A New Procedure for Magic Squares (Part I)

Zig Zag Cross Modified Squares and 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.

This site will explore new squares which are derived in a different manner from the standard Loubère approach. A 5x5 Loubère square which is produced using the staircase method is shown below in I. If all the even numbers in this square are exchanged for their complements the result is II:

I
17 24 1 8 15
2357 14 16
4613 20 22
101219 21 3
11 18 25 2 9
 
II
17 2 1 18 15
2357 12 10
222013 6 4
161419 21 3
11 8 25 24 9

This is equivalent to using the following complementary table where the number sequence is 1 → 24 → 3 → 22...13 → 12 → 15...25.


1 2 3 4 5 6 7 8 9 10 11 12     
13
25 24 23 22 21 20 19 18 17 16 15 14     

Square II, however, is not magic using the standard Loubère approach. It was shown in Part I of new Cross and Mask-Generated Methods how to construct new squares. In this modified method the numbers are added in a different order by zig zagging across two adjacent rows, except for the center row whereby the numbers are added straight, i.e., no zig zagging. First the odd numbers are added starting at one end, then the even numbers starting at the other end to produce A. Adding the latter numbers in a slightly modified form produces B. The yellow cell shows where to place the next number after filling in the center row. Construction of both these squares are shown below. In addition, 4n + 1 squares such as the 5x5 produce mask-generated squares while 4n + 3 squares such as the 7x7 do not.

A
1 22 5 18 9
24 320 716
111213 14 15
10196 23 2
17 8 21 4 25
 
B
1 18 5 22 9
16320 7 24
111213 14 15
10196 23 2
17 8 21 4 25

Both these squares after manipulation produce two types of magic squares. One by the manipulation of individual cells, the other by the use of a symmetrical C2 mask (symmetrical by 180° rotation) of n2 numbers (see Method II below). Moreover, only starting with the squares with regular read-out such as A produces an internal cross, When n is 4n + 1 the crosses generated are ½(n2 + 1) along the center row and ½(n2 + 1) ± ½(n - 1) along the center column. When n is 4n + 3 the crosses are three cells across and sum up to 3(½)(n2 + 1) ± ½(n - 3). The magic squares produced have only partially sequential order and not all the numbers from 1 to n2 are present. Because of the manipulation numbers greater than n2or less than 1 are generated.

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 5x5 Magic Squares

Method I: Reading zig zag from left to right - generation of a cross square
  1. Construct Square 1 by adding odd numbers in a zig zag manner to the cells. At the center row proceed straight (no zig zag) then swich back upon reaching 17 (lower left).
  2. Add the even numbers in reverse manner starting at the lower right, follow the green path, again switching to straight upon reaching the center row. Add 16 to the rightmost cell of the second row and continue adding numbers in a zig zag manner (Square 2).
  3. Adjust the center column by adding and subtracting numbers from the selected cells to generate 2. For example, adding 10 to 5 and subtracting 10 from 21 keeps the center column at 65.
  4. Adjust the center row by adding and subtracting numbers from the selected cells to generate 3 where the center row consists of all 13 and the center column of 15 and 11 which average out to 13.
  5. The non-yellow group of four cells sum to 50 and 54.
1
1 5 9
3 7
1113 15
19 23
1721 25
2
65
1 22 5 18955
24320 7 1670
111213 14 15 65
10196 23 260
17821 4 2575
636465 66 6765
3
65
1 22 15 18965
24315 7 1665
131313 13 13 65
101911 23 265
17811 4 2565
656565 65 6565
Method II: Reading zig zag from left to right - use of mask
  1. Construct Square 1 by adding odd numbers in a zig zag manner to the cells.
  2. Add the even numbers in reverse manner starting at the lower right and following the green path. Add 16 to the leftmost cell of the second row and continue adding numbers in a zig zag manner (Square 2).
  3. Adjust the center row by adding and subtracting numbers from the selected cells to generate 3, containing duplicate cells in light blue.
  4. Use the mask A consisting of the number 1 to remove duplicates. Where the numbers intersect with the numbers of square 2 add the appropriate factor of n2, 25 for "1" and 50 for "2".
  5. Addition of the right factor gives square 3 with S = ½(n3 + 31n) and the modification of 13 numbers.
1
1 5 9
3 7
1113 15
19 23
1721 25
2
65
1 18 5 22955
16320 7 2470
111213 14 15 65
10196 23 260
17821 4 2575
556065 70 7565
3
65
11 18 5 22965
16320 2 2465
111213 14 15 65
10246 23 265
17821 4 1565
656565 65 6565
+
Mask A
1 1 1
1 2
11 1
21
1 1 1
4
140
36 18 30 479140
16320 27 74140
113738 39 15 140
60496 23 2140
173346 4 40140
140140140 140 140140

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


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