A New Procedure for Magic Squares (Part IB)

Equation Generated Equation Generated Centered Sequential 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 until every other cell is filled. (This is a modification of the 9x9 square constructed previously ). Consecutive numbers are then added to the next rows using a (1,down,1,right) knight move as shown below in the examples. However, there is one exception to the rule:

(1) the center row is not filled in until until all the other rows are partially filled in.

Additions to the square going up are performed by filling in the center row followed by the empty cells below the center row. The cells above the filled centered row are filled according to the general sequence formula shown below, i.e., 2 cells to the right, 4 cells to the right and 3 cells to the left as shown in the sequence table derived from the sequence equation. Again this is modified from the 9x9 method where the row adjacent to the center row was initially filled in a reverse manner instead of to the right as in this new method.

After converting the squares into semi-magic ones the square are converted into magic ones 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 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 where 9 = 4n + 1 by adding numbers in a consecutive manner starting at row 1 cell 5 and filling every other the cell (square 1).
  2. Upon reaching the numeral 5 perform a knight break enter 6 at cell 4 and continue filling cells in a normal fashion.
  3. After skipping over the center row add 19 to the eight cell in the 6th row and continue adding numerals in normal fashion in the last row of the square.
  4. From 36 go to the center row and fill in the row consecutively (square 2). Fill in the rest of the square, going backwards this time (square 3), starting at 45 → 46. At 63 go to 64 adjacent to the numeral 2 on the first row.
  5. 1
    451 2 3
    96 7 8
    141011 12 13
    15 16 17 18
     
    20 21 22 19
    25 2627 23 24
    30 31 28 29
    35 3632 33 34
    2
    451 2 3
    96 7 8
    141011 12 13
    15 16 17 18
    37 38 394041 42 4344 45
    20 21 22 19
    25 2627 23 24
    30 31 28 29
    35 3632 33 34
  6. Fill in the rest of row 1 in a backwards manner, then fill in the rest of the square using the sequence table to generate the moves. For example, to go to the second row move 2 cells (knight), followed by filling the row to the right. Continue filling the next rows using first, 4 moves right then 3 moves left.
  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 four duplicates have been generated in pink (Square 4).
  8. 3
    193
    4675661 65 264 3 27792
    72968669 7 708 71 380-11
    147610 7311 7412 751335811
    78 15 7916 8017 811877461-92
    37 38 394041 42 4344 453690
    47 20 482149 22 5019 4632247
    25 52 265327 54 2351 2433534
    57 30 583159 28 5529 56 403-34
    35 62 366332 6033 6134416-47
    369369369 369 369369 369369369 188
    4
    193
    46756693 65 264 3 369
    72968658 7 708 71 369
    1476107322 7412 7513369
    78 15 7916 -1217 811877369
    37 38 394041 42 4344 45369
    47 20 482196 22 5019 46369
    25 52 265361 54 2351 24369
    57 30 583125 28 5529 56 369
    35 62 3663-15 6033 6134369
    369369369 369 369369 369369369 188
    +
  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 square 3 (as in the de la Hire method) that all sums will equal.
Mask B
176 181
176 181
176181
181 176
181 176
176 181
176 181
181 176
181 176
5
726
4 67 1816693 65 264184 726
72968182 587251871726
1476107322 74 18825613726
781579197-12 193 811877726
21838394041 42 4322045726
22320482196 203 501946726
252282653242 54 235124726
573023931201 28 552956726
352433663-15 60 3361210726
726726726 726 726726 726726 726726

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


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