A New Procedure for Magic Squares (Part IB)

Consecutive 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, 2 right) knight break is used to get to the next line. (This knight break is also equivalent to a slant 2 break in the right down direction). The next line is then added from left to righty.
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 regular from left to right and 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, 2 right) and continue adding numbers from left to right (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
     
    4 5
     
    67 8
    2
    60
    1 14 2 1533530
    21922 10 2385-20
    16417 5 18 605
    112412 25 13 85-20
    6197 20 8605
    557060 75 6560
    10-55 -10 0
    3
    70
    1 14 32 15365
    2192 10 2365
    26-127 -5 18 65
    1124-8 25 13 65
    61912 20 865
    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 14 32 15365
2192 10 2365
26-127 -5 18 65
1124-8 25 13 65
61912 20 865
656565 65 6570
+
Mask A
520
20 5
205
520
5 20
4
90
1 14 32 202390
21292 10 2890
46427 -5 18 90
162412 25 13 90
61917 40 890
909090 90 9090

Construction of a 7x7 Magic Square

Method: Reading consecutive from left to right and 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, 2 right) and continue adding numbers from left to right (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
    5 6 7
    19 20 21
    89 1011
    2223 2425
    12 13 14
    6
    189
    1 26 2273 28 4 9184
    154016 41 174218189-14
    29530 6 3173214035
    431944 20 452146238-63
    8339 34 10351114035
    224723 48 244925238-63
    361237 13 381439189-14
    154182161 189 168196 175189
    21-714 -14 7-210
    7
    112
    1 26 21113 28 4 175
    154016 27 174218175
    29530 41 31732175
    641258 -57 520 46175
    8339 69 103511175
    224723 -15 244925175
    361237 -1 381439175
    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 26 651113 28 4 238
154016 27 1710518238
92530 41 31732238
641258 6 520 46238
8339 69 733511238
224723 -15 244988238
36 7537 -1 381439238
238238238 238 238238 238238

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


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