A New Procedure for Magic Squares (Part II)

Consecutive Boustrophedonic 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 first leftmost column and entered across every other cell. Consecutive numbers are then added to the next rows boustrophedonically. The numbers are added to the last cell on the last row/column. in reverse order starting at the next to the last cell in the last row/column. The square which is not magic is modified into a form which can be converted into a magic one by the use of a mask. This mask generates numbers which are added to certain cells in the square to produce a final square 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 the sum equation shown in the New block Loubère Method. :

S = ½(n3 ± an)

Construction of a 9x9 Magic Square

Method: Reading boustrophedonically (like a sidewinder snake) - use of mask
  1. Construct the 9x9 Square 1 where 5 = 4n + 1 by adding numbers in a consecutive manner starting at row 1 cell 1, then swing around at the end (boustrophedonically) until the final cell is reached (square 1).
  2. On reaching the end cell reverse the process and continue adding numbers consecutively and boustrophedonically (Square 2).
  3. 1
    1 2 3 4 5
    98 7 6
    1011 12 13 14
    1817 16 15
    19 2021 22 23
    27 26 25 24
    28 2930 31 32
    36 35 34 33
    37 3839 40 41
    2
    208
    1 812 803 79478 533336
    73974875 7 766 77 405-36
    1072117112 7013 6914 34227
    64186517 6616 671568396-27
    19 63 206221 61 2260 23 35118
    55 27 56 2657 25 5824 59 387-18
    28 54 295330 5231 51323609
    46 36 473548 34 4933 50378-9
    37 45 384439 43 4042 413690
    333405342 396351387 360 378369 189
    36 -36 27-2718 -189 -90
  4. Since all sums of all the columns or rows are not equal to 65 add or subtract the numbers in the last row from those numbers identical in sum in the last columns. At this point six duplicates have been generated (Square 3).
  5. 3
    208
    37 812 803 79478 5369
    73-2774875 7 766 77 369
    1072387112 7013 6914 369
    641865-10 6616 671568369
    19 63 206239 61 2260 23 369
    55 27 56 2657 7 5824 59 369
    28 54 295330 5240 5132369
    46 36 473548 34 4924 50369
    37 45 384439 43 4042 41369
    369369369 369369369 369 369369 189
  6. Generate a mask whereby the sums of the columns and rows are constructed as in the box below. This assures that when each of these values is added to the corresponding cell in square 3 (as in the de la Hire method) that all sums will equal the magic sum.
+
Mask A
180 162
162 180
162 180
180162
162 180
180 162
180 162
162 180
180 162
4
711
217 81 16480 379 4785711
73-277417075 7 76186 77711
172723825112 70 1369 14711
641865-1066 16247 15230 711
19 63 2062201 61 2260 203711
55 27 562657 187 58186 59711
28 234 295330 214 4051 32711
46 198 4735228 3449 2450711
37 45 2184439 43 20242 41711
711711711 711711711 711 711711 711

This completes this section on a new Consecutive Boustrophedonic Mask-Generated Squares (Part II). To return to homepage.


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