A New Procedure for Magic Squares (Part IA)

Consecutive Boustrophedonic 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 first leftmost column and entered across every other cell. Consecutive numbers are then added to the next rows boustrophedonically. The numbers are added to the last cell on the last row/column. in reverse order starting at the next to the last cell in the last row/column. The square which is not magic is modified into a form which can be converted into a magic one by the use of a mask. This mask generates numbers which are added to certain cells in the square to produce a final square 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 1, then swing around at the end (boustrophedonically) until the final cell is reached (square 1).
  2. On reaching the end cell reverse the process and continue adding numbers consecutively and boustrophedonically(Square 2).
  3. Since all sums of all the columns or rows are not equal to 65 add or subtract the numbers in the last row from those numbers identical in sum in the last columns. At this point three duplicates have been generated (Square 3).
  4. 1
    1 2 3
    5 4
    67 8
    10 9
    1112 13
    2
    35
    1 25 2 2435510
    21522 4 2375-10
    6207 19 8 605
    161017 9 18 70-5
    111512 14 13650
    557560 70 6535
    10-105 -5 0
    3
    40
    11 25 2 24365
    21-522 4 2365
    62012 19 8 65
    161017 4 18 65
    111512 14 1365
    656565 65 6535
  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.

3
40
11 25 2 24365
21-522 4 2365
62012 19 8 65
161017 4 18 65
111512 14 1365
656565 65 6535
+
Mask A
55
30 25
25 30
55
25 30
4
120
66 25 2 243120
21-552 29 23120
62012 49 33 120
1612017 4 18 120
111537 14 33120
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 numbers in a consecutive manner starting at row 1 cell, then swinging around at the end (boustrophedonically) until the final cell is reached (square 1).
  2. On reaching this cell reverse the process and continue adding numbers consecutively (Square 2).
  3. Since all sums of all the columns or rows are not equal to 175 add or subtract the numbers in the last row from those numbers identical in sum in the last columns. At this point four duplicates have been generated (Square 3).
  4. 1
    1 2 3 4
    7 6 5
    89 1011
    14 13 12
    1516 1718
    21 20 19
    2223 2425
    2
    91
    1 49 2483 47 4 15421
    43744 6 45546196-21
    8429 41 10401116114
    361437 13 381239189-14
    153516 34 1738181687
    292130 20 311932182-7
    222823 27 2426251750
    154196161 189 168182 17591
    21-2114 -14 7-70
    3
    77
    22 49 2483 47 4 175
    43-1144 6 45546175
    84223 41 104011175
    361437 -1 381239175
    153516 34 243818175
    292130 20 311232175
    222823 27 242625175
    175175175 175 175175 17591
    +
  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 3 (as in the de la Hire method) that all sums will equal 357.
Mask B
84 98
98 84
9884
84 98
9884
98 84
84 98
4
357
106 49 2483 47 102 357
141-1444 6 129546357
84223 139 104095357
361437 83 3811039357
153516 34 12211718357
29119114 20 311232357
22112121 27 242625357
357357357 357 357357 357357

This completes this section on a new Consecutive Boustrophedonic Mask-Generated Squares (Part I). The next section deals with Consecutive Boustrophedonic 7x7 Mask-Generated Squares (Part IB). To return to homepage.


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