A New Procedure for Magic Squares (Part II)

Consecutive Internal 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 second leftmost column and entered across every other cell. Consecutive numbers are then added to the next rows boustrophedonically. When n > 7 at least one row turns into a regular left to right order, increasing by one for each n. In addition every other number in the center row (starting with the second cell) will take on its complement. For example for a 5x5 square the second number 12 becomes 14 and 14 becomes 12 (see the 5x5 example below). 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 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. Don't fill in the center row but proceed to the first cell in the fourth row (the number 6).
  2. On reaching 36 reverse the pattern by adding consecutive numbers (remembering that every other number except for the center cell takes on its complement) filling the center row then proceeding from 45 along the khaki path(Square 2).
  3. 1
    1 2 3 4
    567 8 9
    1011 12 13
    181716 15 14
     
    19 2021 22 23
    27 26 25 24
    28 2930 31 32
    36 35 34 33
    2
    1 2 3 4
    567 8 9
    1011 12 13
    181716 15 14
    37 44 394241 40 4338 45
    19 46 204721 48 2249 23
    54 27 532652 25 5124 50
    28 55 295630 5731 5832
    63 36 623561 34 6033 59
  4. On reaching 63 fill in 64 (ast cell in the first row).
  5. Fill in the first row then add 69 to the khaki cell on the second row. This is the only row that has its entries added in reverse to the other rows as was mentioned above.
  6. Since the columns are all equal to 369 add or subtract the numbers in the last row from the center column values to generate square 4. At this point four duplicates have been generated.
  7. 3
    546
    68 1 672 663 6546434029
    5726717 70 869 931752
    7710761175 12 7413 73421-52
    1878177916 8015 8114398-29
    37 44 394241 40 4338 453690
    19 46 204721 48 2249 2329574
    54 27 532652 25 5124 503627
    28 55 295630 5731 5832376-7
    63 36 623561 34 6033 59443-74
    369369369 369369369 369 369369 552
    4
    546
    68 1 672 953 65464369
    57267159 70 869 9369
    7710761123 12 7413 73369
    18781779-13 8015 8114369
    37 44 394241 40 4338 45369
    19 46 204795 48 2249 23369
    54 27 532659 25 5124 50369
    28 55 295623 5731 5832369
    63 36 6235-13 34 6033 59369
    369369369 369369369 369 369369 552
  8. 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 4 (as in the de la Hire method) that all sums will equal the magic sum.

+
Mask A
183 177 177 183
177 177183 183
183 177 183 177
177 183 183177
177 177 183183
183 177177183
183 183 177 177
183 177 183 177
183 183 177177
5
1089
68 184 67179 2723 2484641089
1827218325459 70 8252 91089
2601877611206 12 74190 731089
187817256-13 263198 81191 1089
214 44 2164241 40 43221 2281089
19 46 204795 231 199226 2061089
54 210 53209236 202 5124 501089
211 232 2125623 23431 58321089
63 36 24535170 34 23733 2361089
108910891089 108910891089 1089 10891089 1089

This completes this section on a new Consecutive Internal 9x9 Mask-Generated Squares (Part II). The next section deals with Consecutive Internal 13x13 Mask-Generated Squares (Part III). To return to homepage.


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