A New Procedure for Magic Squares (Part II)

Zig Zag Consecutive 7x7 Mask-Generated Squares

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:

Construction of a 7x7 Magic Square

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).

1 1 3 5 7 2 4 6 8 10 12 14    9 11 13 16 18 20 15 17 19 21

⇒

2 1 3 5 7 2 4 6 8 10 12 14 22 23 24 25 26 27 28 9 11 13 29 16 31 18 33 20 35 15 30 17 32 19 34 21

⇒

3 119 1 44 3 46 5 48 7 154 21 43 2 45 4 47 6 49 196 -21 8 37 10 39 12 41 14 161 14 22 23 24 25 26 27 28 175 0 36 9 38 11 40 13 42 189 -14 29 16 31 18 33 20 35 182 -7 15 30 17 32 19 34 21 168 7 154 161 168 175 182 189 196 119 21 14 7 0 -7 -14 -21

⇒



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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 Square 4. At this point four duplicates have been generated.


5. 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
   
   
   
   


6. 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).

Construction of a second 7x7 Magic Square

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).

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