A New Procedure for Magic Squares (Part VII) Continuation

Loubère Knight Block trans-Modified 7x7 Squares

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 7x7 regular Loubère square is shown below :
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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, which are then used to generate modified Loubère having numbers greater than n².

The non-magic square may use a variety of methods. For example, adding a constant to a non-magic diagonal and converting that diagonal into one that is readily amenable to a transposition (trans) is one method.

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. Dual modification is then performed as discussed above.

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:

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.
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trans-Conversion of Loubère non-magic Squares to Magic Squares

1. Begin generating the non-magic Loubère square by placing 1 in the center cell of the first row and add six non-consecutive using heptad table I and method of New generalized Loubère procedure.
2. Perform a knight break two cells right one cell up as shown in color (this is done in order to produce a non-magic square).
3. Repeat the process until the square is filled as shown below in squares 1-2.
4. As shown square 2 is non-magic and the last column corresponds to how much must be added or subtracted to the previous column numbers to get to 175.



1             1 8                 3                     5                     2                                                 6                     7                     4 11

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2 175     29 36 43 1 8 15 22 154 +21 38 45 3 10 17 24 31 168 +7 47 5 12 19 26 33 40 182 -7 2 9 16 23 30 37 44 161 +14 13 20 27 34 41 48 6 189 -14 21 28 35 42 49 7 14 196 -21 25 32 39 46 4 11 18 175 0 175 175 175 175 175 175 175 175

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5. The next series of moves (in groups of 2), adding or subtracting numbers from a row converts all the sums to 175 with the generation of duplicate numbers.
6. At this point (square 3) four numbers are duplicates (in blue and orange ).
7. In order to remove these duplicates add n² = 49 to each row so that each row, column and diagonal have exactly one modification.
8. Square 4 is produced whereby all the sums have been converted to the magic sum 224, and S = ½(n³ + 15n).

1. Begin generating the non-magic Loubère square by placing 1 in the center cell of the first row and performing a knight break move down.
2. Fill the square as shown to produce non-magic square 2.
3. As shown square 2 the last row corresponds to how much must be added or subtracted to the previous column numbers to get to 182 (the sums in the last column.



1             1         4                     3                     2                     8         11         7         10         6         9         5

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2 35 40 27 14 1 30 17 4 133 26 13 49 36 16 3 39 182 12 48 35 22 2 38 25 182 47 34 21 8 37 24 11 182 33 20 7 43 23 10 46 182 19 6 42 29 9 45 32 182 5 41 28 15 44 31 18 182 182 189 196 154 161 168 175 182 0 -7 -14 +28 +21 +14 +7

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4. To begin converting this square into a magic one, add 28 to each of the entries of the right blue diagonal.
5. At this point (square 3) all the sums have changed by 28.
6. Transpose (trans), two cells up, all the 7 numbers from the blue diagonal of square 2 to the second adjacent left diagonal (square 4). At this point the highest sum possible is 231. This sum may change.

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3 231 40 27 14 1 30 17 32 161 26 13 49 36 16 31 39 210 12 48 35 22 30 38 25 210 47 34 21 36 37 24 11 210 33 20 35 43 23 10 46 210 19 34 42 29 9 45 32 210 33 41 28 15 44 31 18 210 210 217 224 182 189 196 203 210 28 21 14 56 49 42 35

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4 231 40 27 14 1 2 17 32 133 26 13 49 8 16 31 39 182 12 48 7 22 30 38 25 182 47 6 21 36 37 24 11 182 5 20 35 43 23 10 46 182 19 34 42 29 9 45 4 182 33 41 28 15 44 3 18 182 182 189 196 154 161 168 175 182 0 -7 -14 +28 +21 +14 +7

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7. 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.
8. Adding n² = 49 to (26, 41, 7, 23, 38 and 42), shown in orange , removes four duplicates and gives the semi-magic square 6.
9. In the second modification adding n² = 49 to (9, 10, 11, 12, 13, 14 and 15) gives square 7 where the magic sum is 231 and where S = ½(n³ + 31n).

This completes this section on the new block Loubère Knight Block trans-Modified 7x7 Squares (Part VII). To continue to section VIII A New Procedure for Magic Squares Continuation. To return to homepage.