Octagon Algorithm for Border Wheel Squares (Part IX)

The Eight Node Way - (1,2,5) Squares

The previous sections Part VI and Part VII introduced the Octagon G and H algorithm for the construction of 5×5 and 7×7 Wheel Border Squares (WBS). The type of Wheel Border squares in those sections were of the type (a,b,c) where the a, b and c were consecutive numbers. This section will look at WBS are of the type (a,b,e) where a and b are 1 and 2 and e is 5, i.e., the start of each wheel spoke is (1,2,5) and the difference (Δ) between the terms in two of the spokes is 5, while a third spoke will be modified where one of the Δ terms is equal to 7. This is to ensure that the internal 3×3 square is magic, otherwise without this modification these (a,b,e) squares, unlike the (a,b,c), will be non magic.

To start, the wheel part of the square is first constructed using the (Δ) value of 5 followed by the non wheel portion (i.e, non spokes) which uses either the Octagon I (only the 5th order) and K algorithm to fill in the empty cells. The method can be summarized as follows where regular means with a degree of separation (Δ) between numbers:

By first using the Wheel algorithm, square 5(05)j is filled using Δ=5 for the first central and left diagonal spokes, while the central row is filled with one Δ= 3. The 7(05)ij and the 9(05)ij are both filled as usual except that the center row is first filled using a Δ=5; followed by a Δ=3;. The latter Δ=3 is always added last to ensure that the inner 3×3 square is magic. Once the wheel is complete the Octagon J algorithm is only used to complete the 5th order square. To complete the larger squares, Octagon I is used with the number 3 and Octagon J with numbers 14 and 18 for the 7th order square while Octagon I is used with the numbers 3, 13 and 25 and Octagon J with the numbers 21, 29 and 33 for the 9th order square:

The caption in the tables is used for numbering the squares. For example, 5(05)j corresponds to n=5, 0 is the wheel type, 5 is the Δ and j is the algorithm used, in this case Octagon J; while the caption for 7(05)ij corresponds to n=7, 0 is the wheel type, 5 is the Δ and ij means that both algorithms, Octagon I and J, are used to develop the squares.

By first using the Wheel algorithm, square 5(07)j is filled using Δ=7 for the first central and left diagonal spokes, while the central row is filled with one Δ= 5. The 7(07)ij and the 9(07)ij are both filled as usual while the center row is first filled using a Δ=7; followed by a Δ=5;. The latter Δ=5 is always added last to ensure that the inner 3×3 square is magic. Once the wheel is complete the Octagon J algorithm is only used to complete the 5th order square. To complete the larger squares, Octagon I is used with the number 3 and Octagon J with the numbers 10 and 18 for the 7th order square while Octagon I is used with the numbers 3, 10 and 25 and Octagon J with the numbers 18, 29 and 33 for the 9th order square:

By first using the Wheel algorithm, square 11(05)ij is filled using Δ=5 for the first central and left diagonal spokes, while the central row is filled with one Δ= 3. Square 11(07)ij is filled using Δ=7 for the first central and left diagonal spokes, while the central row is filled with one Δ= 5. The squares are filled as usual with the latter Δs always added last to ensure that the inner 3×3 square is magic. Once the wheels are complete the Octagon I and J algorithms are used to complete the 11th order squares:

It's interesting that the squares use the Octagon J algorithm by employing the set of triangular numbers (0,1,3,6,10,15...) with n(n+1)/2, while the Octagon I algorithm employs the set of natural numbers (1,2,3,4,...) in producing the squares.

This completes the Octagon I and J method. Go to Part X for (a,d,e) squares. Go back to homepage.