A New Procedure for Magic Squares (Part I)

Consecutive Internally Added 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 second leftmost column and entered across every other cell. Consecutive numbers are then added to the next rows boustrophedonically. When n > 7 at least one row turns into a regular left to right order, increasing by one for each n. In addition every other number in the center row (starting with the second cell) will take on its complement. For example for a 5x5 square the second number 12 becomes 14 and 14 becomes 12 (see the 5x5 example below). 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 these squares follow the sum equation shown in the New block Loubère Method. :

S = ½(n3 ± an)

Construction of a 5x5 Magic Square

Method: Reading boustrophedonically (like a sidewinder snake) - use of a mask
  1. Construct the 5x5 Square 1 where 5 = 4n + 1 by adding numbers in a consecutive manner starting at row 1 cell. Don't fill in the center row but proceed to the first cell in the fourth row (the number 6).
  2. On reaching 10 reverse the pattern by adding consecutive numbers (remembering that every other number except for the center cell takes on its complement) 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 consecutively (Square 4).
  3. 1
    1 2
    54 3
     
    67 8
    10 9
    2
    1 2
    54 3
    111413 12 15
    67 8
    10 9
    3
    1 2
    54 3
    111413 12 15
    6167 17 8
    201019 9 18
    4
    95
    23 1 22 22169-4
    5244 25 3614
    111413 12 15 540
    6167 17 8 7611
    201019 9 1866-11
    656565 65 6595
  4. Since the columns are all equal to 65 add or subtract the numbers in the last row from the center column values to generate square 5. At this point two duplicates have been generated.
  5. Generate a mask whereby the sums of the columns and rows are constructed as in the box below. 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 120.

5
95
23 1 18 22165
5248 25 365
111413 12 15 65
61618 17 8 65
20108 9 1865
656565 65 6595
+
Mask A
30 25
3025
25 30
25 30
30 25
6
120
23 1 48 2721120
352433 25 3120
361413 42 15 120
64118 17 38 120
20408 9 43120
120120120 120 120120

Construction of a 7x7 Magic Square

Method: Reading consecutive from left to right boustrophedonically - use of a mask
  1. Construct the 7x7 Square 1 where 7 = 4n + 3 by adding consecutive numbers in a consecutive manner to the cells.At the number 8 proceed in a zig zag manner to number 14 then to the 5th row second cell and enter 9. The center row is not filled at this time.
  2. On reaching 21 reverse the pattern by adding consecutive numbers 22-28 to the center row (remembering that every other number except for the center cell takes on its complement) yellow and khaki path, then filling in the last two rows. On reaching 35 proceed to 36 and fill in the middle rows reversibly in a zig zag manner.
  3. 7
    1 2 3
    76 54
    810 1214
     
    9 11 13
    1817 1615
    19 20 21
    8
    1 2 3
    76 54
    810 1214
    222724 25 262328
    9 11 13
    183117 30 162915
    321933 20 342135
    9
    1 2 3
    76 54
    84110 39 123714
    222724 25 262328
    42940 11 381336
    183117 30 162915
    321933 20 342135
  4. Fill in the last top rows as shown in square 10.

  5. Since the columns are all equal to 175 add or subtract the numbers in the last row from the center column values to generate square 11. At this point two duplicates have been generated (Square 11).
  6. 10
    232
    46 1 45244 3 43 184-9
    7476 48 54941669
    84110 39 12371416114
    222724 25 2623281750
    42940 11 381336189-14
    183117 30 16291515619
    321933 20 342135194-19
    175175175 175 175175 175230
    11
    232
    46 1 45-744 3 43 175
    7476 57 5494175
    84110 53 123714175
    222724 25 262328175
    42940 -3 381336175
    183117 49 162915175
    321933 1 342135175
    175175175 175 175175 175230
    +
  7. Generate a mask whereby the sums of the columns and rows are constructed as in the box below. This assures that when each of these values is added to the corresponding cell in square 10 (as in the de la Hire method) that all sums will equal 287.
Mask B
57 55
5755
57 55
55 57
5557
55 57
5557
12
287
46 1 102-799 3 43 287
710461 57 5494287
654110 53 123769287
772724 25 832328287
42940 -3 386893287
188617 106 162915287
321933 56 347835287
287287287 287 287287 287287

Note that if square 9 is produced by swithching several entries in the last two rows that a cross square is generated. All numbers in the center row are 25 and the average of the sums of the 1st and 2nd, 3rd and 5th and 6th and 7th are each 25.

13
232
46 1 45244 3 43 184-9
7476 48 54941669
84110 39 12371416114
222724 25 2623281750
42940 11 381336189-14
153116 30 17291815619
322133 20 341935194-19
172177174 175 176173 178230
3-21 0 -12-3
14
232
46 1 45-744 3 43 184
7476 57 5494166
84110 53 123714161
252525 25 2525 25175
42940 -3 381336189
153116 49 172918156
322133 1 341935194
175175175 175 175175 175230

This completes this section on a new Consecutive Internally Added Mask-Generated Squares (Part I). The next section deals with Consecutive Internal 9x9 Mask-Generated Squares (Part II). To return to homepage.


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