Algorithm to make pantriagonal diagonal magic cubes

1. Algorithms for orders divisible by 4

   A pantriagonal diagonal magic cube of even order can exist only if the order is higher than 7 and divisible by 4. This algorithm works for orders higher than 11. If the order is divisible by 8, there exists a Nasik magic cube, which satisfies a stronger condition than pantriagonal diagonal.

1.1 Non-associated pantriagonal diagonal magic cubes (m = 4x, m >= 12) (also complete)

A pantriagonal diagonal magic cube a_ijk of order m, where i,j,k = 0,...,m-1, is given by the following equation (a_ijk is also complete):
a_ijk = b_ijk m² + b_jki m + b_kij + 1,

where
b_ijk = T_m(k) or m - 1 - T_m(k),
T_m(x) = x (where x < m/2), 3m/2-1-x (otherwise)    (identical to the definition of T_m(x) for pantriagonal magic cubes).

b_ijk = T_m(k) if i and j satisfy one of the following conditions:

• i mod (m/2) < m/4,   j mod (m/2) < m/4,   (i-j) mod (m/4) is not 1.
• i mod (m/2) < m/4,   j mod (m/2) >= m/4,   (i+j+1) mod (m/4) = 1.
• i mod (m/2) >= m/4,   j mod (m/2) < m/4,   (i+j+1) mod (m/4) = 0.
• i mod (m/2) >= m/4,   j mod (m/2) >= m/4,   (i-j) mod (m/4) is not 0.

• i mod (m/2) < m/4,   j mod (m/2) < m/4,   (i-j) mod (m/4) = 1.
• i mod (m/2) < m/4,   j mod (m/2) >= m/4,   (i+j+1) mod (m/4) is not 1.
• i mod (m/2) >= m/4,   j mod (m/2) < m/4,   (i+j+1) mod (m/4) is not 0.
• i mod (m/2) >= m/4,   j mod (m/2) >= m/4,   (i-j) mod (m/4) = 0.

	blog SEO tool