A New Procedure for Magic Squares (Part V) Continuation

Loubère and Méziriac Knight Block 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 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, the final square is transformed into a modified Loubère or Méziriac square, having numbers greater than n². This is done by modifying a block or grid of numbers on the square. The initial number is added either at the normal Loubère sites, or on the main diagonal. Every number on the diagonal can be modified, up to n - 1. Replacement of all n numbers on the diagonal leads to semi-magic squares. The same for the Méziriac method.

In this new method 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.

Construction of 7x7 Block Knight Modified Loubère and Méziriac Squares

1. Place 1 in the center cell of the first row.
2. Knight break, two cells down one cell left as shown in color.
3. Add consecutive numbers.
4. Repeat the process until the square is filled as shown below in squares 1-2.
5. As shown below this square is not magic because all the columns, rows and diagonals don't sum to a magic sum.
6. Where the sum 133 crosses corresponds to the small square in blue.
7. Add 49 to the number 1. At this point all the numbers except for one sum to 182.
8. Using one example adding n² = 25 to (4,7,16,21,28,29 and 35) gives the magic square 3. Note that all these modified numbers fall outside the blue square. In the previous non-knight Part I method all the modified numbers fall within the blue square.
9. The magic sum is S = ½(n³ + 17n).

1. Place 1 in the center cell of the first row.
2. Knight break, two cells up one cell left as shown in color.
3. Add consecutive numbers.
4. Repeat the process until the square is filled as shown below in squares 1-2.
5. As shown below this square is not magic because all the columns, rows and diagonals don't sum to a magic sum.
6. Where the sum 133 crosses corresponds to the small square in blue.
7. Add 49 to the number 1. At this point all the numbers except for one sum to 182.
8. Adding 2n² = 98 to (16,17,18,19,20,21 and 22) gives the magic square 3. Note that all these modified numbers fall outside the blue square. In the previous non-knight Part I method all the modified numbers fall within the blue square.
9. The magic sum is S = ½(n³ + 31n).

1. Place 1 in the left corner cell of the last row.
2. Knight break, two cells up one cell left as shown in color.
3. Add consecutive numbers.
4. Repeat the process until the square is filled as shown below in squares 1-2.
5. As shown below this square is not magic because all the columns, rows and diagonals don't sum to a magic sum.
6. Where the sum 133 crosses corresponds to the small square in blue.
7. Add 49 to the number 1. At this point all the numbers except for one sum to 182.
8. Adding 2n² = 98 to (16,17,18,19,20,21 and 22) gives the magic square 3. Note that all these modified numbers fall outside the blue square. In the previous non-knight Part I method all the modified numbers fall within the blue square.
9. The magic sum is S = ½(n³ + 31n) identical to sum of method II and that the rows of both squares 2 have their cell numbers in the same order. Either square may be converted to the other by transposing columns and rows.

1. Start out with a Méziriac type square by placing 1 one cell to the right of center.
2. Knight break, three cells up two cell left.
3. Repeat the process until the square is filled as shown below in squares 1-2.
4. As shown below this square is not yet magic.
5. Where the sums 133 cross corresponds to the small square in blue.
6. Add 49 to the number 1. At this point all the numbers except for one sum to 182.
7. Adding n² = 49 to (4,5,8,19,20,38,39) gives the magic square 3 where S = ½(n³ + 17n).
8. To convert this Méziriac Knight Square to a Loubère Knight Square -1 is subtracted from every cell to give square 4 where S is now equal to ½(n³ + 15n).
9. Subtraction of 25 from those cells colored in tan from square 4 gives the Loubère semi-magic Knight square 1b. This square can be generated using the New De la Loubère Method (Part II), except using a variable (3,2) up knight break.

This completes this section on the new block 7x7 Loubère Magic Square Method (Part V). To return to homepage.