A New Procedure for Magic Squares (Part I) - New Magic Constant Equation

Loubère Block 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 as an example:

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Normally the Loubère method involves 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, having numbers greater than n² by modifying a block or grid of numbers. The initial number is added either at the normal Loubère 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.

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.

Construction of 5x5 Block Modified Loubère Squares

1. Place 1 into the lower left hand corner.
2. Add consecutive numbers up to the middle cell.
3. Move, i.e. break, one cell down.
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. Since the numbers 1 and 2 sit on a diagonal which sums to 75, these numbers remain unchanged.
8. To square 3 adding n² = 25 to both 20 and 3 (color change to light green) gives the magic square 4 where S = ½(n³ + 5n).
9. To square 3 adding -n² = -25 outside the block to 23, 18 and 6 (color change to light green) gives the magic square 5 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, one cell down.
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. Since the numbers 1 and 2 sit on a diagonal which sums to 75, these numbers remain unchanged.
8. Adding n² = 25 to both 20 and 3 gives the magic square 4 where S = ½(n³ + 5n).
9. Similarly the square breaking to the left results a square identical to 4 except for rotation (not shown).

1. Place 1 into the lower left hand corner.
2. Add consecutive numbers up to the n - 1 cell.
3. Move, i.e. break, one cell down.
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 80.
6. Where the sums 60 cross corresponds to the small square 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. Also the left diagonal.
8. Adding n² = 25 to both (2,3,13,17) and (3,4,8,20) (color change to light green) gives the magic squares 3 and 4 and where S = ½(n³ + 9n).

1. Place 1 in the middle of the last column.
2. For this 5x5 square add consecutive number 2.
3. Move, i.e. break, one cell down.
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 four numbers do not sit on a diagonal, two possible squares are possible.
8. Adding n² = 25 to both (2 and 1) or (3 and 20) gives the magic squares 3 and 4.

1. Place 1 in the middle of the last column.
2. For this 5x5 square add consecutive numbers 2 and 3.
3. Move, i.e. break, one cell down.
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 55 instead of 80.
6. Where the sums 55 cross corresponds to the small square in blue.
7. Since the numbers to be modified cannot sit on a diagonal, two possible squares are possible.
8. Adding n² = 25 to both (3, 8 and 20) or (1, 2 and 3) gives the magic squares 3 and 4 and where S = ½(n³ + 7n).

1. Place 1 in the middle of the last row.
2. For this 5x5 square add consecutive number 2.
3. Move, i.e. break, one cell down.
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 one of the six number sits on a diagonal which adds to 50 the magic square must modify this number.
8. Adding n² = 25 to 1 and 2 gives the magic square 3.

1. Place 1 in the middle of the last row.
2. For this 5x5 square add consecutive numbers 2 and 3.
3. Move, i.e. break, one cell down.
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 55 instead of 80.
6. Where the sums 50 cross corresponds to the small square in blue.
7. Since one of the six number sits on a diagonal which adds to 55, two magic square are possible.
8. Adding n² = 25 to (1, 2 and 3) or (10, 20 and 21) gives the magic squares 3 and 4.

This completes this section on the new block Loubère Method (Part I). The next section deals with Continuation of block Loubère Method (Part II). To return to homepage.