A New Procedure for Magic Squares (Part IC)

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 center cell of the first row column and entered across every other cell until the partial row is filled. A 2 down, 1 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 in the same manner.
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 follo either of the two sum equation shown: New block Loubère Method and New Consecutive Mask-Generated Method. :

S = ½(n3 ± an)
S = ½(n3 ± an + b)

Construction of a 5x5 Magic Square

Method: Reading consecutive 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 3, on all the requisite numbers to the row, knight break (2 down, 1 right) and continue adding numbers in regular mode.

  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 four duplicates has been generated (in pink, Square 3).
  3. 1
    3 1 2
     
    4 5
     
    78 6
    2
    60
    3 14 1 1523530
    231021 9 2285-20
    17418 5 16 605
    132411 25 12 85-20
    7208 19 6605
    637257 73 5862
    2-76 -8 7
    3
    71
    3 14 31 15265
    23101 9 2265
    19-329 -3 23 65
    1324-9 25 1265
    72013 19 665
    656565 65 6573
  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
71
3 14 31 15265
23101 9 2265
19-329 -3 23 65
1324-9 25 1265
72013 19 665
656565 65 6573
+
Mask A
14 8 68
226 8
88 14 6
148 14
20 16
4
101
17 22 31 2110101
45161 9 30101
19537 11 29 101
1338-1 25 26 101
72033 35 6101
101101101 101 101101

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 4, on reaching the last number in the set (3), knight break (2 down, 1 right) and continue adding numbers using this break (squares 5, 6 and 7).
  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 four duplicats have been generated (Square 8).
  3. 5
    3 1 2
    74 56
    8 9 10
    1213 1411
    6
    3 1 2
    16 17 15
    74 56
    2021 1819
    8 9 10
    24 22 23
    1213 1411
    +
    7
    160
    28 3 25126 2 27 11263
    411642 17 391540210-35
    7294 30 531611263
    204521 43 184419210-35
    33834 9 35103216114
    492446 22 472348259-84
    123713 38 14361116114
    190162185 160 184161 183160
    -1513-10 15 -914-8
  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.
8
140
28 3 256426 2 27 175
411642 -18 391540175
7294 93 5316175
55811 23 958 11175
33834 23 351032175
492446 -62 472348175
123713 52 143611175
175175175 175 175175 175140
+
Mask B
105 3570
105105
14070
10535 70
105 70 35
175 35
105 105
9
385
133 3 256461 2 97 385
411642 -18 14412040385
7294 233 75316385
516346 23 9128 11385
138834 93 351067385
4924221 -62 475848385
1214213 52 1436116385
385385385 385 385385 385385

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


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