A New Procedure for Magic Squares (Part IX) Continuation

Loubère and Méziriac Block Modified 9x9 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.

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

This is done by taking the non-magic squares and converting them into magic ones using a variety of means. These squares are depicted below in methods I and II. The generation of these magic squares involves an almost normal approach:
Numbers are added consecutively starting out with 1 and ending with numbers greater than n² after modification. A break involves translational moves (up, down or sideways).

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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Conversion of a Loubère non-magic Square to a Magic Square

1. Begin generating the non-magic Loubère square by placing 1 at the first cell of the last row and add three consecutive numbers followed by break 2 cells down.
2. Fill in the rest of the square as shown below in squares 1-2.
3. 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 405.



1     12     15                     11     14                                                         10                             9                             8     20             4     7     19             3     6     18             2     5     17             1     13     16

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2 405 42 12 54 15 66 27 69 39 81 405 0 11 43 14 65 26 68 38 80 41 396 +9 52 22 64 25 76 37 79 49 10 414 -9 21 63 24 75 36 78 48 9 51 405 0 62 23 74 35 77 47 8 50 20 396 +9 31 73 34 4 46 7 58 19 61 333 +72 72 33 3 45 6 57 18 60 30 324 +81 32 2 44 5 56 17 59 29 71 315 +90 1 43 13 55 16 67 28 70 40 333 +72 324 324 324 324 405 405 405 405 405 405

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4. At this point (square 3) add n² = 81 to 31, 32, 44 and 55.
5. The next series of moves (in groups of 2), adding or subtracting numbers from a column (square 4) converts all the sums to 405 with the generation of a duplicate number (68).



3 405 42 12 54 15 66 27 69 39 81 405 0 11 43 14 65 26 68 38 80 41 396 +9 52 22 64 25 76 37 79 49 10 414 -9 21 63 24 75 36 78 48 9 51 405 0 62 23 74 35 77 47 8 50 20 396 +9 112 73 34 4 46 7 58 19 61 414 -9 72 114 3 45 6 57 18 60 30 405 0 32 2 125 5 56 17 59 29 71 396 +9 1 43 13 136 16 67 28 70 40 414 -9 405 405 405 405 405 405 405 405 405 405

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4 405 42 12 54 15 66 27 69 39 81 405 11 43 14 65 26 68 38 80 50 405 52 22 64 25 76 37 79 49 1 405 21 63 24 75 36 78 48 9 51 405 62 23 74 35 77 47 8 59 20 405 112 73 34 4 46 7 58 10 61 405 72 114 3 45 6 57 18 60 30 405 32 2 125 5 56 17 68 29 71 405 1 43 13 136 16 67 19 70 40 405 405 405 405 405 405 405 405 405 405 405

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6. In order to remove this duplicate add n² = 81 to each row so that each row, column and diagonal have exactly one modification.
7. Square 4 is produced whereby all the sums have been converted to the magic sum 486, and S = ½(n³ + 27n).

Conversion of a Méziriac non-magic Square to a Magic Square

1. Begin generating the non-magic Méziriac square by placing 1 to the right of center cell of the fifth row and add four consecutive numerals, followed by a right break.
2. Fill the square as shown to produce non-magic square 2.

1 5 6                             14 15                         4                             3 13                         2 12                         1 11                             10                             9                             8                             7

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2 657 5 6 16 26 36 46 56 66 76 333 14 15 25 35 45 55 65 75 4 333 23 24 34 44 54 64 74 3 13 333 32 33 43 53 63 73 2 12 22 333 41 42 52 62 72 1 11 21 31 333 50 51 61 71 81 10 20 30 40 414 59 60 70 80 9 19 29 39 49 414 68 69 79 8 18 28 38 48 58 414 77 78 7 17 27 37 47 57 67 414 369 378 387 396 405 333 342 351 360 333

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3. To begin converting this square into a magic one, add 81 to each of the entries 1, 2, 3, 4 and 5. This modifies all the sums in the last column to 414 and five sums on the next to the last row.
4. The next series of moves (in groups of 2), adding or subtracting numbers from a row (square 4) converts all the sums to 414 with the exception of the right diagonal. Four duplicates, 23, 80, 27 and 29 are produced.

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3 657 86 6 16 26 36 46 56 66 76 414 14 15 25 35 45 55 65 75 85 414 23 24 34 44 54 64 74 84 13 414 32 33 43 53 63 73 83 12 22 414 41 42 52 62 72 82 11 21 31 414 50 51 61 71 81 10 20 30 40 414 59 60 70 80 9 19 29 39 49 414 68 69 79 8 18 28 38 48 58 414 77 78 7 17 27 37 47 57 67 414 450 378 387 396 405 414 423 432 441 414 -36 +36 +27 +18 +9 0 -9 -18 -27

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4 657 86 6 16 26 36 46 56 66 76 414 14 15 25 35 45 55 65 75 85 414 23 24 34 44 54 64 74 84 13 414 32 33 43 53 63 73 83 12 22 414 41 42 52 80 72 82 11 3 31 414 50 51 61 71 81 10 20 30 40 414 23 96 70 80 9 19 29 39 49 414 68 69 79 8 27 28 29 48 58 414 77 78 7 17 27 37 47 57 67 414 414 414 414 414 414 414 414 414 414 414

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5. Adding 3n² = 243 to (23, 33, 34, 80, 55, 42, 29, 66 and 40), shown in orange , removes four duplicates and gives the magic square 5, where the magic sum is 657 and where S = ½(n³ + 65n).

This completes this section on the new block Loubère Knight Block Modified 9x9 Squares (Part IX). To return to homepage.