A KNOWN SEQUENCE FROM MAGIC SQUARE DIAGONALS (Part D)

Multi-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 "Multi-Cross-Overs" on the complementary table. This site will show how to determine the different "Multi-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 and are imcremented by 3 until the diagonal is filled.

Determination of "Multi-Cross-Over pairs"

  1. The function F(n) which is equal to ½(n2 − 4n + 7) is shown in Table Ff.
  2. The sequence produced from this equation is stored as the Sloane sequence A051890 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 "Multi-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 A008586 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 + 2x + 2.
Table Ff
x= ½(n-3)n    ½(n2 -4n + 7) ΔΔΔ
2x2 + 2x + 2
15 6
8
27 144
12
39 264
16
411 424
20
513 624
24
615 864
28
717 1144
32
819 1464
36
921 1824
40
1023 2224
44
1125 2664
48
1227 3144
52
1329 3664
56
1431 4224
60
1533 4824

Multiplex Multi-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 Multi-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 Multi-Cross-Over pairs.

Cross-Over Pairs for the 5x5 Complementary Table

  1. The six Cross-Over double pairs are shown in the complementary table for a 5x5 squarewhich 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 six in the F(n) column:
{1,4}{5,2}
{2,5}{6,3}
{3,6}{7,4}
{4,7} {8,5}
{5,8} {9,6}
{6,9}{10,7}

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

  1. The 14 Multi-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 Multi-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 14 in the F(n) column:
Cross-Over-Point 1
{1,4,7}{8,5,2}
{2,5,8}{9,6,3}
{3,6,9}{10,7,4}
{4,7,10}  {11,8,5}
{5,8,11}  {12,9,6}
{6,9,12}{13,10,7}
{7,10,13}{14,11,8}
{8,11,14}{15,12,9}
{9,12,15}{16,13,10}
{10,13,16}{17,14,11}
{11,14,17}{18,15,12}
{12,15,18}  {19,16,13}
{13,16,19}{20,17,14}
{14,17,20}  {21,18,15}
  
Cross-Over-Point 2
{1,4,7}{8,5,2}
{2,5,8}{9,6,3}
{3,6,9}{10,7,4}
{4,7,10}  {11,8,5}
{5,8,11}  {12,9,6}
{6,9,12}{13,10,7}
{7,10,13}{14,11,8}
{8,11,14}{15,12,9}
{9,12,15}{16,13,10}
{10,13,16}{17,14,11}
{11,14,17}{18,15,12}
{12,15,18}  {19,16,13}
{13,16,19}{20,17,14}
{14,17,20}  {21,18,15}
  
Cross-Over-Point 3
{1,4,7}{8,5,2}
{2,5,8}{9,6,3}
{3,6,9}{10,7,4}
{4,7,10}  {11,8,5}
{5,8,11}  {12,9,6}
{6,9,12}{13,10,7}
{7,10,13}{14,11,8}
{8,11,14}{15,12,9}
{9,12,15}{16,13,10}
{10,13,16}{17,14,11}
{11,14,17}{18,15,12}
{12,15,18}  {19,16,13}
{13,16,19}{20,17,14}
{14,17,20}  {21,18,15}

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

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

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