A New Procedure for Magic Squares (Part VI) Continuation

Loubère and Méziriac Knight Block trans-Modified 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 5x5 regular Loubère square is shown below on the left and the regular Méziriac on the right :

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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, 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:

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 and Méziriac semi-magic Squares to Magic Squares

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. 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.



1         1             5             4                                 3         6 2

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2 90 14 10 1 22 18 65 9 5 21 17 13 65 4 25 16 12 8 65 24 20 11 7 3 65 19 15 6 2 23 65 70 75 55 60 65 65

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3 95 19 10 1 22 18 70 9 10 21 17 13 70 4 25 21 12 8 70 24 20 11 12 3 70 19 15 6 2 28 70 75 80 60 65 70 90

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7. 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.
8. 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.
9. 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),

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.



1             2                     8                     7                     6             1                                 5     9             4                     3

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2 84 46 36 19 2 41 24 14 182 35 18 8 40 23 13 45 182 17 7 39 29 12 44 34 182 6 38 28 11 1 33 16 133 37 27 10 49 32 22 5 182 26 9 48 31 21 4 43 182 15 47 30 20 3 42 25 182 182 182 182 182 133 182 182 175

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3 85 47 36 19 2 41 24 14 183 35 19 8 40 23 13 45 183 17 7 40 29 12 44 34 183 6 38 28 12 1 33 16 134 37 27 10 49 33 22 5 183 26 9 48 31 21 5 43 183 15 47 30 20 3 42 26 183 183 183 183 183 134 183 183 182



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8. 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.
9. Add 49 to the numeral 1 to convert sums 133 to 182.
10. Adding n² = 49 to (3, 4, 9, 12, 13, 17 and 18) gives the magic square 5.
11. The magic sum is S = ½(n³ + 17n).

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.