A New Procedure for Magic Squares (Part II)

Consecutive Internal 9x9 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 9x9 Magic Square

1. Construct the 9x9 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 36 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 45 along the khaki path(Square 2).



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

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2 1 2 3 4 5 6 7 8 9 10 11 12 13 18 17 16 15 14 37 44 39 42 41 40 43 38 45 19 46 20 47 21 48 22 49 23 54 27 53 26 52 25 51 24 50 28 55 29 56 30 57 31 58 32 63 36 62 35 61 34 60 33 59

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3. On reaching 63 fill in 64 (ast cell in the first row).
4. Fill in the first row then add 69 to the khaki cell on the second row. This is the only row that has its entries added in reverse to the other rows as was mentioned above.
5. Since the columns are all equal to 369 add or subtract the numbers in the last row from the center column values to generate square 4. At this point four duplicates have been generated.

3 546 68 1 67 2 66 3 65 4 64 340 29 5 72 6 71 7 70 8 69 9 317 52 77 10 76 11 75 12 74 13 73 421 -52 18 78 17 79 16 80 15 81 14 398 -29 37 44 39 42 41 40 43 38 45 369 0 19 46 20 47 21 48 22 49 23 295 74 54 27 53 26 52 25 51 24 50 362 7 28 55 29 56 30 57 31 58 32 376 -7 63 36 62 35 61 34 60 33 59 443 -74 369 369 369 369 369 369 369 369 369 552

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4 546 68 1 67 2 95 3 65 4 64 369 5 72 6 71 59 70 8 69 9 369 77 10 76 11 23 12 74 13 73 369 18 78 17 79 -13 80 15 81 14 369 37 44 39 42 41 40 43 38 45 369 19 46 20 47 95 48 22 49 23 369 54 27 53 26 59 25 51 24 50 369 28 55 29 56 23 57 31 58 32 369 63 36 62 35 -13 34 60 33 59 369 369 369 369 369 369 369 369 369 369 552



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6. 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 the magic sum.

This completes this section on a new Consecutive Internal 9x9 Mask-Generated Squares (Part II). The next section deals with Consecutive Internal 13x13 Mask-Generated Squares (Part III). To return to homepage.