A New Procedure for Magic Squares (Part IC)

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. 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 1 where 13 = 4n + 1 by adding numbers in a consecutive manner starting at row 1 cell 7 and filling every other the cell (square 1).
  2. Upon reaching the numeral 7 perform a knight break enter 8 at row 2 cell 6 and continue filling 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 gp to the center row and fill in the row consecutively.
  5. 1
    5671 2 34
    12138 9 10 11
    192014 15 161718
    26 21 2223 2425
    332728 29 303132
    34 35 3637 3839
     
    41 42 4344 4540
    484950 51 524647
    55 56 5758 5354
    626364 65 596061
    69 70 7166 6768
    767778 72 737475
    2
    5671 2 34
    12138 9 10 11
    192014 15 161718
    26 21 2223 2425
    332728 29 303132
    34 35 3637 3839
    79 80 81828384 85 86 8788899091
    41 42 4344 4540
    484950 51 524647
    55 56 5758 5354
    626364 65 596061
    69 70 7166 6768
    767778 72 737475
  6. Fill in the bottom of the square starting at 91 and continuing to 130.
  7. 3
    5671 2 34
    12138 9 10 11
    192014 15 161718
    26 21 2223 2425
    332728 29 303132
    34 35 3637 3839
    79 80 81828384 85 86 8788899091
    93 41 9442 95439644 9745984092
    481004910150102 51103 52104469947
    107 55 10856 1095711058 1115310554106
    621146311564116 65117 591126011361
    121 69 12270 1237112466 1186711968120
    761287712978130 72125 731267412775
  8. From 130 go to row 1 cell 12 and fill in 131 in a backwards manner. 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 row using six moves, 1 down and 5 right then filling the row to the right.
  9. The next three moves involve 2 knight moves to the right and a final 3 moves total to the left.
  10. 4
    565
    5136613571341 133 2132 31314 829276
    14312137131388139 9 14010 141 11142104362
    1914820 14914144 15145 161461714718998107
    155 26 15621 1502215123 15224153251541212-107
    331622715728158 29159 3016031161321167-62
    164 34 16535 1663616737 16838169391631381-276
    79 80 81828384 85 86 878889909111050
    93 41 9442 95439644 9745984092920185
    481004910150102 51103 52104469947952153
    107 55 10856 1095711058 1115310554106108916
    621146311564116 65117 5911260113611121-16
    121 69 12270 1237112466 11867119681201258-153
    761287712978130 72125 7312674127751290-185
    110511051105 1105 11051105 110511051105 110511051105 1105559

  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
    565
    513661357134277 133 2132 31314 1105
    14312137131388201 9 14010 141 111421105
    1914820 14914144 122145 161461714718 1105
    155 26 15621 150224423 15224153251541105
    331622715728158 -33159 301603116132 1105
    164 34 16535 16636-10937 16838169391631105
    79 80 81828384 85 86 87888990911105
    93 41 9442 954328144 97459840921105
    481004910150102 204103 52104469947 1105
    107 55 10856 1095712658 11153105541061105
    621146311564116 49117 591126011361 1105
    121 69 12270 12371-2966 11867119681201105
    761287712978130 -113125 731267412775 1105
    110511051105 1105 11051105 110511051105 110511051105 1105559
  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
546 540
546 540
546540
540 546
546 540
540 546
546 540
546540
540546
540 546
540546
540546
540546
+ 5
6
2191
51366135553134817 133 2132 31314 2191
68912137131388201 9 14010 141 5511422191
191482069514684 122145 161461714718 2191
155 26 156561 150224423 152570153251542191
331622715728158 513159 3016057116132 2191
704 34 16535 16636-10937 168381695851632191
79 626 81828384 85 86 627888990912191
93 41 9442 9543281590 974598406322191
4810058910150648 204103 52104469947 2191
107 55 10856 6495712658 11153651541062191
621146311564116 49657 6051126011361 2191
121 609 66870 12371-2966 11867119681202191
761287712978130 -113125 7366674127621 2191
219121912191 2191 21912191 219121912191 219121912191 21912191

This completes this section on a new 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