A New Procedure for Magic Squares (Part IX) Continuation

Loubère and Méziriac Block Modified 9x9 Squares

A stairs

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.

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, to generate an intermediate non-magic square the final square is transformed into a modified Loubère or Méziriac, having numbers greater than n2.

This is done by taking the non-magic squares and converting them into magic ones using a variety of means. These squares are depicted below in methods I and II. The generation of these magic squares involves an almost normal approach:
Numbers are added consecutively starting out with 1 and ending with numbers greater than n2 after modification. A break involves translational moves (up, down or sideways).

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.

Conversion of a Loubère non-magic Square to a Magic Square

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 at the first cell of the last row and add three consecutive numbers followed by break 2 cells down.
  2. Fill in the rest of the square as shown below in squares 1-2.
  3. 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 405.

  4. 1
    12 15
    11 14
    10
    9
    8 20
    4 719
    3 618
    2 5 17
    1 13 16
    2
    405
    42 1254 15 66276939 814050
    11 4314 65 26683880 41396+9
    52 2264 25 76377949 10414-9
    21 6324 75 3678489 514050
    62 2374 35 7747850 20396+9
    31 7334 4 4675819 61333+72
    72 333 45 6571860 30324+81
    32 244 5 56175929 71315+90
    1 4313 55 16672870 40333+72
    324324324 324 405405 405 405405 405
  5. At this point (square 3) add n2 = 81 to 31, 32, 44 and 55.
  6. The next series of moves (in groups of 2), adding or subtracting numbers from a column (square 4) converts all the sums to 405 with the generation of a duplicate number (68).

  7. 3
    405
    42 1254 15 66276939 814050
    11 4314 65 26683880 41396+9
    52 2264 25 76377949 10414-9
    21 6324 75 3678489 514050
    62 2374 35 7747850 20396+9
    112 7334 4 4675819 61414-9
    72 1143 45 6571860 304050
    32 2125 5 56175929 71396+9
    1 4313 136 16672870 40414-9
    405405405 405 405405 405 405405 405
    4
    405
    42 1254 15 66276939 81405
    11 4314 65 26683880 50405
    52 2264 25 76377949 1405
    21 6324 75 3678489 51405
    62 2374 35 7747859 20405
    112 7334 4 4675810 61405
    72 1143 45 6571860 30405
    32 2125 5 56176829 71405
    1 4313 136 16671970 40405
    405405405 405 405405 405 405405 405
  8. In order to remove this duplicate add n2 = 81 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 486, and S = ½(n3 + 27n).
5
486
42 1254 15 662715939 81486
11 4314 65 261583880 50486
52 2264 25 157377949 1486
21 6324 156 3678489 51486
62 23155 35 7747859 20486
112 15434 4 4675810 61486
72 1143 45 65718141 30486
32 2125 5 56176829 152486
82 4313 136 16671970 40486
486486486 486 486486 486 486486 486

Conversion of a Méziriac non-magic Square to a Magic Square

Method II: Start at first row center (1 ⇒ (2,1) down knight break)

  1. Begin generating the non-magic Méziriac square by placing 1 to the right of center cell of the fifth row and add four consecutive numerals, followed by a right break.
  2. Fill the square as shown to produce non-magic square 2.
  3. 1
    5 6
    14 15 4
    3 13
    212
    111
    10
    9
    8
    7
    2
    657
    5 616 26 36465666 76333
    14 1525 35 45556575 4333
    23 2434 44 5464743 13333
    32 3343 53 6373212 22333
    41 4252 62 7211121 31333
    50 5161 71 81102030 40414
    59 6070 80 9192939 49414
    68 6979 8 18283848 58414
    77 787 17 27374757 67414
    369378387 396 405333 342 351360 333
  4. To begin converting this square into a magic one, add 81 to each of the entries 1, 2, 3, 4 and 5. This modifies all the sums in the last column to 414 and five sums on the next to the last row.
  5. The next series of moves (in groups of 2), adding or subtracting numbers from a row (square 4) converts all the sums to 414 with the exception of the right diagonal. Four duplicates, 23, 80, 27 and 29 are produced.

  6. 3
    657
    86 616 26 36465666 76414
    14 1525 35 45556575 85414
    23 2434 44 54647484 13414
    32 3343 53 63738312 22414
    41 4252 62 72821121 31414
    50 5161 71 81102030 40414
    59 6070 80 9192939 49414
    68 6979 8 18283848 58414
    77 787 17 27374757 67414
    450378387 396 405414 423 432441 414
    -36 +36+27 +18 +90-9-18 -27
    4
    657
    86 616 26 36465666 76414
    14 1525 35 45556575 85414
    23 2434 44 54647484 13414
    32 3343 53 63738312 22414
    41 4252 80 7282113 31414
    50 5161 71 81102030 40414
    23 9670 80 9192939 49414
    68 6979 8 27282948 58414
    77 787 17 27374757 67414
    414414414 414 414414 414 414414 414
  7. Adding 3n2 = 243 to (23, 33, 34, 80, 55, 42, 29, 66 and 40), shown in orange , removes four duplicates and gives the magic square 5, where the magic sum is 657 and where S = ½(n3 + 65n).
5
657
86 616 26 364656309 76657
14 1525 35 452986575 85657
23 24277 44 54647484 13657
32 27643 53 63738312 22657
41 4252 323 7282113 31657
50 5161 71 81102030 256657
266 9670 80 9192939 49657
68 6979 8 272827248 58657
77 787 17 270374757 67657
657657657 657 657657 657 657657 657

This completes this section on the new block Loubère Knight Block Modified 9x9 Squares (Part IX). To return to homepage.


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