A New Procedure for Magic Squares (Part I)

Consecutive 5x5 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 leftmost column and entered across every other cell. Consecutive numbers are then added to the next rows boustrophedonically or in regular left to right order. 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 a modified sum equation (the non-modified form was shown in the New block Loubère Method). This equation is to be used when n = 5:

Construction of 5x5 Magic Square I

1. Construct Square 1 by adding numbers in a consecutive manner to the cells. Don't fill in the center row but proceed to the fourth cell in the last row (the number 6). Then proceed to 8 by two reverse readouts.
2. On reaching 10 reverse the pattern by adding consecutive numbers, 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 (Squares 2 and 3).

1 1 2 3 4 5    7 6 10 9 8

⇒

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

⇒

3 38 1 22 2 21 3 49 16 25 4 24 5 23 81 -16 11 12 13 14 15 65 0 18 7 17 6 16 64 1 10 20 9 19 8 66 -1 65 65 65 65 65 32

⇒



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3. Construct a mask according to the following logical method:




• We start by subtracting each of the diagonals(32,38) from 65 (square 3) to give 33 and 27, respectively and which will be used as what I call the "de la Hire constants".
  
  Addition of the sum of these two numbers, 33 + 27 = 60 to 65 gives 125 a magic pre-sum.


• If we subtract the the two diagonals from the sum 125 we obtain the following two equations:
  
  125 = 38 + 87 and 125 = 32 + 93.


• These can be broken down into:
  
  125 = 38 + 2(27) + 33
  
  125 = 32 + 27 + 2(33) and
  
  125 = 65 + 27 + 33
  
  
• However, only by converting the pre-sum 125 into 158 (by the addition of 33 to each of the above equations) can we obtain a solution such that the following conditions are obeyed:
  
  The right diagonal: 158 = 38 + 120 = 38 + 2(27) + 2(33)
  
  The left diagonal: 158 = 32 + 126 = 32 + 27 + 3(33)
  
  The rows and columns: 158 = 65 + 27 + 2(33).


• Generate the mask using the 27 and 33 factors adding these factors to the appropriate cells in square 4 to generate square 5. The factors 27 or 43 may be added alone to a cell or in combination 27 + 33 or 33 + 33 as shown in mask A.


• Square 5 has a magic sum equal to 158, i.e., S = 158 = ½(n³ + 37n + 4).




4. Where the grey sums on the penultimate right hand column intersect the grey sums in the next to the last row adjust the values in these cells by adding and subtracting the values in the last row and columns to generate 4. At this point three duplicates have been generated.
5. Generate a mask whereby the sums of the columns and rows are constructed as in the box above. 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 158.

Construction of 5x5 Magic Square II

1. Construct Square 1 by adding consecutive numbers in a consecutive manner to the cells. Don't fill in the center row but proceed to the fourth cell in the fourth row (the number 6). However at this point place the numbers into the cells as was done in the above example II.
2. On reaching 10 reverse the pattern by adding consecutive numbers, 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 in a zig zag manner(Squares 2 and 3).

6 1 2 3 5 4    7 6 10 9 8

⇒

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

⇒

8 37 1 21 2 22 3 49 16 25 5 24 4 23 81 -16 11 12 13 14 15 65 0 18 7 17 6 16 64 1 10 20 9 19 8 66 -1 65 65 65 65 65 33

⇒



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3. Construct a mask according to the following logical method:




• We start by subtracting each of the diagonals(33,37) from 65 (square 3) to give 32 and 28, respectively and which will be used as what I call the "de la Hire constants".
  
  Addition of the sum of these two numbers, 32 + 28 = 60 to 65 gives 125 a magic pre-sum.


• If we subtract the the two diagonals from the sum 125 we obtain the following two equations:
  
  125 = 37 + 88 and 125 = 33 + 92.


• These can be broken down into:
  
  125 = 37 + 2(28) + 32
  
  125 = 32 + 28 + 2(32) and
  
  125 = 65 + 28 + 32
  
  
• However, only by converting the pre-sum 125 into 157 (by the addition of 32 to each of the above equations) can we obtain a solution such that the following conditions are obeyed:
  
  The right diagonal: 157 = 37 + 120 = 37 + 2(27) + 2(33)
  
  The left diagonal: 157 = 33 + 124 = 33 + 28 + 3(32)
  
  The rows and columns: 157 = 65 + 28 + 2(32).


• Generate the mask using the 27 and 33 factors adding these factors to the appropriate cells in square 4 to generate square 5. The factors 27 or 43 may be added alone to a cell or in combination 28 + 32 or 32 + 32 as shown in mask A.


• Square 5 has a magic sum equal to 157, i.e., S = 157 = ½(n³ + 37n + 4).




4. Where the grey sums on the penultimate right hand column intersect the grey sums in the next to the last row adjust the values in these cells by adding and subtracting the values in the last row and columns to generate 4. At this point three duplicates have been generated.
5. Generate a mask whereby the sums of the columns and rows are constructed as in the box above. 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 157.

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