A New Procedure for Magic Squares (Part IV)

Consecutive 9x9 Mask-Generated Boustrophedonic 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 shown in New block Loubère Method):

Construction of a 9x9 Magic Square

1. Construct Square 1 by adding consecutive numbers numbers to every other cells. Do not fill in the center row at this moment.
2. After reaching number 36 jump to the center row and fill it consecutively (Square 2).
3. After reaching 45 go down one cell and continue filling the cells up to 59 (Square 3). At this point fill in in the number 60 into the second cell of the last row and continue to the numeber 63.

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

⇒

2 1 2 3 4 5 9 8 7 6 10 11 12 13 14 15 16 17 18 37 38 39 40 41 42 43 44 45 22 21 20 19 23 24 25 26 27 31 30 29 28 36 35 34 33 32

⇒

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

4. From 63 go up the column and insert 64 into the cell and continue filling in a reverse manner on all of the four top rows.


5. At this point not all columns, rows or diagonals sum to 369.


6. Where the grey sums on the next to the last right hand column right 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 5. At this point five duplicates have been generated.



4 193 1 64 2 65 3 66 4 67 5 277 92 72 9 71 8 70 7 69 6 68 380 -11 10 73 11 74 12 75 13 76 14 358 11 81 18 80 17 79 16 78 115 77 461 -92 37 38 39 40 41 42 43 44 45 369 0 50 22 49 21 48 20 47 19 46 322 47 23 51 24 52 25 53 26 54 27 335 34 59 31 58 30 57 29 56 28 55 403 -34 36 63 35 62 34 61 33 60 32 416 -47 369 369 369 369 369 369 369 369 369 185

⇒

5 193 1 64 2 65 95 66 4 67 5 369 72 9 71 8 59 7 69 6 68 369 10 73 11 74 23 75 13 76 14 369 81 18 80 11 -13 116 78 15 77 369 37 38 39 40 41 42 43 44 45 369 50 22 49 21 95 20 47 19 46 369 23 51 24 52 59 53 26 54 27 369 59 31 58 30 23 29 56 28 55 369 36 63 35 62 -13 61 33 60 32 369 369 369 369 369 369 369 369 369 369 185

+

7. A possible mask for for converting square 5 into square 6 adds either 176 or 184 of mask A to the appropriate cells of square 5 (see box below).




• We start by subtracting each of the diagonals(185,193) from square 7 from 369 to give 183 and 176, respectively and which will be used as what I call the "de la Hire constants".
  Addition of the sum of these two numbers, 183 + 176 = 360 to 369 gives 729 a magic pre-sum.
• If we subtract the sum 775 from the two diagonals we obtain the following two sums:
  729 = 185 + 544 and 729 = 193 + 536.
• The following solutions are obtained:
  The left diagonal: 729 = 185 + 544 = 185 + 176 + 2(184)
  The right diagonal: 729 = 193 + 536 = 193 + 2(176) + 184
  The rows and columns: 729 = 369 + 176 + 184.
• However, four cells in the center column of square 5 must be changed for all duplicates to disappear. In order for no duplicates to occur the equations must be changed such that S = 1089 and the equations change accordingly:
  The left diagonal: 1089 = 185 + 906 = 185 + 2(176) + 3(184)
  
  The right diagonal: 1089 = 193 + 894 = 193 + 3(176) + 2(184)
  
  The rows and columns: 1089 = 369 + 720 = 369 + 2(176) + 2(184).


• Generate the mask using the 176 and 184 factors and adding these factors to the appropriate cells in square 5 to generate square 6. Sometimes duplicates are formed so go back and tweak the mask a little (move some of the cell numbers around) until no duplicates are generated.
• Square 6 has a magic sum equal to 1089, i.e., S = 1089 = ½(n³ + 161n).

This completes this section on a consecutive 9x9 Mask-Generated Methods (Part IV). To return to homepage.