A New Procedure for Magic Squares (Part III)

Consecutive 9x9 Mask-Generated 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
    67 8 9
    101112 13 14
    1516 17 18
       
    22 21 20 19
    27 2625 24 23
    31 30 29 28
    32 3334 35 36
    2
    1 2 3 45
    67 8 9
    101112 13 14
    1516 17 18
    37 38 394041 42 4344 45
    22 21 20 19
    27 2625 24 23
    31 30 29 28
    32 3334 35 36
    3
    1 2 3 45
    67 8 9
    101112 13 14
    1516 17 18
    37 38 394041 42 4344 45
    50 22 492148 20 4719 46
    27 54 265325 52 2451 23
    59 31 583057 2956 2855
    32 60 336134 62 3563 36
  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
    195
    1 67 266 365 464527792
    72671770 8 699 68380-11
    1076117512 74 1373 14358 11
    8115801679 1778 1877461-92
    37 38 394041 42 4344 453690
    50 22 492148 20 4719 46322 47
    27 54 265325 52 2451 2333534
    59 31 583057 2956 2855403-34
    32 60 336134 62 3563 36416-47
    369369369 369 369369 369 369369 183
    5
    195
    1 67 266 9565 4645369
    72671759 8 699 68369
    1076117523 74 1373 14369
    81158016-13 1778 1877369
    37 38 394041 42 4344 45369
    50 22 492195 20 4719 46369
    27 54 265359 52 2451 23369
    59 31 583023 2956 2855369
    32 60 3361-13 62 3563 36369
    369369369 369 369369 369 369369 183
    +
  9. A possible mask for for converting square 5 into square 6 adds either 174 or 186 of mask A to the appropriate cells of square 5 (see box below).


Mask A
174 174 186186
186174186 174
186 186 174 174
174186174 186
174 186 186 174
174 186 174186
186 174 174 186
186 174174 186
186186 174 174
6
1089
175 67 266 26965 190641911089
7219271181245 8 69183 681089
10761977523 260 18773 188 1089
25520125416-13 17264 18771089
37 212 3922641 228 43218 451089
50 196 4920795 20 221205 461089
213 54 2005359 226 2451 2091089
245 31 58204197 2956 214551089
32 60 21961173 236 3563 2101089
108910891089 1089 10891089 1089 10891089 1089

This completes this section on a consecutive 9x9 Mask-Generated Methods (Part III). Next section deals with consecutive a 9x9 Mask-Generated Boustrophedonic Method (reading right then left) Part IV. To return to homepage.


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