A New Procedure for Magic Squares (Part IB)

Centered Sequential 9x9 Zig Zag Mask-Generated Squares

A mask

A Discussion of the New Method

Skip the discussion go to examples

In this method the numbers on a 4n + 1 square are placed consecutively starting from the center cell in the top row and entered in a zig zag manner. Additions to the square are done by leaving the center row unfilled and continuing the zig zag pattern below the center row. The center row is then filled in consecutively followed by the filling in of the bottom and top rows again in a zig zag manner. Breaking involves moving down 2 cells to the next row and continuing addition of numbers to the right.

After the square is filled, the square is converted into a semi-magic one by addition or subtraction i,e. by the differences of numbers in the last row. The square is then converted by means of a numerical mask into a magic square. Moreover, this new square may have negative numbers or numbers greater than n2.

In addition, it will also be shown that the sums of these squares follows either of the two sum equations shown in the New block Loubère Method and Consecutive 5x5 Mask Generated squares:

S = ½(n3 ± an)
S = ½(n3 ± an + b)

Construction of a 9x9 Magic Square

Method: Sequential Readout - use of a mask
  1. Construct the 9x9 Square 1 by adding numbers in a consecutive zigzag manner starting at row 1 cell 5 and filling every other cell (square 1). Note that on reaching the final cell in the row the addition is continued at the other end of the row.
  2. Upon reaching the numeral 9 a 2 down cell break is performed to numeral 10 and addition to the cells in a normal fashion.
  3. To skip over the center row at numeral 18, a cell break of 4 down is performed and addition to the cells in a normal fashion.
  4. From 36 go to the center row and fill in the row consecutively (square 2).
  5. 1
    681 3 5
    79 2 4
    161811 13 15
    17 10 12 14
     
    27 20 22 24
    26 1921 23 25
    28 30 32 34
    36 2931 33 35
    2
    681 3 5
    79 2 4
    161811 13 15
    17 10 12 14
    37 38 394041 42 4344 45
    27 20 22 24
    26 1921 23 25
    28 30 32 34
    36 2931 33 35
  6. After filling in the center row the rest of the square is filled starting at 46 and continuing in zigzag fashion (square 3), Thence to 64 (second row) and filling and completing the square.
  7. Since only the columns are equal to the magic sum while the row and diagonal sums are not, add or subtract the numbers in the last column to those of the center column. At this point six duplicates have been generated in pink or the blue line (Square 4).
  8. 3
    178
    6688701 72 365 5 29871
    67769971 2 644 66 35910
    1678188011 7313 7515379-10
    77 17 7910 8112 741476440-71
    37 38 394041 42 4344 453690
    47 27 492051 22 5324 4633930
    26 48 195021 52 2354 2531851
    57 28 593061 32 6334 56 420-51
    36 58 296031 6233 5535399-30
    369369369 369 369369 369369369 196
    4
    178
    66887072 72 365 5 369
    67769981 2 644 66 369
    167818801 7313 7515369
    77 17 7910 1012 741476369
    37 38 394041 42 4344 45369
    47 27 492081 22 5324 46369
    26 48 195072 52 2354 25369
    57 28 593010 32 6334 56 369
    36 58 29601 6233 5535369
    369369369 369 369369 369369369 196
    +
  9. 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 square4 (as in the de la Hire method) that all sums will equal.
Mask B
173 191191 173
173173191 191
191173 191173
191173191 173
173191 173191
364 173191
191 191173173
173173 191 191
191 173191173
5
1097
1796887072263 19465 178 1097
24672429272 2 64195 66 1097
1626918253192 73186 75151097
268 17 252201 1012 74187761097
37 211 2304041 215 4344 2361097
47 391 492081 22 53197 2371097
26 48 210241245 52 19654 251097
57 28 59203183 223 63225 56 1097
227 58 29601 235224 552081097
109710971097 1097 10971097 109710971097 1097

This completes this section on a new Centered Sequential Mask-Generated Squares (Part IB). The next section deals with Centered 13x13 Sequential Mask-Generated Zig ZagSquares (Part IC). To return to homepage.


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