A New Procedure for Magic Squares (Part IA)

Consecutive Boustrophedonic Knight Break 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 until the end of the row is reached. A 2 down, 1 left knight break is used to get to the next line. The next line is then added in reverse order to the previous, i.e. , boustrophedonically. The square obtained, 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, on reaching the end of the row knight break (2 down, 1 left) and continue adding numbers in the reverse mode (squares 1 and 2).
  2. 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 in the center row. Then add or subtract the numbers in the last column to those in the center column. At this point no duplicates has been generated (Square 3).
  3. 1
    1 2 3
     
    5 4
     
    67 8
    60
    1 15 2 1433530
    211022 9 2385-20
    16517 4 18 605
    112512 24 13 85-20
    6207 19 8605
    557560 70 6560
    10-105 -5 0
    3
    70
    1 15 32 14365
    21102 9 2365
    26-527 -1 18 65
    1125-8 24 13 65
    62012 19 865
    656565 65 6570
  4. 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 a magic sum.

3
70
1 15 32 14365
21102 9 2365
26-527 -1 18 65
1125-8 24 13 65
62012 19 865
656565 65 6570
+
Mask A
520
20 5
205
520
5 20
4
90
1 15 32 192390
21302 9 2890
46027 -1 18 90
162512 24 13 90
62017 39 890
909090 90 9090

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 1, on reaching the end of the row knight break (2 down, 1 left) and continue adding numbers in the reverse mode (squares 5 and 6).
  2. 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 in the center row. Then add or subtract the numbers in the last column to those in the center column. At this point one duplicate has been generated (Square 7).
  3. 5
    1 2 3 4
    1516 1718
    7 6 5
    21 20 19
    89 1011
    2223 2425
    14 13 12
    6
    190
    1 28 2273 26 4 9184
    154216 41 174018189-14
    29730 6 3153214035
    432144 20 451946238-63
    8359 34 10331114035
    224923 48 244725238-63
    361437 13 381239189-14
    154196161 189 168182 175189
    21-2114 -14 7-70
    7
    112
    1 28 21113 26 4 175
    154216 27 174018175
    29730 41 31532175
    64058 -57 5212 46175
    8359 69 103311175
    224923 -15 244725175
    361437 -1 381239175
    175175175 175 175175 175112
    +
  4. 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 7 (as in the de la Hire method) that all sums will equal to a magic sum.
Mask B
63
63
63
63
63
63
63
8
238
1 28 21113 26 67 238
154279 27 174018238
297030 41 31532238
64058 6 5212 46238
71359 69 103311238
224923 -15 2411025238
361437 -1 1011239238
238238238 238 238238 238238

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


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