A New Procedure for Magic Squares (Part VI) Continuation

Loubère and Méziriac Knight Block trans-Modified 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, with the generation of a modified Loubère or Méziriac having numbers greater than n2.

This is done taking the semi-magic square and adding a constant to a semi-magic diagonal and converting that diagonal into one that is readily amenable to conversion to a magic sum and performing a transposition (trans).

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). 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.

trans-Conversion of Loubère and Méziriac semi-magic Squares to Magic Squares

Method I: Start on the first row center (1 ⇒ (2,1) down knight break)
  1. Begin generating the semi-magic Loubère square by placing 1 in the center cell of the first row and add four consecutive numbers.
  2. Knight break, two cells down one cell left as shown in color.
  3. Repeat the process until the square is filled as shown below in squares 1-2.
  4. As shown square 2 is semi-magic.
  5. To convert this square into a magic one add 3 to each of the entries of the left blue diagonal.
  6. At this point (square 3) there are two adjacent diagonals with four numbers identical and all sums except for one are 68.
  7. Transpose (trans), one cell to the right, all the 5 numbers from the blue diagonal of square 2 to the adjacent right diagonal (square 4). At this point two rows and one diagonal sum to the possible sum of 90. Note that if a true transposition of numbers were done, i.e., a switch of the two diagonals then column 4 would sum to 40.
  8. Using one example adding n2 = 25 to (20,5,15,10) gives the magic square 5.
  9. The magic sum is S = ½(n3 + 11n).
1
1
5
4
6 3
2
2
65
20 23 1 91265
2258 11 1965
4715 18 21 65
61417 25 365
131624 2 1065
656565 65 6575
3
68
23 23 1 91268
2288 11 1968
4718 18 21 68
61417 28 368
131624 2 1368
686868 68 6890
4
65
23 20 1 91265
2285 11 1965
4718 15 21 65
61417 28 2590
101624 2 1365
656565 65 9090
5
90
23 45 1 91290
22830 11 1990
4718 40 21 90
61417 28 2590
351624 2 1390
909090 90 9090
Method II: Start at first row center (1 ⇒ (2,1) up knight break)
  1. Begin generating the non-magic Loubère square by placing 1 in the center cell of the first row and add four consecutive numbers.
  2. Knight break, two cells up one cell right as shown in color.
  3. Repeat the process until the square is filled as shown below in squares 1-2.
  4. As shown square 2 is not magic mainly because the sum of the columns are not all 65.
  5. To begin converting this square into a magic one add 5 to each of the entries of the left blue diagonal.
  6. At this point (square 3) all the sums have changed by 5 except for the blue diagonal which is 90.

  7. 1
    1
    5
    4
    3
    6 2
    2
    90
    14 10 1 221865
    9521 17 1365
    42516 12 8 65
    242011 7 365
    19156 2 2365
    707555 60 6565
    3
    95
    19 10 1 221870
    91021 17 1370
    42521 12 8 70
    242011 12 370
    19156 2 2870
    758060 65 7090
  8. Transpose (trans), one cell up, all the 5 numbers from the blue diagonal of square 2 to the adjacent right diagonal (square 4). At this point one row, one column and both diagonals sum to the possible sum of 90.
  9. The next series of moves (in groups of 2), adding or subtracting numbers from a row converts most sums to 65 with the generation of 3 duplicate numbers.
  10. Adding n2 = 25 to (5, 9, 12 and 16) removes three duplicates and gives the magic square 6, whereby all the sums have been converted to the magic sum 90 and S = ½(n3 + 11n),
4
90
19 5 1 221865
91016 17 1365
42521 7 8 65
242011 12 2390
14156 2 2865
707555 60 9090
5
90
19 5 1 221865
91016 17 1365
-12521 12 8 65
242011 12 2390
14516 2 2865
656565 65 9090
6
90
19 30 1 221890
341016 17 1390
-12521 37 8 90
242011 12 2390
14541 2 2890
909090 90 9090
Method III: Start at right of center (1 ⇒ (3,1) up knight break)
  1. Begin generating the semi-magic Méziriac square by placing 1 to the right of the center cell.
  2. Knight break, three cells up one cell left as shown in color.
  3. Add seven consecutive numbers.
  4. Repeat the process until the square is filled as shown below in squares 1-2.
  5. As shown square 2 is not magic mainly because the sum of the columns are not all 182.
  6. To begin converting this square into a magic one add 1 to each of the entries of the left blue diagonal.
  7. At this point (square 3) all the sums have changed by 1 except for the blue diagonal which is 182.

  8. 1
    2
    8
    7
    6 1
    5
    9 4
    3
    2
    84
    46 36 192 412414 182
    35 18 840 231345182
    17 7 3929 124434182
    6 38 2811 13316133
    37 27 1049 32225182
    26 9 4831 21443182
    15473020 3 4225182
    182182182 182 133182 182 175
    3
    85
    47 36 192 412414 183
    35 19 840 231345183
    17 7 4029 124434183
    6 38 2812 13316134
    37 27 1049 33225183
    26 9 4831 21543183
    15473020 3 4226183
    183183183 183 134183 183 182
  9. Transpose (trans), one cell diagonally up, all the 5 numbers from the blue diagonal of square 2 to the third right diagonal (square 4). At this point one row, one column and both diagonals sum to the possible sum of 182.
  10. Add 49 to the numeral 1 to convert sums 133 to 182.
  11. Adding n2 = 49 to (3, 4, 9, 12, 13, 17 and 18) gives the magic square 5.
  12. The magic sum is S = ½(n3 + 17n).
4
84
47 36 182 412414 182
35 19 839 231345182
17 7 4029 114434182
6 38 2812 503216182
37 27 1049 33224182
25 9 4831 21543182
15463020 3 4226182
182182182 182 182182 182 182
5
231
47 36 672 412414 231
35 19 839 236245231
66 7 4029 114434231
6 38 2861 503216231
37 27 1049 332253231
25 58 4831 21543231
15463020 52 4226231
231231231 231 231231 231 231

This completes this section on the a new procedure for Loubère Squares (Part VI). To continue to section VII A New Procedure for Magic Squares (part VII). To return to homepage.


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