A New Procedure for Magic Squares (Part II)

7x7 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.

This site will explore new squares which are derived in a different manner from the standard Loubère approach. A 7x7 Loubère square which is produced using the staircase method is shown below in I. If all the even numbers in this square are exchanged for their complements the result is II:

I
30 39 48 1 1019 28
38477 9 1827 29
4668 17 2635 37
51416 25 3436 45
13 15 24 33 4244 4
21 23 32 41 433 12
32 31 40 49 211 20
 
II
20 39 2 1 4019 22
292732 9 747 12
44442 17 2435 37
451416 25 3436 5
13 15 26 33 86 46
38 3 43 41 1823 21
18 31 10 49 4811 30

This is equivalent to using the following complementary table where the number sequence is 1 → 24 → 3 → 22...25 → 26 → 27...49.


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24     
25
49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26     

Square II, however, is not magic using the standard Loubère approach but a new approach which gives two types of squares is. Taking the complementary table numbers and placing them in the same order into a 7x7 square produces A. If the numbers are placed boustrophedonically (reading in regular order then in backward order) the result is B:

A
1 48 3 46 544 7
42940 11 3813 36
153417 32 1930 21
282326 25 2427 22
29 20 31 18 3316 35
14 37 12 39 1041 8
43 6 45 4 472 49
 
B
1 48 3 46 544 7
361338 11 409 42
153417 32 1930 21
222724 25 2623 28
29 20 31 18 3316 35
8 41 10 39 1237 14
43 6 45 4 472 49

Both these squares after manipulation produce two types of magic squares. One by the manipulation of individual cells, the other by the use of a symmetrical C2 mask (symmetrical by 180° rotation) of n2 numbers (see Method II below). Moreover, the 4n + 3 produce squares with internal cross three cells wide. These squares will are developed in new 7x7 and 11x11 Cross Squares Methods (Part III). The magic squares produced have only partially sequential order and not all the numbers from 1 to n2 are present. Because of the manipulation numbers greater than n2or less than 1 are generated.

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

S = ½(n3 ± an)

Construction of 7x7 Magic Squares

Method I: Reading from left to right
  1. Recopy square A with the sum and row columns (in gray).
  2. As shown below this square is not magic because all the columns and rows don't sum to 175.
  3. Adjust the center column by adding and subtracting numbers from the selected cells to generate 2.
  4. Adjust the center row by adding and subtracting numbers from the selected cells, containing duplicate cells in green.
  5. Use the mask A consisting of the numbers 1 and 2 to remove duplicates. Where the numbers intersect with the numbers of square 2 add the appropriate factor of n2, 49 for "1" and 98 for "2".
  6. Addition of the right factor gives square 3 with S = ½(n3 + 57n).
1
175
1 48 3 46 544 7154 -21
42940 11 3813 3618914
153417 32 1930 21168-7
282326 25 2427 22175 0
29 20 31 18 3316 351827
14 37 12 39 1041 8161-14
43 6 45 4 472 49196 21
172177174 175 176173 178175   
-3 2 -1 0 1-2 3
2
175
1 48 3 67 544 7175
42940 -3 3813 36175
153417 39 1930 21175
312127 25 2329 19175
29 20 31 11 3316 35175
14 37 12 53 1041 8175
43 6 45 -17 472 49175
175175175 175 175175 175175
Square 2 +
Mask A
2 1 1
12 1
2 2
11 11
2 2
1 2 1
1 12
3
371
1 48 101 116 593 7 371
9110740 46 3813 36371
1133417 39 11730 21371
807027 25 2378 68 371
29 20 129 11 3316 133371
14 37 12 102 10139 57371
43 55 45 32 1452 49371
371371371 371 371371 371371
Method II: Reading Boustrophedonically
  1. Recopy square A with the sum and row columns (in gray).
  2. Invert rows 2, 4 and 6 and color the left diagonal blue.
  3. As shown below this square is not magic because all the columns and rows don't sum to 175.
  4. Adjust the blue cells by adding and subtracting numbers from the selected cells to generate 3. In addition, the cells containing 22, 23, 26 and 28 on the blue diagonal are duplicates of ther cells.
  5. Use the mask B consisting of numbers 1, 2 and 3 to remove duplicates. Where the numbers intersect with the numbers of square 2 add the appropriate factor of n2, 49 for "1" and 98 for "2" and 147 for "3".
  6. Addition of the right factor gives square 4 with S = ½(n3 + 57n).
1
175
1 48 3 46 544 7154 -21
42940 11 3813 3618914
153417 32 1930 21168-7
282326 25 2427 22175 0
29 20 31 18 3316 351827
14 37 12 39 1041 8161-14
43 6 45 4 472 49196 21
172177174 175 176173 178175   
-3 2 -1 0 1-2 3
2
175
1 48 3 46 544 7154 -21
361338 11 409 4218914
153417 32 1930 21168-7
222724 25 2623 28175 0
29 20 31 18 3316 351827
8 41 10 39 1237 14161-14
43 6 45 4 472 49196 21
154189168 175 182161 196175   
-21 14 -7 0 7-14 21
3
175
22 48 3 46 544 7175
36-138 11 409 42175
153424 32 1930 21175
222724 25 2623 28175
29 20 31 18 2616 35175
8 41 10 39 1251 14175
43 6 45 4 472 28175
175175175 175 175175 175175
+
Mask B
1 3
2 2
31
2 2
1 3
2 2
3 1
4
371
71 48 3 46 15244 7 371
36-138 109 40107 42371
1623473 32 1930 21371
2212524 25 26121 28371
29 20 31 18 7516 182371
8 139 10 137 1251 14371
43 6 192 4 472 77371
371371371 371 371371 371371

This completes this section on the new 7x7 Mask-Generated Methods (Part II). The next section deals with new 9x9 and 13x13 Cross Squares Methods (Part III). To return to homepage.


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