A New Procedure for Magic Squares (Part IV)

Consecutive 9x9 Mask-Generated Boustrophedonic Squares

A mask

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):

S = ½(n3 ± an)

Construction of a 9x9 Magic Square

Method: Reading from left to right - use of mask
  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.
  4. 1
    1 2 3 45
    98 7 6
    101112 13 14
    1516 17 18
       
    22 21 20 19
    23 2425 26 27
    31 30 29 28
    36 3534 33 32
    2
    1 2 3 45
    98 7 6
    101112 13 14
    1516 17 18
    37 38 394041 42 4344 45
    22 21 20 19
    23 2425 26 27
    31 30 29 28
    36 3534 33 32
    3
    1 2 3 45
    98 7 6
    101112 13 14
    1516 17 18
    37 38 394041 42 4344 45
    50 22492148 20 4719 46
    23 51 245225 53 2654 27
    59 31 583057 2956 2855
    36 63 356234 61 3360 32
  5. 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.

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

  7. 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.

  8. 4
    193
    1 64 265 366 467527792
    72971870 7 696 68380-11
    1073117412 75 1376 14358 11
    8118801779 1678 11577461-92
    37 38 394041 42 4344 453690
    50 22 492148 20 4719 46322 47
    23 51 245225 53 2654 2733534
    59 31 583057 2956 2855403-34
    36 63 356234 61 3360 32416-47
    369369369 369 369369 369 369369 185
    5
    193
    1 64 265 9566 4675369
    72971859 7 696 68369
    1073117423 75 1376 14369
    81188011-13 11678 1577369
    37 38 394041 42 4344 45369
    50 22 492195 20 4719 46369
    23 51 245259 53 2654 27369
    59 31 583023 2956 2855369
    36 63 3562-13 61 3360 32369
    369369369 369 369369 369 369369 185
    +
  9. 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).


Mask A
176 176 184 184
184176184 176
184 184 176 176
176184176 184
176 184 184 176
176 184 176184
184 176 176 184
184 176176 184
184184 176 176
6
1089
177 64 265 27166 188671891089
7219371184243 7 69182 681089
10731957423 259 18976 190 1089
25720225617-13 16262 15771089
37 214 3922441 226 43220 451089
50 198 4920595 20 22319 2301089
207 51 2005259 229 26238 271089
243 31 58206199 2956 212551089
36 63 21962171 237 3360 2081089
108910891089 1089 10891089 1089 10891089 1089

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


Copyright © 2010 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com