A New Procedure for Magic Squares (Part IE)

Centered Sequential Mask-Generated Squares

A mask

Skip the discussion go to examples

A Discussion of the New Method

This method differs from the Centered Sequential Mask-Generated Squares (Part IA) which employs 4n + 1 squares. The 4n + 3 squares are produced via an alternative novel route using a sequence of numbers (either positive or negative) which tells us the direction and number of steps to move during a break.

  1. We begin by placing the numeral 1 in the center cell of the top row and consecutively entering the next numeral into every other cell as was shown in the previous page. After partially filling in the first row (1 down,1 right) or the equivalent (1 right,1 down) knight move is used to get to the next row (see the examples below).
  2. Upon reaching the row adjacent to the center row, the numbers are filled in a zig zag pattern as was shown in Zig Zag Consecutive 7x7 Mask-Generated Squares leaving the center row unfilled in the first pass. After filling in the zig zag rows continue as before filling bottom of the square as was previously done employing (1,down,1,right) knight moves as breaks. From the last row go to the center row and fill in the center row, followed by the bottom rows. (Best seen in the examples). From the bottom row we jump back to the zig zag rows and fill these in in reverse order, followed by inserting the next number into the last cell in the first row.
  3. The first move is to add the numbers in reverse order until the first row is filled in. This ensures that the square is primed to use the new break sequences which may be either either positive (move in the right direction) or negative numbers (move in the left direction). In addition the size of the numeral tells how many steps to travel in that direction including the (1 down) break. The rest of the square is filled in using these numerical sequences and are filled in the direction shown in the last column of the sequence table 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 the Break Sequence Table

Table S is generated by using the following general sequence:

{ ½(n + 1), [½(n - 3), - ½(n + 1)]r }

where n is an odd number of the type 4m + 3 and r is a repeating sequence equal to ¼(n - 7).

 
nr
7
111
152
193
234
Sequence Table
Number of cells to move per breakDirection of row fillings
(4)R
(6, 4, -6)R, L, R
(8, 6, -8, 6, -8)R, L, R, L, R
(10, 8, -10, 8, -10, 8, -10)R, L, R, L, R, L, R
(12, 10, -12, 10, -12, 10, -12, 10, -12)R, L, R, L, R, L, R, L, R

Construction of a 15x15 Magic Square

Method: Sequential Readout - use of a mask
  1. Construct the 15x15 Square 1 where 15 = 4n + 3 by adding numbers in a consecutive manner starting at row 1 cell 6 and filling every other the cell (square 1).
  2. Upon reaching the numeral 7 generate a (1 down, 1 right) knight break move and continue filling cells in a normal fashion.
  3. Continue filling in rows adding numbers sequentially using the (1 down, 1 right) knight break moves until 45 is reached.
  4. Go down one cell and fill in 46 (in light tan color) and add the next sequential numbers in a zig zag pattern as was shown for the 7x7 and 11x11 square, continuing from 60 to 61 (in light green color).
  5. Partially fill all the last rows.
  6. 1
    567 1 2 34
    13141589 10 1112
    21221617 18 19 20
    29 30 23 2425262728
    37 3132 33 343536
    45 383940 41 424344
    46 485052 54 565860
     
    47 4951 53 55 5759
    62 636465 66 676861
    70 7172 73 74 7569
    78 798081 82 837677
    86 8788 89 90 8485
    94 959697 98 919293
    102 103104 105 99 100101
  7. Fill in the center the center row (square 2), followed by the last six rows in the normal fashion as above.
  8. 2
    5671 2 34
    13141589 10 1112
    21221617 18 19 20
    29 30 23 2425262728
    37 3132 33 343536
    45 383940 41 424344
    46 485052 54 565860
    106 107 108109110111112 113 114115 116117118119120
    47 4951 53 55 5759
    62 122 631236412465 125 66126 671276812161
    130 70 1317113272133 73 13474 1357512869129
    78 138 791398014081 141 82142 831367613777
    146 86 1478714888149 89 15090 1438414485145
    94 154 951559615697 157 98151 911529215393
    162 102 163103164104165 105 15899 159100160101161
  9. Go from 165 to 166 and zigzag backwards filling each cell sequentially to 180.
  10. From 180 go to cell 15 on the first row and fill in 181. Move leftwards in the first row and fill in row one. This primes the square for the new reverse sequences.
  11. Use row 3 of the sequence table above to input the rest of the numbers as follows:

    1. 8 steps (right) from 188 → 189 and moving to the right to fill in the row.
    2. 6 steps(right) 195 → 196 and moving to the left to fill in the row.
    3. 8 steps (left) 203 → 204 and moving to the right to fill in the row.
    4. 6 steps(right) 210 → 211 and moving to the left to fill in the row.
    5. 8 steps (left) 218 → 219 and moving to the right to fill in the row.
  12. At this point all columns at this point sum to 1695, while the row sums are to be adjusted (by + or - values) according to the last cell in square 3 in order to sum to 1695.
  13. 3
    2422
    1885187 61867 1851 184 2183 31824181 1504191
    1319314194151958 1899 190 10191 11192121436259
    201212002219916198 17197 18 196 19 20320202 1729-34
    29 209 30210 23204 2420525 20626207 2720828166134
    215 37 2143121332 212 33211 342183521736216 1954-259
    45 225 3821939 22040 221 41222 4222343224441886-191
    46 179 481775017552 173 54171 561695816760163560
    106 107 108109110111112 113 114115 116117118119120 16950
    180 47 1784917651174 53 17255 1705716859166 1755-60
    62 122 631236412465 125 66126 671276812161 1384311
    130 70 1317113272133 73 13474 1357512869129 1556139
    78 138 791398014081 141 82142 831367613777 160986
    146 86 1478714888149 89 15090 1438414485145 1781-86
    94 154 951559615697 157 98151 911529215393 1834-139
    162 102 163103164104165 105 15899 159100160101161 2006-311
    169516951695 1695 16951695 169516951695 169516951695 169516951695 2416
  14. Square 5 shows the result of the adjustment with the generation of 5 pink duplicates.
  15. 4
    2422
    1885187 61867 185192 184 2183 31824181 1695
    1319314194151958 4489 190 10191 11192121695
    201212002219916198 -17197 18 196 19 20320202 1695
    29 209 30210 23204 2423925 20626207 27208281695
    215 37 2143121332 212 -226211 342183521736216 1695
    45 225 3821939 22040 30 41222 4222343224441695
    46 179 481775017552 233 54171 561695816760 1695
    106 107 108109110111112 113 114115 116117118119120 1695
    180 47 1784917651174 -7 17255 1705716859166 1695
    62 122 631236412465 436 66126 671276812161 1695
    130 70 1317113272133 212 13474 1357512869129 1695
    78 138 791398014081 227 82142 831367613777 1695
    146 86 1478714888149 3 15090 1438414485145 1695
    94 154 951559615697 18 98151 911529215393 1695
    162 102 163103164104165 -206 15899 159100160101161 1695
    169516951695 1695 16951695 169516951695 169516951695 169516951695 2416
    +
  16. At 105 go a distance of six units (5 right, 1 down) as directed in the second line of the sequence table and fill in the row in the right direction.
  17. At 110 go a distance of four units (1 down, 3 right) as directed in the second line of the sequence table and fill in the row in the left direction.
  18. At 116 go a distance of 6 units(1 down, 5 left) as directed in the second line of the sequence table and fill in the row in the right direction according to the last row of the sequence table. At this point all columns at this point sum to 1695, while the row sums are to be adjusted according to the last cell in square 10 in order to sum to 1695.
  19. Square 11 shows the result of the adjustment with the generation of 4 pink duplicates.
  20. 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 A
727 721
727 721
727 721
721 727
721 727
721727
727721
721 727
721727
721 727
727 721
727721
721 727
727721
727 721
+
5
3143
1885187 61867 912192 184 2183 7241824181 3143
1392014194151958 4489190 10191 11 913123143
928212002219916198 -17197 739 196 19 20320202 3143
29 209 751210 23204 2423925 206753207 27208283143
215 37 2143121332 933 -226211 342183521736943 3143
766 225 3894639 22040 30 41222 4222343224443143
46 179 7751775089652 233 54171 5616958167603143
106 107 108109831111112 113 841115 116117118119120 3143
180 47 17877090351174 -7 17255 1705716859166 3143
62 122 631236412465 1157 66126 6712779512161 3143
130 70 13171132799133 212 13474 1357584969129 3143
78 138 791398014081 954 803142 831367613777 3143
146 807 1478714888149 3 15090 14384144812145 3143
94 154 951559615697 18 98151 9187992153814 3143
162 102 163103164104165 -206 158826 880100160101161 3143
314331433143 3143 31433143 314331433143 314331433143 314331433143 3143

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


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