A New Procedure for Magic Squares (Part I)

Consecutive 5x5 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 these squares follow a modified sum equation (the non-modified form was shown in the New block Loubère Method). This equation is to be used when n = 5:

S = ½(n3 ± an + b)

Construction of 5x5 Magic Square I

Method: Reading consecutive from left to right - use of mask
  1. Construct Square 1 by adding numbers in a consecutive manner to the cells. Don't fill in the center row but proceed to the fourth cell in the last row (the number 6). Then proceed to 8 by two reverse readouts.
  2. On reaching 10 reverse the pattern by adding consecutive numbers, 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 (Squares 2 and 3).
  3. 1
    1 2 3
    4 5
    7 6
    109 8
    2
    1 2 3
    4 5
    111213 14 15
    7 6
    109 8
    3
    38
    1 22 2 2134916
    25424 5 2381-16
    111213 14 15 650
    18717 6 16 641
    10209 19 866-1
    656565 65 6532
  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 158.
4
38
1 22 18 21365
2548 5 2365
111213 14 15 65
18718 6 16 65
10208 19 865
656565 65 6532
+
Mask A
27 66
33 60
  33 60
3327 33
3327 33
5
158
28 22 84 213158
25378 65 23158
114513 14 75 158
513418 39 16 158
432035 19 41158
158158158 158 158158

Construction of 5x5 Magic Square II

Method: Reading consecutive from left to right boustrophedonically - use of mask
  1. Construct Square 1 by adding consecutive numbers in a consecutive manner to the cells. Don't fill in the center row but proceed to the fourth cell in the fourth row (the number 6). However at this point place the numbers into the cells as was done in the above example II.
  2. On reaching 10 reverse the pattern by adding consecutive numbers, 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 in a zig zag manner(Squares 2 and 3).
  3. 6
    1 2 3
    5 4
      
    7 6
    109 8
    7
    1 2 3
    5 4
    111213 14 15
    7 6
    109 8
    8
    37
    1 21 2 2234916
    25524 4 2381-16
    111213 14 15 650
    18717 6 16 641
    10209 19 866-1
    656565 65 6533
  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 157.
9
37
1 21 18 22365
2558 4 2365
111213 14 15 65
18718 6 16 65
10208 19 865
656565 65 6533
+
Mask B
28 64
32 60
  32 60
3228 32
3228 32
10
157
29 21 82 223157
25378 64 23157
114413 14 75 157
503518 38 16 157
422036 19 40157
157157157 157 157157

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


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