A New Procedure for Magic Squares (Part VIII) Continuation

Méziriac Knight Block non-trans and trans Modified 7x7 Squares

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 :
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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:

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

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.



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

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2 280     46 4 11 18 25 32 39 175 +7 3 10 17 24 31 38 45 168 +14 9 16 23 30 37 44 2 161 +21 22 29 36 43 1 18 15 154 +28 28 35 42 49 7 14 21 196 -14 34 41 48 6 13 20 17 189 -7 40 47 5 12 19 26 33 182 0 182 182 182 182 133 182 182 182

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4. Add 49 to 1 to convert all sums in the last row to 182. This also changes 154 to 203.
5. 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.
6. 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 .
7. Square 5 is produced whereby all the sums have been converted to the magic sum 280, and S = ½(n³ + 31n).

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

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.



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

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

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4. Add 49 to 1 to convert all sums in the last column to 182. This also changes 154 to 203.
5. To begin converting this square into a magic one, add 21 to each of the entries of the right diagonal (in blue).
6. At this point (square 4) all the sums have changed by 21.

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3 35 46 40 34 28 22 9 3 182 39 33 27 21 15 2 45 182 32 26 20 14 8 44 38 182 25 19 13 7 50 37 31 182 18 12 6 49 43 30 24 182 11 5 48 42 36 23 17 182 4 47 41 35 29 16 10 182 175 182 189 196 203 161 168 182 +7 0 -7 -14 -21 +21 +14

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4 182 46 40 34 28 22 9 24 203 39 33 27 21 15 25 45 203 32 26 20 14 29 44 38 203 25 19 13 28 50 37 31 203 18 12 27 49 43 30 24 203 11 26 48 42 36 23 17 203 25 47 41 35 29 16 10 203 196 203 210 217 224 182 189 203 +7 0 -7 -14 -21 +21 +14

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7. 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.
8. 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).
9. 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).

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.