A NEW SEQUENCE FROM MAGIC SQUARE DIAGONALS (Part B)

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SPOKE SHIFT MAGIC SQUARES - CALCULATION OF DIAGONAL PAIRS

A magic square is an arrangement of numbers 1,2,3,... n2 where every row, column and diagonal add up to the same magic sum S and n is also the order of the square. A magic square having all pairs of cells diametrically equidistant from the center of the square and equal to the sum of the first and last terms of the sequence n2 + 1 is also called associated or symmetric. In addition, the center of this type of square must always contain the middle number of the sequence, i.e., ½(n2 + 1).

These new squares can produce different diagonal pairs which generate many "Cross-Overs" on the complementary table. This site will show how to determine the different "Cross-Over pairs" via use of a new sequence generated from either of two different equations, where x is based on the natural numbers 1,2,3,4... or one based on the order of an odd square n = 5,7,9,11... The numbers that form the diagonals of the magic squares may begin at any position of the complementary tables and are imcremented by 3 until the diagonal is filled.

Determination of "Cross-Over pairs"

  1. The function F(n) which is equal to ½(n2 − 7n + 20) is shown in Table Fd.
  2. n stands for the size of the odd square and F(n) is a sequence whose values determine the number of "Cross-Over pairs" with increasing n for magic square diagonals.
  3. Column 4 shows the differences between the F(n)s under the Δ column which just happens to be the Sloane sequence A016813 in the oeis database.
  4. Column 5 shows ΔΔs of the Sloane sequence A016813. The column shows a pattern of only 4s.
  5. The second method for obtaining F(n) is via a second equationinvolving x. This requires converting n to x where x= ½(n − 3). This generates the natural numbers 1,2,3,4,...which we can plug into the equation 2x2 − x + 4.
Table Fd
x= ½(n-3)n    ½(n2 -7n + 20) ΔΔΔ
2x2 - x + 4
15 5
5
27 104
9
39 194
13
411 324
17
513 494
21
615 704
25
717 954
29
819 1244
33
921 1574
37
1023 1944
41
1125 2354
45
1227 2804
49
1329 3294
53
1431 3824
57
1533 4394

Multiplex Cross-Over Pairs and the last position on a Complementary Table

In general the last multiplex (double for 2, triple for 3, quadruple for 4, etc.) on a complementary table is equal to ½(n2 − n) while its complement is ½(n2 + n + 2). The numbers from which the Cross-Over pairs are chosen start at position 1 and end at position ½(n2 − n). In addition, the numbers between ½(n2 − n) and its complement ½(n2 + n + 2) are used for setting up the central column of a magic square (i.e., those that start at ½(n2 − n + 2) and end at ½(n2 + n) ). These latter numbers are not used in the calculation of the Cross-Over pairs.

Cross-Over Pairs for the 5x5 Complementary Table

  1. The five Cross-Over double pairs are shown in the complementary table for a 5x5 square. The numbers in color are the points where the two pairs cross over.
  2. The Cross-Over pairs are used to fill up the diagonals of an nxn magic square. The last five numbers for example as in a 5x5 square {11,12,13,14,15} are not included in these pairs since they are not part of the diagonal numbers.
  3. The numbers associated with these pairs and their complements (not shown) account for the five in the F(n) column:
{1,4}{3,6}
{2,5}{4,7}
{3,6}{5,8}
{4,7}  {6,9}
{5,8}  {7,10}

Cross-Over Pairs for the 7x7 Complementary Table

  1. The ten Cross-Over triple pairs are shown in the complementary table for a 7x7 square. The numbers in color are the points where the two pairs cross over.
  2. The Cross-Over pairs are used to fill up the diagonals of an nxn magic square. The last seven numbers for example as in a 7x7 square {22,23,24,25,26,27,28} are not included in these pairs since they are not part of the diagonal numbers.
  3. The numbers associated with these pairs and their complements (not shown) account for the ten in the F(n) column:
{1,4,7}{6,9,12}
{2,5,8}{7,10,13}
{3,6,9}{8,11,14}
{4,7,10}  {9,12,15}
{5,8,11}  {10,13,16}
{6,9,12}{11,14,17}
{7,10,13}{12,15,18}
{8,11,14}{13,16,19}
{9,12,15}{14,17,20}
{10,13,16}{15,18,21}

Cross-Over Pairs for the 9x9 Complementary Table

  1. The 19 quadruple Cross-Over pairs are shown in the complementary table for a 9x9 square. The numbers in color are the points where the two pairs cross over.
  2. The Cross-Over pairs are used to fill up the diagonals of an nxn magic square. The last seven numbers for example as in a 9x9 square {37,38,39,40,41,42,43,44,45} are not included in these pairs since they are not part of the diagonal numbers.
  3. The numbers associated with these pairs and their complements (not shown) account for the 19 in the F(n) column:
{1,4,7,10}{9,12,15,18}
{2,5,8,11}{10,13,16,19}
{3,6,9,12}{11,14,17,20}
{4,7,10,13}{12,15,18,21}
{5,8,11,14}  {13,16,19,22}
{6,9,12,15}  {14,17,20,23}
{7,10,13,16}{15,18,21,24}
{8,11,14,17}{16,19,22,25}
{9,12,15,18}{17,20,23,26}
{10,13,16,19}{18,21,24,27}
{11,14,17,20}{19,22,25,28}
{12,15,18,21}{20,23,26,29}
{13,16,19,22}{21,24,27,30}
{14,17,20,23}{22,25,28,31}
{15,18,21,24}{23,26,29,32}
{16,19,22,25}{24,27,30,33}
{17,20,23,26}{25,28,31,34}
{18,21,24,27}{26,29,32,35}
{19,22,25,28}{27,30,33,36}

This completes the Part D of Spoke Shift functions.
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Copyright © 2014 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com