A New Procedure for Magic Squares (Part I)

Zig Zag Consecutive 5x5 and 9x9 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 leftmost column and zigged zagged up and down between two rows as was done in the Zig-Zag and Mask generated 9x9 square method Zig Zag 9x9 Cross and Mask-Generated Methods (Part II) using odd numbers to start and even numbers to finish.

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 a 5x5 Magic Square

Method: Reading zig zag from left to right - use of mask
  1. Construct Square 1 by adding consecutive numbers in a zig zag manner to the cells. Don't fill in the center row but proceed to the first cell the last row (the number 6).
  2. On reaching 10 reverse the pattern by adding consecutive numbers, filling the center row then proceeding from 15 to 16 along the yellow path, and filling in the last two rows. On reaching 20 proceed to 21 and fill up the top two rows in a zig zag manner(Squares 2 and 3).
  3. 1
    1 3 5
    2 4
      
    7 9
    68 10
    2
    1 3 5
    2 4
    111213 14 15
    7 9
    68 10
    3
    35
    1 22 3 2455510
    21223 4 2575-10
    111213 14 15 650
    16718 9 20 70-5
    6178 19 10605
    556065 70 7535
    1050 -5 -10
  4. Where the grey sums on the penultimate right hand column intersect the grey sums in the next to the last row adjust the values in these cells by adding and subtracting the values in the last row and columns to generate 4. At this point four duplicates (in green) have been generated.
  5. Generate a mask whereby the sums of the columns and rows are each 50 and the right and left diagonals are 80 and 75, respectively. This assures that when each of these values is added to the corresponding cell in square 4 (as in the de la Hire method) that all sums will equal 115. In addition the corresponding equation for the sum of this square is S = ½(n3 + 21n).
4
35
11 22 3 24565
21223 4 1565
111213 14 15 65
16718 4 20 65
6228 19 1065
656565 65 6540
+
Mask A
25 20 5
50
  525 20
2525
25 25
5
115
36 42 3 2410115
21223 54 15115
111738 14 35 115
41743 4 20 115
6478 19 35115
115115115 115 115115

Construction of a 9x9 Magic Square

Method: Reading zig zag from left to right - use of mask
  1. Construct Square 1 by adding consecutive numbers numbers in a zig zag manner to the cells as was done to the 5x5 square above (square 1).
  2. After reaching number 36 jump to the center row and fill it consecutively (Square 2).
  3. After reaching 45 jump to the last row and zig zag fill up to 54 (Square 3).

  4. 1
    1 3 5 79
    24 6 8
    101214 16 18
    1113 15 17
       
    20 22 24 26
    19 2123 25 27
    29 31 33 35
    28 3032 34 36
    2
    1 3 5 79
    24 6 8
    101214 16 18
    1113 15 17
    37 38 394041 42 4344 45
    20 22 24 26
    19 2123 25 27
    29 31 33 35
    28 3032 34 36
    3
    1 3 5 79
    24 6 8
    101214 16 18
    1113 15 17
    37 38 394041 42 4344 45
    20 22 24 26
    19 2123 25 27
    46 29 483150 3352 3554
    28 47 304932 51 3453 36
  5. From 54 go across to 55 and fill up to 63 (Square 4), followed by 64, 72 and 73 following the zig zag path. (Square 5 and 6).

  6. 4
    1 3 5 79
    24 6 8
    101214 16 18
    1113 15 17
    37 38 394041 42 4344 45
    55 20 572259 24 6126 63
    19 56 215823 60 2562 27
    46 29 483150 3352 3554
    28 47 304932 51 3453 36
    5
    1 3 5 79
    24 6 8
    1065126714 69 1671 18
    6411661368 1570 1772
    37 38 394041 42 4344 45
    55 20 572259 24 6126 63
    19 56 215823 60 2562 27
    46 29 483150 3352 3554
    28 47 304932 51 3453 36
  7. At this point not all columns, rows or diagonals sum to 369.

  8. Where the grey sums on the penultimate right hand column intersect the grey sums in the next to the last row adjust the values in these cells by adding and subtracting the values in the last row and columns to generate 7. At this point five duplicates (in light orange) have been generated.

  9. 6
    189
    1 74 376 578 780933336
    73275477 6 798 81405-36
    1065126714 69 1671 18342 27
    6411661368 1570 1772396-27
    37 38 394041 42 4344 453690
    55 20 572259 24 6126 63387 -18
    19 56 215823 60 2562 2735118
    46 29 483150 3352 3554378-9
    28 47 304932 51 3453 363609
    333342351 360 369378 387 396405 189
    36 27 1890 -9 -18-27 -36
    7
    207
    37 74 376 578 7809369
    73275477 6 798 45369
    1092126714 69 1671 18369
    6411661368 1570 -1072 369
    37 38 394041 42 4344 45369
    55 20 572259 24 4326 63369
    19 56 395823 60 2562 27369
    46 29 483150 2452 3554369
    28 47 305832 51 3453 36369
    369369369 369 369369 369 369369 225
    +
  10. A possible mask for use with square 7 adds either 81 or 99 (each a multiple of 9) of mask B to the appropriate cells of square 7. The number 81 was picked so as to use numbers >= to n2. 91 was pulled out a hat in order that the sum of 81 + 99 with 369 added to 549. The diagonals were similarly adjusted using these two numbers.
  11. Addition of the right factors gives square 8 with S = 549 = ½(n3 + 41n) and the modification of 18 numbers.
  12. Mask B
    81 99
    81 99
    99 81
    9981
    81 99
    81 99
    9981
    99 81
    99 81
    8
    549
    118 74 376 5177 7809549
    73275477 6 7989 144549
    10921267113 69 9771 18549
    64110669468 1570 -1072 549
    37 137 394041 42 43143 45549
    55 20 572259 105 14226 63549
    19 56 13858104 60 2562 27549
    145 29 1473150 2452 3554549
    28 47 3015732 51 3453 117549
    549549549 549 549549 549 549549 549
  13. Alternatively Mask C may be produced which uses a more logical construction.
Mask C
144 144 162
144 144 162
162 144 144
162144 144
144 144 162
144 144 162
144 162 144
144 162 144
162 144 144
9
819
181 74 14776 167 787809819
7327514877 150 79170 45819
10921746714 69 16071 162819
226116613212 1570 13472 819
37 182 3918441 42 4344 207819
55 164 5722203 24 20526 63819
19 56 1835823 222 16962 27819
190 191 483150 16852 3554819
28 47 3022032 51 34197 180819
819819819 819 819819 819 819819 819

This completes this section on a new zig zag consecutive 5x5 and 9x9 Mask-Generated Methods (Part I). The next section deals with new zig zag 7x7 Mask-Generated Methods (Part II). To return to homepage.


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