A New Procedure for Magic Squares (Part II)

Consecutive 7x7 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 squares when n ≥ 7 follow the sum equation that was shown in New block Loubère Method.

Construction of 7x7 Magic Square I

1. Construct Square 1 by adding numbers in consecutively forward to the cells. At the number 8 skip the center row and fill in consecutely every other ceell. Proceed from 14 to 15 and fill in the last two rows up to the number 21.
2. On reaching 21 go back and fill in the center row. Proceed from 28 to 29 along the yellow path, and fill in the last two rows. On reaching 35 proceed to 36 and fill up the third and fifth rows consecutively (Squares 2 and 3). Fill in the last two top rows in a forward manner.



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

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

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3 154 1 45 2 44 3 43 4 142 33 49 5 48 6 47 7 46 208 -33 8 41 10 39 12 37 14 161 14 22 23 24 25 26 27 28 175 0 42 9 40 11 38 13 36 189 -14 18 31 17 30 16 29 15 156 19 35 21 34 20 33 19 32 194 -19 175 175 175 175 175 175 175 140

⇒

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




• We start by subtracting each of the diagonals(140,154) from 175 (square 3) to give 21 and 35, respectively and which will be used as what I call the "de la Hire constants".
  Addition of the sum of these two numbers, 21 + 35 = 56 to 175 gives 231 a magic pre-sum.


• If we subtract the two diagonals from the sum 231 we obtain the following two equations:
  231 = 140 + 91 and 231 = 154 + 77.


• These can be broken down into the following equations such that the following conditions are obeyed:
  The left diagonal: 231 = 140 + 21 + 2(35)
  The right diagonal: 231 = 154 + 2(21) + 35 and
  The rows and columns: 231 = 175 + 21 + 35
  
• Generate a mask using the 21 and 35 factors adding these factors to the appropriate cells in square 4 to generate square 5.


• Square 5 has a magic sum equal to 231, i.e., S = 231 = ½(n³ + 17n).


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 231 in square 5.

Construction of a 7x7 Magic Square II

1. Construct Square 6 by adding consecutive numbers boustrophedonically, then boustrophedonically then switching over to forward to forward. At the number 8 skip the center row and fill in consecutely every other cell. Proceed from 14 to 15 and fill in the last two rows up to the number 21.
2. On reaching 21 go back and fill in the center row. Proceed from 28 to 29 along the yellow path, and fill in the last two rows. On reaching 35 proceed to 36 and fill up the third and fifth rows consecutively (Squares 2 and 3). Fill in the last two top rows boustrophedonically.



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

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

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8 152 1 43 2 44 3 45 4 142 33 49 7 48 6 47 5 46 208 -33 8 41 10 39 12 37 14 161 14 22 23 24 25 26 27 28 175 0 42 9 40 11 38 13 36 189 -14 18 31 17 30 16 29 15 156 19 35 21 34 20 33 19 32 194 -19 175 175 175 175 175 175 175 142

⇒

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


• We start by subtracting each of the diagonals(142,152) from 175 (square 3) to give 23 and 33, respectively and which will be used as what I call the "de la Hire constants".
  Addition of the sum of these two numbers, 23 + 33 = 56 to 175 gives 231 a magic pre-sum.


• If we subtract the the two diagonals from sum 231 we obtain the following two equations:
  231 = 140 + 91 and 231 = 152 + 79.


• These can be broken down into the following equations such that the following conditions are obeyed:
  The left diagonal: 231 = 142 + 23 + 2(33)
  The right diagonal: 231 = 152 + 2(23) + 33 and
  The rows and columns: 231 = 175 + 23 + 33
  
• Generate a mask using the 23 and 33 factors adding these factors to the appropriate cells in square 9 to generate square 10.


• Square 5 has a magic sum equal to 231, i.e., S = 231 = ½(n³ + 17n).


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 9. 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 9 (as in the de la Hire method) that all sums will equal 231 in square 10.

This completes this section on a new consecutive 7x7 Mask-Generated Methods (Part II). The next section deals with new consecutive 9x9 Mask-Generated Methods (Part III). To return to homepage.