946 4 11181750
    175175175 175 175175 175 175
  6. The next series of moves (in groups of 2), adding or subtracting numbers from a row converts all the sums to 175 with the generation of duplicate numbers.
  7. At this point (square 3) four numbers are duplicates (in blue and orange ).
  8. In order to remove these duplicates add n2 = 49 to each row so that each row, column and diagonal have exactly one modification.
  9. Square 4 is produced whereby all the sums have been converted to the magic sum 224, and S = ½(n3 + 15n).
3
175
29 36 641 81522 175
38 45 317 172431175
47 5 1212 263340175
2 9 1623 305144175
13 20 2734 41346175
21 28 1442 49714175
25323946 4 1118175
175175175 175 175175 175 175
4
224
29 85 641 81522 224
38 45 366 172431224
47 5 6112 263340224
2 9 1623 795144224
13 20 2734 41836224
21 28 1442 49763224
74323946 4 1118224
224224224 224 224224 224 224
Method II: Start at first row center (1 ⇒ (2,1) down knight break)
A double Modification
  1. Begin generating the non-magic Loubère square by placing 1 in the center cell of the first row and performing a knight break move down.
  2. Fill the square as shown to produce non-magic square 2.
  3. As shown square 2 the last row corresponds to how much must be added or subtracted to the previous column numbers to get to 182 (the sums in the last column.

  4. 1
    1 4
    3
    2
    8 11
    7 10
    6 9
    5
    2
    35
    40 27 141 30174 133
    26 13 4936 16339182
    12 48 3522 23825182
    47 34 218 372411182
    33 20 743 231046182
    19 6 4229 94532182
    5412815 44 3118182
    182189196 154 161168 175 182
    0-7-14+28 +21 +14+7
  5. To begin converting this square into a magic one, add 28 to each of the entries of the right blue diagonal.
  6. At this point (square 3) all the sums have changed by 28.
  7. Transpose (trans), two cells up, all the 7 numbers from the blue diagonal of square 2 to the second adjacent left diagonal (square 4). At this point the highest sum possible is 231. This sum may change.

  8. 3
    231
    40 27 141 301732 161
    26 13 4936 163139210
    12 48 3522 303825210
    47 34 2136 372411210
    33 20 3543 231046210
    19 34 4229 94532210
    33412815 44 3118210
    210217224 182 189196 203 210
    28211456 49 4235
    4
    231
    40 27 141 21732 133
    26 13 498 163139182
    12 48 722 303825182
    47 6 2136 372411182
    5 20 3543 231046182
    19 34 4229 9454182
    33412815 44 318182
    182189196 154 161168 175 182
    0-7-14+28 +21 +14+7
  9. The next series of moves (in groups of 2), adding or subtracting numbers from a column converts all the sums to 182 with the generation of duplicate numbers in green.
  10. Adding n2 = 49 to (26, 41, 7, 23, 38 and 42), shown in orange , removes four duplicates and gives the semi-magic square 6.
  11. In the second modification adding n2 = 49 to (9, 10, 11, 12, 13, 14 and 15) gives square 7 where the magic sum is 280 and where S = ½(n3 + 31n).
5
231
40 27 1429 231732 182
26 13 498 163139182
12 41 722 303832182
47 6 736 373811182
5 20 3543 231046182
19 34 4229 9454182
33412815 44 318182
182182182 182 182182 182 182
6
280
40 27 1429 231781 231
75 13 498 163139231
12 41 722 308732231
47 6 5636 373811231
5 20 3543 721046231
19 34 4278 9454231
33902815 44 318231
231231231 231 231231 231 231
7
280
40 27 6329 231781 280
75 62 498 163139280
61 41 722 308732280
47 6 5636 373860280
5 20 3543 725946280
19 34 4278 58454280
33902864 44 318280
280280280 280 280280 280 280

This completes this section on the new block Loubère Knight Block trans-Modified 7x7 Squares (Part VII). To continue to section VIII A New Procedure for Magic Squares Continuation. To return to homepage.


Copyright © 2009 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com