A New Procedure for Magic Squares (Part II)

Zig Zag Consecutive 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.

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 7x7 Magic Square

Method: Reading zig zag from left to right - use of a 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 in the last row (the number 15).
  2. On reaching number 8 zig zag over the center row (don't fill just yet).
  3. On reaching 21 reverse the pattern by adding consecutive numbers, filling the center row then proceeding from 22 to 28 along the light green path, and filling in the last two rows. On reaching 35 proceed to 36 and fill the two rows hugging the middle rows in a zig zag manner as before then go to 43 and fill up the rest of the rows in the square (Squares 2 and 3).
  4. 1
    1 3 5 7
    2 4 6
    810 12 14
    9 11 13
    16 18 20
    1517 19 21
    2
    1 3 5 7
    2 4 6
    810 12 14
    222324 25 26 2728
    9 11 13
    291631 18 3320 35
    153017 32 1934 21
    3
    119
    1 44 3 465 48715421
    43245 4 47649196-21
    83710 39 1241 1416114
    222324 25 26 27281750
    36938 11 40 1342189-14
    291631 18 3320 35182-7
    153017 32 1934 211687
    154161168 175 182189 196119
    21147 0 -7-14-21
  5. 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 Square 4. At this point four duplicates have been generated.

  6. Construct the equations for the mask as follows:


    • Add the diagonal sums to 175 and generate two pre-sums 315 and 289.
    • 315 happens to be the best bet since the following three equations are obtained having the factors 42 and 49.
      The right diagonal: 315 = 119 + 196 = 119 + 4(49)
      The left diagonal: 315 = 140 + 175 = 140 + 3(42) + 49
      The rows and columns: 315= 175 + 140 = 175 + 2(49) + 42

  7. Generate a mask using the 42 and 49 factors adding these factors to the appropriate cells in square 4 to generate square 5. Sometimes duplicates are formed so go back and tweak the mask a little (move some of the cell numbers around) until no duplicates are generated. This assures that when each of these factors is added to the corresponding cell in square 4 (as in the de la Hire method) a sum equal to 315 is obtained. In addition, the corresponding equation for the sum of this square is S = ½(n3 + 41n).
4
119
22 44 3 465 487154
43245 4 47628196
85110 39 1241 14161
222324 25 26 2728175
36938 11 40 -142189
291631 18 2620 35182
153024 32 1934 21168
154161168 175 182189 196140
+
Mask A
42 4949
42 4949
49 42 49
4249 49
49 42 49
4949 42
4949 42
5
315
22 86 3 9554 487315
43287 4 475577315
575110 81 1290 14315
642324 74 75 2728315
36987 11 82 -191315
786531 18 2662 35315
157973 32 1934 63315
315315315 315 315315 315315

Construction of a second 7x7 Magic Square

Method: Reading zig zag from left to right - use of a second mask (a more logical approach)
  1. Construct the equations for the mask as follows:


    • Subtract the diagonal sums from 175 and generate two factors 56 and 35.
    • The sum of these two factors when added to 175 gives a pre-sum of 266.
      The right diagonal: 266 = 119 + 147 = 119 + 35 + 2(56)
      The left diagonal: 266 = 140 + 126 = 140 + 2(35) + 56
      The rows and columns: 266= 175 + 91 = 175 + 35 + 56


  2. Generate a mask using the 35 and 56 factors adding these numbers to the appropriate cells in square 4 to generate square 6. Sometimes duplicates are formed so go back and tweak the mask a little (move some of the cell numbers around) until no duplicates are generated. This assures that when each of these factors is added to the corresponding cell in square 4 (as in the de la Hire method) a sum equal to 266 is obtained. In addition, the corresponding equation for the sum of this square is S = ½(n3 + 27n).
4
119
22 44 3 465 487154
43245 4 47628196
85110 39 1241 14161
222324 25 26 2728175
36938 11 40 -142189
291631 18 2620 35182
153024 32 1934 21168
154161168 175 182189 196140
+
Mask A
35 56
5635
3556
56 35
35 56
56 35
5635
6
266
57 100 3 465 487266
43245 4 476263266
85110 39 1276 70266
222380 60 26 2728266
364438 11 96 -142266
291631 74 6120 35266
713059 32 1934 21266
266266266 266 266266 266266

This completes this section on a new zig zag consecutive 7x7 Mask-Generated Methods (Part II). To return to homepage.


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