A New Procedure for Magic Squares (Part III)

Méziriac Block Modified Squares

A Discussion of the New Method

An important general principle for generating odd magic squares by the 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 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 5x5 regular Méziriac square is shown below as an example:

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Normally the Méziriac method involves a stepwise approach of consecutive numbers, i.e., 1,2,3...n². For example, A new generalized Méziriac 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 Méziriac, having numbers greater than n² by modifying a block or grid of numbers. The initial number is added either at the normal Méziriac sites, i.e., (the center of the last row or last column) 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 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.

Construction of 5x5 and 7x7 Block Modified Méziriac Squares

1. Place 1 into the first cell right of center.
2. Add the next consecutive number.
3. Move, i.e. break, two cells left.
4. Repeat the process until the square is filled as shown below in squares 1-3.
5. As shown below this square is not magic because the two columns and two rows (in grey) sum to 50 instead of 75.
6. Where the sums 50 cross corresponds to the small square in blue.
7. Only the numbers 1 and 2 may be changed to 75.
8. Adding n² = 25 to both 1 and 2 (color change to light green) gives the magic square 3 where S = ½(n³ + 5n).

1. Place 1 one cell up from the center of the main diagonal.
2. Add consecutive numbers up to the right hand corner.
3. Move, i.e. break, two cells left.
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 the two columns and two rows (in grey) sum to 50 instead of 75.
6. Where the sums 50 cross corresponds to the small square in blue.
7. Since the numbers 1 and 2 sit on a diagonal which sums to 75, these numbers remain unchanged.
8. Adding n² = 25 to both 8 and 15 gives the magic square 3 where S = ½(n³ + 5n).
9. Starting over and breaking two cells right instead of left gives another magic square 4.

1. Place Place 1 one cell up from the center of the main diagonal.
2. Add consecutive numbers up to the n - 1 cell.
3. Move, i.e. break, two cells left.
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 the two columns and two rows (in grey) sum to 35 and 60 instead of 85.
6. Where the sums 60 cross corresponds to the small squares in blue.
7. Since the numbers 1, 2, 3 and 4 sit on a diagonal which sums to 35, two of these numbers will be modified.
8. Adding n² = 25 to either (3,4,13,15) or (10,17,13,15) (color change to light green) gives the magic squares 3 and 4 where where S = ½(n³ + 9n).

Construction of 7x7 Block Modified Méziriac Squares

1. Generate the square and its sums.
2. As shown below this square is not magic because the three columns and three rows (in grey) sum to 147 instead of 196.
3. Where the sums 147 cross corresponds to the small square in blue.
4. For example two squares 2 and 3 where the set (1,2,3) or (2,4,42) are modified by the addition of n² = 49 (color change to light green).
5. The magic sum S = ½(n³ + 7n).

1. Generate the square and its sums.
2. As shown below this square is not magic because the three columns and three rows (in grey) sum to 112 and 161 instead of 210.
3. Where the sums 112 and 161 cross corresponds to the small squares in blue.
4. For example two squares 2 and 3 where the set (2,3,14,17,21) or (4,5,11,25,26) are modified by the addition of n² = 49 (color change to light green).
5. The magic sum S = ½(n³ + 11n).

This completes this section on the new block Méziriac Method (Part III). To return to Part II. To return to homepage.