Part P1

Picture of a wheel

How to Spoke Shift 5x5 Magic Squares

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 series 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 series, i.e., ½(n2 + 1).

This site introduces a new methods used for the construction of border wheel type squares except that the initial spoke parts are added in a somewhat different manner than in the original wheel method. The method consists of forming an internal 3x3 magic square, then generating all subsequent border magic squares. as was done in the original method. The difference between this type of square and the original is that the left diagonal numbers don't have to be chosen from the consecutive group ½(n2-n+2) to ½(n2+n) but may be chosen from any other consecutive group of numbers. However, the spoke on the consisting of the central column is no longer the adjacent numbers (11,12,13,14,15). The only numbers retained are (11,13,15) with the remaining two (12,14) being replaced by another complementary pair from the list:

1 2 3 4 5 6 7 8 9 10 11 12
25 24 23 22 21 20 19 18 17 16 15 14

This produces a magic wheel square whose central column is no longer dependent on ½(n2-n+2) to ½(n2+n). In addition, the symbol Δ will be used to specify a number added to or subtracted from the constants a, b or c to generate the first internal 3x3 magic square. In this case Δ is, consequently, obtained from the first number at the left bottom cell of the 3x3 square, i.e., 10 on the left diagonal, to give 13 − 9 = 4.

In addition, both 4n + 1 and 4n + 3 squares may be filled with the entire complement set.

3x3 template
c+Δ a b+Δ
a+2Δ b c
b-Δ c+2Δ a+Δ

A 5x5 Magic Square Using the Pairs {6,7} and {12,9}

  1. To the center column of the internal 3x3 square fill numbers ½(n2-1) to ½(n2+3) in consecutive order starting at the bottom middle cell of the 3x3 internal square and proceeding to the top middle cell of the 3x3 internal square using the numbers listed in the complementary table described above, for example using n = 5. For a 5x5 square the numbers in the center column correspond to 11 → 13 → 15 starting from the 4th row (Square A1).
  2. With 9, 10 and their complements generate a 3x3 square using Δ=4, b=13 and a=15 so that the sum of each column, row and diagonal of the 3x3 square sums up to 39. (Square A2).
  3. Generate Square A2 by adding consecutive numbers to the two diagonals and the central column and row (the "spoke") and include their complements from the complement list above. Squares with n > 5 will be shown to have the numbers in the central row to the left of center increasing in value. This particular n = 5 square shows number in the center row to the left of center decreasing to the left.
  4. To begin filling up the square add up the entries on the first row and subtract from 65 (the magic sum for a 5x5 square). This affords the value 22 See Figure A3.
  5. Repeat for row 5 giving a value of 30.
  6. Then repeat for columns 1 and 5 obtaining, respectively, 27 and 25.
  7. Fill the 1st and 5th rows with the pairs/complements from the complement list corresponding to the requisite pair (8,12) and enter into Square A4.
  8. Fill the 1st and 5th columns with the pairs/complements from the complement list corresponding to the requisite pair (2,1) and also enter into Square A4.
  9. Figure A shows the connectivity between numbers in the complementary table where the red and orange bars are the "spoke" numbers.
  10. Picture of squares
    Figure A
  11. The result of these operations is a wheel with a shifted spoke where the numbers in the diagonal of the regular wheel 12 → 13 → 14 are no longer in the central column.
  12. The square that is produced via this method (A4) is a border square, since the internal 3x3 square has a Sum of 39 while the external has a Sum of 65.
7 15 17
23 13 3
9 11 19
6 21 1622
7 15 17
22 23 13 3 2
9 11 19
10 5 2030
27 25
6 8 21 14 16
2 7 15 17 24
22 23 13 3 4
25 9 11 19 1
10 18 5 12 20
1 2 3 4 5 6 7 8 9 10 11 12
25 24 23 22 21 20 19 18 17 16 15 14

This completes Part P1 of 5x5 Magic Square Wheel Spoke Shift method. To go to Part P2 of an odd 7x7 square.
Go back to homepage.

Copyright © 2014 by Eddie N Gutierrez. E-Mail: