A New Procedure for Magic Squares (Part VIII) Continuation

Méziriac Knight Block non-trans and trans 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 Méziriac 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 de Méziriac 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 7x7 regular Méziriac square is shown below :

4 29 12 37 20 45 28
351136 19 44 27 3
104218 43 26 2 34
411749 25 1 33 9
16 48 24 7 32 8 40
47 23 6 31 14 39 15
22 5 30 13 38 21 46

Normally the 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, to generate an intermediate non-magic square the final square is transformed into a modified Méziriac, having numbers greater than n2.

This is done by taking the non-magic squares and converting them into magic ones using transposition (trans) and non-transposition routes. These squares are shown below in methods I and II. Accordingly, 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). These square types were covered in 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.

non-trans-Conversion of de Méziriac non-magic Squares to Magic Squares

Method I: Start on the first row center (1 ⇒ (2,1) up knight break)
  1. Begin generating the non-magic de Méziriac square by placing 1 to the right of the center cell and performing a (1,2) up knight break.
  2. Add consecutive numbers and repeat until the square is filled. This produces the non-magic square 2.
  3. As shown square 2 is non-magic since the last column corresponds to how much must be added or subtracted to the previous row numbers to get to 182.

  4. 1
    411
    3 10
    9 2
    18
    7
    6
    5
    2
    280
    46 4 1118 253239 175+7
    3 10 1724 313845168+14
    9 16 2330 37442161+21
    22 29 3643 11815154+28
    28 35 4249 71421196-14
    34 41 486 132017189-7
    4047512 19 26331820
    182182182 182 133182 182 182
  5. Add 49 to 1 to convert all sums in the last row to 182. This also changes 154 to 203.
  6. The next series of moves (in groups of 2), adding or subtracting numbers from a row converts all the sums to 182 with the generation of duplicate numbers in green, except for 1 which has no duplicate.
  7. In order to remove these duplicates add 2n2 = 98 to each row (each duplicate must be included in the modification) so that each row, column and the left diagonal have exactly one modification, i.e, numbers in pink .
  8. Square 5 is produced whereby all the sums have been converted to the magic sum 280, and S = ½(n3 + 31n).
3
280
46 4 1118 253239 175+7
3 10 1724 313845168+14
9 16 2330 37442161+21
22 29 3643 501815203-21
28 35 4249 71421196-14
34 41 486 132017189-7
4047512 19 26331820
182182182 182 182182 182 182
4
280
46 4 1118 323239 182
3 10 1738 313845182
30 16 2330 37442182
1 29 3643 501815182
28 35 4235 71421182
34 41 486 62017182
4047512 19 2633182
182182182 182 182182 182 182
5
280
46 4 1118 3213039 280
3 10 17136 313845280
128 16 2330 37442280
1 29 13443 501815280
28 133 4235 71421280
34 41 486 1042017280
4047512 19 26131280
280280280 280 280280 280 280

trans-Conversion of de Méziriac non-magic Squares to Magic Squares

Method II: Start at first row center (1 ⇒ (2,1) down knight break)
  1. Begin generating the non-magic de Méziriac square by placing 1 to the right of the center cell and performing a (1,2) up knight break.
  2. Add consecutive numbers and repeat until the square is filled. This produces the non-magic square 2.
  3. As shown square 2 is non-magic since the last row corresponds to how much must be added or subtracted to the previous column numbers to get to 182.

  4. 1
    93
    2
    8
    7 1
    6
    11 5
    4 10
    2
    35
    46 40 3428 2293 182
    39 33 2721 15245182
    32 26 2014 84438182
    25 19 137 13731133
    18 12 649 433024182
    11 5 4842 362317182
    4474135 29 1610182
    175182189 196 154161 168 182
    +70-7-14 +28 +21+14
  5. Add 49 to 1 to convert all sums in the last column to 182. This also changes 154 to 203.
  6. To begin converting this square into a magic one, add 21 to each of the entries of the right diagonal (in blue).
  7. At this point (square 4) all the sums have changed by 21.

  8. 3
    35
    46 40 3428 2293 182
    39 33 2721 15245182
    32 26 2014 84438182
    25 19 137 503731182
    18 12 649 433024182
    11 5 4842 362317182
    4474135 29 1610182
    175182189 196 203161 168 182
    +70-7-14 -21 +21+14
    4
    182
    46 40 3428 22924 203
    39 33 2721 152545203
    32 26 2014 294438203
    25 19 1328 503731203
    18 12 2749 433024203
    11 26 4842 362317203
    25474135 29 1610203
    196203210 217 224182 189 203
    +70-7-14 -21 +21+14
  9. Transpose (trans), three cells up, all the 7 numbers from the blue diagonal of square 3 to the blue diagonal of square 5. At this point the highest sum possible may be 182. This sum may change.
  10. The next series of moves (in groups of 2), adding or subtracting numbers from a column converts all the sums to 182 with the generation of duplicate numbers in green (square 6).
  11. Adding n2 = 49 to (6, 11, 17, 29, 30, 33 and 35), shown in pink , removes five duplicates and gives the magic square 7, where the magic sum is 231 and where S = ½(n3 + 17n).
5
182
46 40 347 22924 182
39 33 621 152345182
32 5 2014 294438182
4 19 1328 503731182
18 12 2749 43303182
11 26 4842 36217182
25474135 8 1610182
175182189 196 203161 168 182
+70-7-14 -21 +21+14
6
182
46 40 347 13024 182
39 33 621 152345182
32 5 2014 294438182
11 19 628 503731182
18 12 2735 433017182
11 26 4842 36217182
25474135 8 1610182
182182182 182 182182 182 182
7
231
46 40 347 17924 231
39 82 621 152345231
32 5 2014 784438231
11 19 5528 503731231
18 12 2735 433066231
60 26 4842 36217231
25474184 8 1610231
231231231 231 231231 231 231

This completes this section on the A New Procedure for Méziriac Magic Squares Continuation 7x7 Squares (Part VIII). To continue to section IX-New Procedure for Magic Squares. To return to homepage.


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