A New Procedure for Magic Squares (Part I)

Consecutive Internally Added Mask-Generated Squares

A Discussion of the New Method

Magic squares such as the Loubère have a center cell which must always contain the middle number of a series of consecutive numbers, 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.

In this method the numbers on the square are placed consecutively starting from the second leftmost column and entered across every other cell. Consecutive numbers are then added to the next rows boustrophedonically. When n > 7 at least one row turns into a regular left to right order, increasing by one for each n. In addition every other number in the center row (starting with the second cell) will take on its complement. For example for a 5x5 square the second number 12 becomes 14 and 14 becomes 12 (see the 5x5 example below). The final square is composed of numbers which may not be in serial order. For example, negative numbers or numbers greater than n² may be present in the square.

In addition, it will also be shown that the sums of these squares follow the sum equation shown in the New block Loubère Method. :

Construction of a 5x5 Magic Square

1. Construct the 5x5 Square 1 where 5 = 4n + 1 by adding numbers in a consecutive manner starting at row 1 cell. Don't fill in the center row but proceed to the first cell in the fourth row (the number 6).
2. On reaching 10 reverse the pattern by adding consecutive numbers (remembering that every other number except for the center cell takes on its complement) filling the center row then proceeding from 15 to 16 along the yellow path, and filling in the last two rows. On reaching 20 proceed to 21 and fill up the top two rows consecutively (Square 4).

1 1 2 5 4 3   6 7 8 10 9

⇒

2 1 2 5 4 3 11 14 13 12 15 6 7 8 10 9

⇒

3 1 2 5 4 3 11 14 13 12 15 6 16 7 17 8 20 10 19 9 18

⇒

4 95 23 1 22 2 21 69 -4 5 24 4 25 3 61 4 11 14 13 12 15 54 0 6 16 7 17 8 76 11 20 10 19 9 18 66 -11 65 65 65 65 65 95

⇒

3. Since the columns are all equal to 65 add or subtract the numbers in the last row from the center column values to generate square 5. At this point two duplicates have been generated.
4. Generate a mask whereby the sums of the columns and rows are constructed as in the box below. This assures that when each of these values is added to the corresponding cell in square 4 (as in the de la Hire method) that all sums will equal 120.

Construction of a 7x7 Magic Square

1. Construct the 7x7 Square 1 where 7 = 4n + 3 by adding consecutive numbers in a consecutive manner to the cells.At the number 8 proceed in a zig zag manner to number 14 then to the 5th row second cell and enter 9. The center row is not filled at this time.
2. On reaching 21 reverse the pattern by adding consecutive numbers 22-28 to the center row (remembering that every other number except for the center cell takes on its complement) yellow and khaki path, then filling in the last two rows. On reaching 35 proceed to 36 and fill in the middle rows reversibly in a zig zag manner.

7 1 2 3 7 6 5 4 8 10 12 14   9 11 13 18 17 16 15 19 20 21

⇒

8 1 2 3 7 6 5 4 8 10 12 14 22 27 24 25 26 23 28 9 11 13 18 31 17 30 16 29 15 32 19 33 20 34 21 35

⇒

9 1 2 3 7 6 5 4 8 41 10 39 12 37 14 22 27 24 25 26 23 28 42 9 40 11 38 13 36 18 31 17 30 16 29 15 32 19 33 20 34 21 35

⇒

3. Fill in the last top rows as shown in square 10.


4. Since the columns are all equal to 175 add or subtract the numbers in the last row from the center column values to generate square 11. At this point two duplicates have been generated (Square 11).

10 232 46 1 45 2 44 3 43 184 -9 7 47 6 48 5 49 4 166 9 8 41 10 39 12 37 14 161 14 22 27 24 25 26 23 28 175 0 42 9 40 11 38 13 36 189 -14 18 31 17 30 16 29 15 156 19 32 19 33 20 34 21 35 194 -19 175 175 175 175 175 175 175 230

⇒

11 232 46 1 45 -7 44 3 43 175 7 47 6 57 5 49 4 175 8 41 10 53 12 37 14 175 22 27 24 25 26 23 28 175 42 9 40 -3 38 13 36 175 18 31 17 49 16 29 15 175 32 19 33 1 34 21 35 175 175 175 175 175 175 175 175 230

+

5. Generate a mask whereby the sums of the columns and rows are constructed as in the box below. This assures that when each of these values is added to the corresponding cell in square 10 (as in the de la Hire method) that all sums will equal 287.

Note that if square 9 is produced by swithching several entries in the last two rows that a cross square is generated. All numbers in the center row are 25 and the average of the sums of the 1^st and 2^nd, 3^rd and 5^th and 6^th and 7^th are each 25.

This completes this section on a new Consecutive Internally Added Mask-Generated Squares (Part I). The next section deals with Consecutive Internal 9x9 Mask-Generated Squares (Part II). To return to homepage.