A New Procedure for Magic Squares (Part V) Continuation

Loubère and Méziriac Knight Block Modified 7x7 Squares

A Loubere knight square

A Discussion of the New Method

An important general principle for generating odd magic squares by the De La Loubère method is that the center cell must always contain the middle number of the series of numbers used, 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.

The 5x5 regular Loubère square is shown below on the left and the regular Méziriac on the right :

17 24 1 8 15
2357 14 16
4613 20 22
101219 21 3
11 18 25 2 9
 
3 16 9 22 15
20821 14 2
72513 1 19
24125 18 6
11 4 17 10 23

Normally the Loubère and Méziriac methods involve a stepwise approach of consecutive numbers, i.e., 1,2,3...n2. For example, A new generalized Loubère procedure where the initial number is placed anywhere on any n - 1 cells of the first row. However, in this new method although the numbers are added consecutively starting with 1 and ending with n2, the final square is transformed into a modified Loubère or Méziriac square, having numbers greater than n2. This is done by modifying a block or grid of numbers on the square. The initial number is added either at the normal Loubère sites, or on the main diagonal. Every number on the diagonal can be modified, up to n - 1. Replacement of all n numbers on the diagonal leads to semi-magic squares. The same for the Méziriac method.

In this new method the numbers are added consecutively starting out with 1 and ending with numbers greater than n2. Whenever a break occurs a knight move is performed instead of a normal translational move (up, down or sideways). The knight move is either a (2,1) or a variable knight move , e.g (3,2), for Méziriac squares squares or Loubère squares. Both these square types were covered in Loubère Knight and Méziriac Knight square methods.

Moreover, it must be stated here that the magic sum has been modified from the known equation S = ½(n3 + n) to the general equation:

S = ½(n3 ± an)

which takes into account these new squares. The variable a, an odd number, is equal to 1,3,5,7 or ... and may take on + or - values. For example when a = 1 the normal magic sum S is implied. When a takes on different odd values S gives the magic sum of a modified magic square. It will be shown that the addition or subtraction of n2 to some of the cells in the square gives rise to a new magic square.

Construction of 7x7 Block Knight Modified Loubère and Méziriac Squares

Method I: Start on the first row center (1 ⇒ (2,1) down knight break)
  1. Place 1 in the center cell of the first row.
  2. Knight break, two cells down one cell left as shown in color.
  3. Add consecutive numbers.
  4. Repeat the process until the square is filled as shown below in squares 1-2.
  5. As shown below this square is not magic because all the columns, rows and diagonals don't sum to a magic sum.
  6. Where the sum 133 crosses corresponds to the small square in blue.
  7. Add 49 to the number 1. At this point all the numbers except for one sum to 182.
  8. Using one example adding n2 = 25 to (4,7,16,21,28,29 and 35) gives the magic square 3. Note that all these modified numbers fall outside the blue square. In the previous non-knight Part I method all the modified numbers fall within the blue square.
  9. The magic sum is S = ½(n3 + 17n).
1
1 4
3
2
8
7
9 6
5
2
133
27 30 401 4147 133
36 39 493 131626182
38 48 212 222535182
47 8 1121 243437182
7 10 2023 334346182
9 19 2932 42456182
18283141 44 515182
182182182 133 182182 182 182
3
231
27 30 4050 53147 231
36 39 493 136526231
38 48 212 222584231
47 8 1170 243437231
56 10 2023 334346231
9 19 7832 42456231
18773141 44 515231
231231231 231 231231 231 231
Method II: Start at lower left hand corner (1 ⇒ (2,1) up knight break)
  1. Place 1 in the center cell of the first row.
  2. Knight break, two cells up one cell left as shown in color.
  3. Add consecutive numbers.
  4. Repeat the process until the square is filled as shown below in squares 1-2.
  5. As shown below this square is not magic because all the columns, rows and diagonals don't sum to a magic sum.
  6. Where the sum 133 crosses corresponds to the small square in blue.
  7. Add 49 to the number 1. At this point all the numbers except for one sum to 182.
  8. Adding 2n2 = 98 to (16,17,18,19,20,21 and 22) gives the magic square 3. Note that all these modified numbers fall outside the blue square. In the previous non-knight Part I method all the modified numbers fall within the blue square.
  9. The magic sum is S = ½(n3 + 31n).
1
7 1
6
5
4
9 3
2
8
2
280
7 19 311 132537 133
18 30 4912 24436182
36 48 1123 42517182
47 10 2941 41635182
9 28 403 223446182
27 39 221 334515182
3882032 44 1426182
182182182 133 182182 182 182
3
280
7 117 3150 132537 280
116 30 4912 24436280
36 48 1123 425115280
47 10 2941 411435280
9 28 403 1203446280
27 39 2119 334515280
38811832 44 1426280
280280280 280 280280 280 280
Method III: Start at lower left hand corner (1 ⇒ (2,1) up knight break)
  1. Place 1 in the left corner cell of the last row.
  2. Knight break, two cells up one cell left as shown in color.
  3. Add consecutive numbers.
  4. Repeat the process until the square is filled as shown below in squares 1-2.
  5. As shown below this square is not magic because all the columns, rows and diagonals don't sum to a magic sum.
  6. Where the sum 133 crosses corresponds to the small square in blue.
  7. Add 49 to the number 1. At this point all the numbers except for one sum to 182.
  8. Adding 2n2 = 98 to (16,17,18,19,20,21 and 22) gives the magic square 3. Note that all these modified numbers fall outside the blue square. In the previous non-knight Part I method all the modified numbers fall within the blue square.
  9. The magic sum is S = ½(n3 + 31n) identical to sum of method II and that the rows of both squares 2 have their cell numbers in the same order. Either square may be converted to the other by transposing columns and rows.
1
6
5
4
3 9
2
8
1 7
2
280
12 24 436 183049 182
23 42 517 364811182
41 4 1635 471029182
3 22 3446 92840182
21 33 4515 27392182
32 44 1426 38820182
1132537 7 1931133
133182182 182 182182 182 182
3
280
110 24 436 183049 280
23 42 517 3648109280
41 4 1635 4710829280
3 22 3446 1072840280
21 33 45113 27392280
32 44 11226 38820280
501112537 7 1931280
280280280 280 280280 280 280
Method IV:Start one cell right of center (1 ⇒ (3,2) up knight break)
Interconversion of a Méziriac Knight Square to a Loubère Type Knight Square
  1. Start out with a Méziriac type square by placing 1 one cell to the right of center.
  2. Knight break, three cells up two cell left.
  3. Repeat the process until the square is filled as shown below in squares 1-2.
  4. As shown below this square is not yet magic.
  5. Where the sums 133 cross corresponds to the small square in blue.
  6. Add 49 to the number 1. At this point all the numbers except for one sum to 182.
  7. Adding n2 = 49 to (4,5,8,19,20,38,39) gives the magic square 3 where S = ½(n3 + 17n).
  8. To convert this Méziriac Knight Square to a Loubère Knight Square -1 is subtracted from every cell to give square 4 where S is now equal to ½(n3 + 15n).
  9. Subtraction of 25 from those cells colored in tan from square 4 gives the Loubère semi-magic Knight square 1b. This square can be generated using the New De la Loubère Method (Part II), except using a variable (3,2) up knight break.
1
2
8
7
16
5
4 9
3
2
133
46 24 236 144119 182
23 8 3513 401845182
7 34 1239 174429182
33 11 3816 1286133
10 37 2249 27532182
43 21 4826 4319182
2047253 30 1542182
182182182 182 133182 182 182
3
231
46 24 236 144168 231
23 57 3513 401845231
7 34 1288 174429231
33 11 8716 50286231
10 37 2249 275432231
43 21 4826 53319231
6947253 30 1542231
231231231 231 231231 231 231
4
224
45 23 135 134067 224
22 56 3412 391744224
6 33 1187 164328224
32 10 8615 49275224
9 36 2148 265331224
42 20 4725 52308224
6846242 29 1441224
224224224 224 224224 224 224
1b
126
45 23 135 134018 175
22 7 3412 391744175
6 33 1138 164328175
32 10 3715 49275175
9 36 2148 26431175
42 20 4725 3308175
1946242 29 1441175
175175175 175 175175 175 175

This completes this section on the new block 7x7 Loubère Magic Square Method (Part V). To return to homepage.


Copyright © 2009 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com