A New Procedure for Magic Squares (Part II)

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 entered across every other cell. Consecutive numbers are then added to the next rows boustrophedonically or in regular left to right order. 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.

In addition, it will also be shown that the sums of squares when n ≥ 7 follow the sum equation that was shown in New block Loubère Method.

S = ½(n3 ± an)

Construction of 7x7 Magic Square I

Method: Reading consecutive from left to right - use of mask
  1. Construct Square 1 by adding numbers in consecutively forward to the cells. At the number 8 skip the center row and fill in consecutely every other ceell. Proceed from 14 to 15 and fill in the last two rows up to the number 21.
  2. On reaching 21 go back and fill in the center row. Proceed from 28 to 29 along the yellow path, and fill in the last two rows. On reaching 35 proceed to 36 and fill up the third and fifth rows consecutively (Squares 2 and 3). Fill in the last two top rows in a forward manner.

  3. 1
    1 2 3 4
    5 6 7
    810 1214
    9 11 13
    1817 1615
    21 20 19
    2
    1 2 3 4
    5 6 7
    810 1214
    222324 25 262728
    9 11 13
    183117 30 162915
    352134 20 331932
    3
    154
    1 45 2443 43 414233
    49548 6 47746208-33
    84110 39 12371416114
    222324 25 2627281750
    4294011 3813 36189-14
    183117 30 16291515619
    352134 20 331932194-19
    175175175 175 175175 175140
  4. Construct a mask according to the following logical method:


  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 4. At this point three duplicates have been generated.
  6. Generate a mask whereby the sums of the columns and rows are constructed as in the box above. 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 231 in square 5.
4
154
1 45 2773 43 4175
49548 -27 47746175
84110 53 123714175
222324 25 262728175
42940-3 3813 36175
183117 49 162915175
352134 1 331932175
175175175 175 175175 175140
+
Mask A
35 21
35 21
   21 35
21 35
21 35
21 35
35 21
5
231
36 45 2773 64 4231
494048 -27 68746231
84110 74 473714231
432324 25 266228231
42961-3 3813 71231
185217 84 162915231
352169 1 331953231
231231231 231 231231 231231

Construction of a 7x7 Magic Square II

Method: Reading consecutive from left to right boustrophedonically - use of mask
  1. Construct Square 6 by adding consecutive numbers boustrophedonically, then boustrophedonically then switching over to forward to forward. At the number 8 skip the center row and fill in consecutely every other cell. Proceed from 14 to 15 and fill in the last two rows up to the number 21.
  2. On reaching 21 go back and fill in the center row. Proceed from 28 to 29 along the yellow path, and fill in the last two rows. On reaching 35 proceed to 36 and fill up the third and fifth rows consecutively (Squares 2 and 3). Fill in the last two top rows boustrophedonically.

  3. 6
    1 2 3 4
    7 6 5
    810 1214
       
       9 11 13
    1817 1615
    21 20 19
    7
    1 2 3 4
    7 6 5
    810 1214
    222324 25 262728
    9 11 13
    183117 30 162915
    352134 20 331932
    8
    152
    1 43 2443 45 414233
    49748 6 47546208-33
    84110 39 12371416114
    222324 25 2627281750
    4294011 3813 36189-14
    183117 30 16291515619
    352134 20 331932194-19
    175175175 175 175175 175142
  4. Construct a mask according to the following logical method:

  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 9. At this point three duplicates have been generated.
  6. Generate a mask whereby the sums of the columns and rows are constructed as in the box above. This assures that when each of these values is added to the corresponding cell in square 9 (as in the de la Hire method) that all sums will equal 231 in square 10.
9
152
1 43 2773 45 4175
49748 -27 47546175
84110 53 123714175
222324 25 262728175
42940-3 3813 36175
183117 49 162915175
352134 1 331932175
175175175 175 175175 175142
+
Mask B
33 23
2333
  33 23
23 33
23 33
23 33
33 23
10
231
34 43 2773 68 4231
49748 -27 703846231
87410 76 123714231
452324 25 262761231
42963-3 7113 36231
185417 82 162915231
352167 1 331955231
231231231 231 231231 231231

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


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