A New Procedure for Magic Squares (Part III)

Consecutive Internal 13x13 Mask-Generated Squares

A mask

A Discussion of the New Method

Magic squares such as the Loubère have a center cell which must always contain the middle number of a series of consecutive numbers, 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.

In this method the numbers on the square are placed consecutively starting from the second leftmost column and entered across every other cell. Consecutive numbers are then added to the next rows boustrophedonically. When n > 7 at least one row turns into a regular left to right order, increasing by one for each n. In addition every other number in the center row (starting with the second cell) will take on its complement. For example for a 13x13 square the second number 82 becomes 90 and 90 becomes 82 (see the 13x13 example below). The final square is composed of numbers which may not be in serial order. For example, negative numbers or numbers greater than n2 may be present in the square. This method is being repeated for the 13x13 to show that a second row in non-boustrophedonic order is needed for the next higher 4n +1 square.

In addition, it will also be shown that the sums of these squares follow the sum equation shown in the New block Loubère Method. :

S = ½(n3 ± an)

Construction of a 13x13 Magic Square

Method: Reading boustrophedonically (like a sidewinder snake) - use of mask
  1. Construct the 13x13 square (as was shown for the 9x9 square in Part II) first down and then up as shown in squares 1 and 2.
  2. Rows 2 and 4 follow regular reading order as opposed to the rest of the rows which follow boustrophedonic order (like a snake). The last row shows the difference of the sum of the grey row from a sum of a typical 13x13 magic square, viz., S = 1105. The light grey entries, 1643 and and 1659, correspond to the diagonal sums.
  3. 1
    1 2 3 456
    789 10 11 12 13
    1415 16 17 18 19
    202122 23 24 25 26
    2728 29 30 31 32
    393837 36 35 34 33
       
    40 4142 43 44 45 46
    52 51 50 49 48 47
    53 5455 56 57 58 59
    65 64 63 62 61 60
    66 6768 69 70 71 72
    78 77 76 75 74 73
    2
    1643
    137 1 1362 1353 134413351326131959146
    71438 1429141 10140 11 139 12 138 13919192
    150141491514816 14717 14618 145 19 1441128-23
    2015621 15522 15423 153 24152 25 151 26108223
    163271622816129 16030 15931 158 32 1571297-192
    39164381653716636 16735 168 34 169 33 1251-146
    79 90 818883 86 85 8487 8289 809111050
    40 92 41934294 4395 44 96 45 97 46868237
    104 52 1035110250 10149 100 48 99 47 981004101
    53 105 541065510756 10857 109 58 110 59103768
    117 65 11664115 63 11462 113 61 112 60 1111173-68
    66 118 671196812069 12170 122 71 123 721206-101
    130 78 1297712876 12775 126 74 125 73 1241342-237
    110511051105 110511051105 1105 11051105 11051105 11051105 1659
  4. Since the columns are all equal to 1105 add or subtract the numbers in the last row from the center column values to generate square 3. At this point all sums are 1105 except for the diagonals. Also six duplicates have been generated.
  5. 3
    1643
    137 1 1362 1353 280413351326131 1105
    714381429141 202140 11139 12 138 131105
    150141491514816 12417 14618 145 19 1441105
    201562115522 15446 15324152 25 151 261105
    163271622816129 -3230 15931 158 32 1571105
    391643816537166-110 16735 168 34 169 33 1105
    79 90 818883 86 85 8487 8289 80911105
    40 92 41934294 28095 44 96 45 97 461105
    104 52 1035110250 20249 100 48 99 47 981105
    53 105 5410655107124 10857 109 58 110 591105
    117 65 11664115 63 4662 113 61 112 60 1111105
    66 118 6711968120-32 12170 122 71 123 721105
    130 78 1297712876 -11075 126 74 125 73 1241105
    110511051105 110511051105 1105 11051105 11051105 11051105 1659
  6. To convert all sums to a magic Sum we generate a mask whereby the sums of the columns and rows are constructed as in the box below. This assures that when each of these values is added to the corresponding cell in square 3 (as in the de la Hire method) that all sums will equal the magic sum.

  7. Generate the mask using the 538 and 554 factors adding these factors to the appropriate cells in square 3 to generate square 4. This will take a while especially if duplicates are generated.
  8. Mask A
    538 554 538 554554 538
    538 538 554 554 554538
    538 554 538554538 554
    538 554 538 554538 554
    554 538 538538 554554
    554 554 538 554 538 538
    554538 538 538554 554
    538 554 538554538 554
    538 554538 554 554538
    554 554 538554 538 538
    554 538 538554 554538
    554554 538 538538 554
    538 554 554 538538554
    + Square 3
  9. Square 4 has a magic sum equal to 4851, i.e., S = 4851 = ½(n3 + 505n). The cell entries in color correspond to the color s of the factors from the Mask A.
4
4381
137 539 136556 673 38344133559132 5541314381
5456815621429 695202140 11693 12 138 5514381
150146871570216 66257168418 145 573 1444381
5587102115522 69246 70724690 579151 264381
1632716258269929 5063069731 712 586 1574381
593718576 71937166-110 16735 706 572 169 33 4381
633 90 619626 83 8685 622641 8289 80 6454381
40 630 419342648 28095 44634 599 635 6004381
642 52 10351656 588 75649 654 48 99 585 984381
53 105 60810655661 124 646611 109 596 110 5974381
671 65 116602115 63 584616 113 61 112 614 6494381
66 672 621119606 658-32 65970 676 71 123 724381
130 78 129615682 76 44475664 74 663 73 6784381
438143814381 438143814381 4381 43814381 43814381 43814381 4381

This completes this section on a new Consecutive 13x13 Internally Added Mask-Generated Squares (Part III). To return to homepage.


Copyright © 2010 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com