A New Procedure for Magic Squares (Part ID)

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 and entered across every other cell until the partial row is filled. A (2 right, 1 down) knight break is used to get to the next line. (This knight break is also placing half of the numbers going down and the other half going up). 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 in 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 right, 1 down) 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
    2 1 3
     4 5
    67 8
     9 10
    1112 13
    2
    35
    2 14 1 1533530
    16417 5 18605
    6197 20 8 605
    21922 10 23 85-20
    112412 25 1385-20
    567059 75 6536
    9-56 -10 0
    3
    46
    2 14 31 15365
    16422 5 1865
    151418 10 8 65
    2192 10 2365
    1124-8 25 1365
    656565 65 6547
  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
46
2 14 31 15365
16422 5 1865
151418 10 8 65
2192 10 2365
1124-8 25 1365
656565 65 6547
+
Mask A
37 18
18 37
18 37
37 18
37 18
4
120
39 32 31 153120
16440 42 18120
331418 10 45 120
21462 28 23 120
112429 25 31120
120120120 120 120120

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 right, 1 down) 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
    2 1 3
    45 67
    8 9 10
    1112 1314
    15 16 17
    1819 2021
    22 23 24
    6
    259
    25 2 26 127 3 28 11263
    4295 30 6317 11263
    32833 9 341035 16114
    113612 37 13381416114
    391540 16 411742 210-35
    184319 44 204521 210-34
    462247 23 482449 259-84
    175155182 160 189168 196
    020-8 15 -137-18
  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.
7
288
25 2 266427 3 28 175
4295 93 6317175
32833 23 341035175
11565 66 -145 -7175
391540 -19 411742175
184319 9 204521175
462247 -61 482449175
175175175 175 175175 175288
+
Mask B
113
113
113
113
113
113
113
8
288
25 115 266427 3 28 288
429118 93 6317288
145833 23 341035288
11565 66 -1158 -7288
391540 -19 4117155288
184319 22 204521288
462247 -61 1612449288
288288288 288 288288 288288

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


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