A NEW SEQUENCE FROM MAGIC SQUARE DIAGONALS (Part F)

Single-Cross-Over Pairs

Picture of a wheel

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 "Single-Cross-Over" on the complementary table. This site will show how to determine the different "Single-Cross-Over pairs" via use of a known 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. The left diagonal is incremented by 1 then by 2 for the last number until the diagonal is filled. The right diagonal is first incremented by 2 then by 1 until that diagonal is filled.

Determination of "Single-Cross-Over pairs"

  1. The function F(n) which is equal to ½(n2 − 2n − 5) is shown in Table Fh.
  2. The sequence produced from this equation is new and not stored in the in the oeis database.
  3. n stands for the size of the odd square and F(n) is a sequence whose values determine the number of "Single-Cross-Over pairs" with increasing n for magic square diagonals.
  4. Column 4 shows the differences between the F(n)s under the Δ column which just happens to be the Sloane sequence A016825 in the oeis database.
  5. Column 5 shows ΔΔs of the Sloane sequence A008586. The column shows a pattern of only 4s.
  6. The second method for obtaining F(n) is via a second equation involving 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 + 4x − 1.
Table Fh
x= ½(n-3)n    ½(n2 - 3n - 5) ΔΔΔ
2x2 + 4x - 1
15 5
10
27 154
14
39 294
18
411 474
22
513 694
26
615 954
30
717 1254
34
819 1594
38
921 1974
42
1023 2394
46
1125 2854
50
1227 3354
54
1329 3894
58
1431 4474
62
1533 5094

Single-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 Single-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 Single-Cross-Over pairs.

Cross-Over Pairs for the 5x5 Complementary Table

  1. The seven Cross-Over double pairs are shown in the complementary table for a 5x5 square which have only one crossover point as opposed to those where n > 5. 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 seven in the F(n) column:
{1,3}{4,2}
{2,4}{5,3}
{3,5}{6,4}
{4,6} {7,5}
{5,7} {8,6}
{6,8}{9,7}
{7,9}{10,8}

The magic squares using these diagonals have previously been reported and are shown in Example 1 and Example 2.

Single-Cross-Over Pairs for the 7x7 Complementary Table

  1. The 15 Single-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 and are shown as three separate tables.
  2. The Single-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 15 in the F(n) column:
Cross-Over-Point
{1,2,4}{3,5,6}
{2,3,5}{4,6,7}
{3,4,6}{5,7,8}
{4,5,7}  {6,8,9}
{5,6,8}  {7,9,10}
{6,7,9}{8,10,11}
{7,8,10}{9,11,12}
{8,9,11}{10,12,13}
{9,10,12}{11,13,14}
{10,11,13}{12,14,15}
{11,12,14}{13,15,16}
{12,13,15}  {14,16,17}
{13,14,16}{15,17,18}
{14,15,17}  {16,1819}
{15,16,18}{17,19,20}

Single-Cross-Over Pairs for the 9x9 Complementary Table

  1. The 29 quadruple Multi-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 and all five crossover points are shown as five separate tables.
  2. The Single-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 29 in the F(n) column:
Cross-Over-Point
{1,2,3,5}{4,6,7,8}
{2,3,4,6}{5,7,8,9}
{3,4,5,7}{6,8,9,10}
{4,5,6,8}{7,9,10,11}
{5,6,7,9}  {8,10,11,12}
{6,7,8,10}  {9,11,12,13}
{7,8,9,11}{10,12,13,14}
{8,9,10,12}{11,13,14,15}
{9,10,11,13}{12,14,15,16}
{10,11,12,14}{13,15,16,17}
{11,12,13,15}{14,16,17,18}
{12,13,14,16}{15,17,18,19}
{13,14,15,17}{16,18,19,20}
{14,15,16,18}{17,19,20,21}
{15,16,17,19}{18,20,21,22}
{16,17,18,20}{19,21,22,23}
{17,18,19,21}{20,22,23,24}
{18,19,20,22}{21,23,24,25}
{19,20,21,23}{22,24,25,26}
{20,21,22,24}{23,25,26,27}
{21,22,23,25}{24,26,27,28}
{22,23,24,26}{25,27,28,29}
{23,24,25,27}{26,28,29,30}
{24,25,26,28}{27,29,30,31}
{25,26,27,29}{28,30,31,32}
{26,27,28,30}{29,31,32,33}
{27,28,29,31}{30,32,33,34}
{28,29,30,32}  {31,33,34,35}
{29,30,31,33}  {32,34,35,36}

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