A New Procedure for Magic Squares (Part IB)

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. 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 are two exceptions to the rule:

(1) the center row is not filled in until until all the other rows are partially filled in and
(2) the row adjacent to the center row (on going down the square) has its entries filled in a different order, i.e., not using the (1,down,1,right) knight move.

Additions to the square going up are performed, first filling in the center row then filling in the empty cells. Note that since two types of odd squares exists, i.e., the 4n + 1 and the 4n + 3, two methods for filling in the squares are possible. The former example using n = 5 and 9 are shown below.

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 3 and filling every other the cell (square 6).
  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 performing exception (2) skip over center row and add 19 to the second 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 7).
  5. 6
    451 2 3
    96 7 8
    141011 12 13
    18 17 16 15
     
    19 20 21 22
    24 2526 27 23
    29 30 31 28
    34 3536 32 33
    7
    451 2 3
    96 7 8
    141011 12 13
    18 17 16 15
    37 38 394041 42 4344 45
    19 20 21 22
    24 2526 27 23
    29 30 31 28
    34 3536 32 33
  6. Fill in the rest of the square, going backwards this time (square 8), starting at 45 → 46. At 63 go to 64 adjacent to the numeral 2 on the first row.
  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 9).
  8. 8
    188
    4665671 64 265 3 27792
    71972668 7 698 70 380-11
    1476107311 7412 751335811
    81 18 7717 7816 791580461-92
    37 38 394041 42 4344 453690
    47 19 482049 21 5022 4632247
    24 52 255326 54 2751 2333534
    57 29 583059 31 5528 56 403-34
    34 62 356336 6032 6133416-47
    369369369 369 369369 369369369 190
    9
    188
    46656793 64 265 3 369
    71972657 7 698 70 369
    1476107322 7412 7513369
    81 18 7717 -1416 791580369
    37 38 394041 42 4344 45369
    47 19 482096 21 5022 46369
    24 52 255360 54 2751 23369
    57 29 583025 31 5528 56 369
    34 62 3563-11 6032 6133369
    369369369 369 369369 369369369 190
    +
  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
181 179
179181
181179
179181
360
179181
181 179
179 181
181 179
10
729
4 66 524893 64 265182 729
7118872 6 238769870729
195761897322 74 127513729
81187717165 197 791580729
3738394041 42 4034445729
4719482096 21 50201227729
245220623260 54 275123729
23629583025 31 5520956729
342433563-11 239 326133729
729729729 729 729729 729729 729729

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


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