A New Procedure for Magic Squares (Part I)

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 but a new approach which gives two types of squares is. Taking the complementary table numbers and placing them in the same order into a 5x5 square produces A. If the numbers are placed boustrophedonically (reading in regular order then in backward order) the result is B:

A
1 24 3 22 5
20718 9 16
111413 12 15
10178 9 6
21 4 23 2 25
 
B
1 24 3 22 5
16918 7 20
111413 12 15
6198 17 10
21 4 23 2 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 one cell across whose value is ½(n2 + 1), while when n is 4n + 3 the crosses are three cells across and sum up to 3(½)(n2 + 1). 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.

The mask method borrows from de la Hire, who developed a particular method for 5x5 magic squares of using a square of primary numbers (1,2,3,4,5) and a square of root numbers (0,5,10,15,20). These form a sequence which is not repeated in the same order in any similarly placed cells. Adding the numbers of the primary square with the corresponding cell numbers of the root square gives a magic square as shown in the following three squares. ( Borrowed from Magic Square and Cubes by W.S. Andrews). The mask method, however, varies significantly from de la Hire in that all the cells are not occupied with primary numbers and the numbers in the cells correspond to factors of n2 (see methods I and II below).

Primary
1 5 4 3 2
321 5 4
543 2 1
215 4 3
4 3 2 1 5
+
Root
0 10 20 5 15
20515 0 10
15010 20 5
10205 15 0
5 15 0 10 5
de la Hire Square
1 15 24 8 17
23716 5 14
20413 22 6
122110 19 3
9 18 2 11 25

In addition, it will also be shown that the sums of these squares follow the new sum equation, shown previously in a New Block Loubère Method:

S = ½(n3 ± an)

Construction of 5x5 Magic Squares

Method I: Reading from left to right - generation of a cross square
  1. Recopy square A with the sum and row columns (in grey).
  2. As shown below this square is not magic because all the columns and rows don't sum to 65.
  3. Adjust the center column by adding and subtracting numbers from the selected cells to generate 2. For example, adding 10 to 3 and subtracting 10 from 23 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 non-yellow group of four cells sum to 52 a multiple of 13 which is also obtained from the equation ½(n2 + 1)  (½(n - 1))2 .
1
65
1 24 3 22555
20718 9 1670
111413 12 15 65
10178 19 660
21423 2 2575
636665 64 6765
2
65
1 24 13 22565
20713 9 1665
111413 12 15 65
101713 19 665
21413 2 2565
636665 64 6765
3
65
1 24 13 22565
20713 9 1665
131313 13 13 65
101713 19 665
21413 2 2565
656565 65 6565
Method II: Reading Boustrophedonically
  1. Recopy square B with the sum and row columns (in grey).
  2. As shown below this square is not magic because all the columns and rows don't sum to 65.
  3. Adjust the blue cells by adding and subtracting numbers from the selected cells to generate 2. For example, adding 10 to 1 and subtracting 10 from 25 keeps the center column at 65. In addition, the cells in green are duplicates of cells on the blue diagonal.
  4. Use the mask B consisting of numbers 1 and 2 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 + 41n) and the modification of 14 numbers.
1
65
1 24 3 22555
16918 7 2070
111413 12 15 65
6198 17 2060
21423 2 2575
557065 60 7565
2
65
11 24 3 22565
16418 7 2065
111413 12 15 65
6198 22 2065
21423 2 1565
656565 65 6565
+
Mask B
1 1 2
11 2
2 2
2 1 1
2 1 1
3
165
36 49 53 225165
412918 57 20165
611413 12 65 165
6698 47 35165
21473 27 40165
165165165 165 165165

This completes this section on the new 5x5 Cross and Mask-Generated Methods (Part I). The next section deals with new 7x7 Cross and Mask-Generated Methods (Part II). To return to homepage.


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