A New Procedure for Magic Squares (Part IB)

Consecutive Boustrophedonic 7x7 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. At the end the square is filled in by adding numbers to the second cell on the first row and proceeding as before. 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 7x7 Magic Square I

Method: Reading consecutive from left to right boustrophedonically - use of a mask
  1. Construct the 7x7 Square 1 where 7 = 4n + 3 by adding numbers in a consecutive manner starting at row 1 cell, then swinging around at the end (boustrophedonically) until the final cell is reached (square 1).
  2. On reaching this cell go to cell 6 row 1 (next to last) and continue adding consecutive numbers in reverse. This method is equivalent to adding the requisite numbers on a line and then using a (1 down, 1 left) break as shown for the 4 → 5 move.
  3. Since the sums of all the columns or rows are not equal to 175 add or subtract the numbers in the last row to the center row and add or subtract the lst column to the center column. At this point three duplicates (in pink) have been generated (Square 3).
  4. 1
    1 2 3 4
    7 6 5
    89 1011
    14 13 12
    1516 1718
    21 20 19
    2223 2425
    2
    91
    1 28 2273 26 4 9184
    29730 6 3153214035
    8359 34 10331114035
    361437 13 381239189-14
    154216 41 174018189-14
    432144 20 451946238-63
    224923 48 244725238-63
    154196161 189 168182 17591
    21-2114 -14 7-70
    3
    63
    1 28 21113 26 4 175
    29730 41 31532175
    8359 69 103311175
    57-751 -15 455 39175
    154216 27 174018175
    432144 -22 451946175
    224923 -15 244725175
    175175175 175 175175 17563
    +
  5. 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 10 (as in the de la Hire method) that all sums will be equal to a magic sum.

Mask A
112
112
112
112
112
112
112
4
287
113 28 21113 26 4 287
29730 41 3111732287
8359 69 1033123287
57-751 97 45539287
1515416 27 174018287
432144 -22 1571946287
2249135 -15 244725287
287287287 287 287287 287287

Construction of a 7x7 Magic Square II

Method: Reading consecutive from left to right boustrophedonically - use of a mask
  1. Construct the 7x7 Square 1 where 7 = 4n + 3 by adding numbers in a consecutive manner starting at row 1 cell, then swinging around at the end (boustrophedonically) until the final cell is reached (square 1).
  2. On reaching this cell go to cell 2 row 1 (second cell) and continue adding consecutive numbers in reverse. This method is equivalent to adding the requisite numbers on a line and then using a (1 down, 2 right) break as shown for the 4 → 5 move.
  3. Since the sums of all the columns or rows are not equal to 175 add or subtract the numbers in the last row to the center row and add or subtract the lst column to the center column. At this point three duplicates (in pink) have been generated (Square 3).
  4. 1
    1 2 3 4
    5 6 7
    89 1011
    12 13 14
    1516 1718
    19 20 21
    2223 2425
    2
    91
    1 26 2273 28 4 9184
    29530 6 3173214035
    8339 34 10351114035
    361237 13 381439189-14
    154016 41 174218189-14
    431944 20 452146238-63
    224723 48 244925238-63
    154182161 189 168196 17591
    21-714 -14 7-210
    3
    63
    1 26 21113 28 4 175
    29530 41 31732175
    8339 69 103511175
    57551 -15 45-7 39175
    154016 27 174218175
    431944 -43 452146175
    224723 -15 244925175
    175175175 175 175175 17563
    +
  5. 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 10 (as in the de la Hire method) that all sums will be equal to a magic sum.
Mask B
112
112
112
112
112
112
112
4
287
1 26 21113 28 116 287
2911730 41 31732287
8339 69 1014711287
57551 97 45-739287
1274016 27 174218287
431944 -43 1572146287
2247135 -15 244925287
287287287 287 287287 287287

This completes this section on a new Consecutive Boustrophedonic 7x7 Mask-Generated Squares.The next section deals with Consecutive Boustrophedonic 9x9 Mask-Generated Squares (Part II).To return to homepage.


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