A New Procedure for Magic Squares (Part IC)

Centered Sequential 13x13 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 until every other cell is filled. As was shown for 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 13x13 Magic Square

Method: Sequential Readout - use of a mask
  1. Construct the 13x13 Square in a consecutive zigzag manner starting at row 1 cell 7 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 12 a 2 down cell break is performed to numeral 13 and addition to the cells in a normal fashion.
  3. From 39 go down two cells skipping over the center row and add 40 then continuing adding numerals in normal fashion in the last row of the square.
  4. From 78 go to the center row and fill in the row consecutively.
  5. 1
    810121 3 57
    91113 2 4 6
    222426 15 171921
    23 25 1416 1820
    363827 29 313335
    37 39 2830 3234
     
    51 40 4244 4648
    505241 43 454749
    65 54 5658 6062
    645355 57 596163
    66 68 7072 7476
    786769 71 737577
    2
    810121 3 57
    91113 2 4 6
    222426 15 171921
    23 25 1416 1820
    363827 29 313335
    37 39 2830 3234
    79 80 81828384 85 86 8788899091
    51 40 4244 4648
    505241 43 454749
    65 54 5658 6062
    645355 57 596163
    66 68 7072 7476
    786769 71 737577
  6. Fill in the bottom of the square starting at 91 and continuing to 130.
  7. 3
    810121 3 57
    91113 2 4 6
    222426 15 171921
    23 25 1416 1820
    363827 29 313335
    37 39 2830 3234
    79 80 81828384 85 86 8788899091
    93 51 9540 97429944 101461034892
    509452964198 43100 451024710449
    107 65 10954 1115611358 1156011762106
    641085311055112 57114 591166110563
    121 66 12368 1257012772 1297411876120
    781226712469126 71128 731307511977
  8. From 130 go to row 1 cell 10 and fill in 131.
  9. Continue filling in the square in a zigzag manner.
  10. 4
    530
    813610138121401142 3 131 51337866239
    135913711139131412 143 4 132 6134100699
    22150241522615415156 171451914721104857
    149 23 15125 1531415516 14418146201481162-57
    361643816627168 29157 3115933161351204-99
    163 37 16539 1672816930 15832160341621344-239
    79 80 81828384 85 86 8788899091 11050
    93 51 9540 97429944 101461034892951154
    509452964198 43100 451024710449921184
    107 65 10954 1115611358 11560117621061133-28
    64108531105511257114 591166110563107728
    121 66 12368 1257012772 12974118761201289-184
    78122671246912671128 7313075119771259-154
    110511051105 1105 11051105 110511051105 110511051105 1105569
  11. 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 of Square 4 to generate Square 5.
  12. At this point four duplicates have been generated in pink.
  13. 5
    530
    81361013812140240142 3 131 513371105
    135913711139132402 143 4 132 61341105
    22150241522615472156 1714519147211105
    149 23 15125 153149816 14418146201481105
    361643816627168 -70157 3115933161351105
    163 37 16539 16728-7030 15832160341621105
    79 80 81828384 85 86 8788899091 1105
    93 51 9540 974225344 1014610348921105
    509452964198227 100 4510247104491105
    107 65 10954 111568558 11560117621061105
    64108531105511285114 5911661105631105
    121 66 12368 12570-5772 12974118761201105
    781226712469126-83128 7313075119771105
    110511051105 1105 11051105 110511051105 110511051105 1105569
  14. Generate a mask below 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 5 (as in the de la Hire method) that all sums will equal.
Mask A
575 575536 536
575575 536536
536536 575575
536575536 575
536536 575575
575 536 575 536
575575 536536
536 536 575 575
536575 536575
575536 575 536
536575 536575
575536536 575
575 575536 536
+ 5
6
3327
813658513812140 815142 539 131 54113373327
135584137586139 13240538 143 4 132 5421343327
2215024152562 69072156 17145574147576 3327
685 598 151 561 153149816 71918146201483327
57216457416627168 -70157 60673433161353327
163 37 165614 16728 466605 15832160346983327
79 80 8182658659 85 622 8762489 9091 3327
93 51 9540 6334278944 10146103623667 3327
50945889641673227 100 58110247679 493327
682 65 10954 111592 66058 11560653621063327
646445311063011285114 5965263610563 3327
696 602 123604 12570-57647 12974118761203327
7812264212469126-83128 7370575655 6133327
332733273327 3327 33273327 332733273327 332733273327 33273327

This completes this section on a new Centered Sequential 13x13 Zig Zag Mask-Generated Squares (Part IC). The next section deals with Centered 7x7 Sequential Mask-Generated Zig Zag Squares (Part ID). To return to homepage.


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