A New Procedure for Magic Squares (Part I)

Cross Modified Squares and 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.

This site will explore new squares which are derived in a different manner from the standard Loubère approach. A 5x5 Loubère square which is produced using the staircase method is shown below in I. If all the even numbers in this square are exchanged for their complements the result is II:

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This is equivalent to using the following complementary table where the number sequence is 1 → 24 → 3 → 22...13 → 12 → 15...25.

Square II, however, is not magic using the standard Loubère approach but a new approach which gives two types of squares is. Taking the complementary table numbers and placing them in the same order into a 5x5 square produces A. If the numbers are placed boustrophedonically (reading in regular order then in backward order) the result is B:

Both these squares after manipulation produce two types of magic squares. One by the manipulation of individual cells, the other by the use of a symmetrical C₂ mask (symmetrical by 180° rotation) of n² numbers (see Method II below).

Moreover, only starting with the squares with regular read-out such as A produces an internal cross, When n is 4n + 1 the crosses generated are one cell across whose value is ½(n² + 1), while when n is 4n + 3 the crosses are three cells across and sum up to 3(½)(n² + 1). The magic squares produced have only partially sequential order and not all the numbers from 1 to n² are present. Because of the manipulation numbers greater than n²or less than 1 are generated.

In addition, it will also be shown that the sums of these squares follow the new sum equation shown previously in a New Block Loubère Method:

Construction of 5x5 Magic Squares

1. Recopy square A with the sum and row columns (in grey).
2. As shown below this square is not magic because all the columns and rows don't sum to 65.
3. Adjust the center column by adding and subtracting numbers from the selected cells to generate 2. For example, adding 10 to 3 and subtracting 10 from 23 keeps the center column at 65.
4. Adjust the center row by adding and subtracting numbers from the selected cells to generate 3 where the non-yellow group of four cells sum to 52 a multiple of 13 which is also obtained from the equation ½(n² + 1)  (½(n - 1))² .

1. Recopy square B with the sum and row columns (in grey).
2. As shown below this square is not magic because all the columns and rows don't sum to 65.
3. Adjust the blue cells by adding and subtracting numbers from the selected cells to generate 2. For example, adding 10 to 1 and subtracting 10 from 25 keeps the center column at 65. In addition, the cells in green are duplicates of cells on the blue diagonal.
4. Use the mask B consisting of numbers 1 and 2 to remove duplicates. Where the numbers intersect with the numbers of square 2 add the appropriate factor of n², 25 for "1" and 50 for "2".
5. Addition of the right factor gives square 3 with S = ½(n³ + 41n) and the modification of 14 numbers.

This completes this section on the new 5x5 Cross and Mask-Generated Methods (Part I). The next section deals with new 7x7 Cross and Mask-Generated Methods (Part II). To return to homepage.