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A pantriagonal diagonal magic cube of **even** order can exist only if the order is higher than 7 and divisible by 4. If the order is divisible by 8, there exists a associated pantriagonal diagonal magic cube. See also algorithms to make Nasik magic cubes.

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} m^{2} + 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.

For m = 12, b

0 | 0 | 11 | 0 | 11 | 11 | 0 | 0 | 11 | 0 | 11 | 11 |

11 | 0 | 0 | 11 | 11 | 0 | 11 | 0 | 0 | 11 | 11 | 0 |

0 | 11 | 0 | 11 | 0 | 11 | 0 | 11 | 0 | 11 | 0 | 11 |

11 | 11 | 0 | 11 | 0 | 0 | 11 | 11 | 0 | 11 | 0 | 0 |

11 | 0 | 11 | 0 | 11 | 0 | 11 | 0 | 11 | 0 | 11 | 0 |

0 | 11 | 11 | 0 | 0 | 11 | 0 | 11 | 11 | 0 | 0 | 11 |

0 | 0 | 11 | 0 | 11 | 11 | 0 | 0 | 11 | 0 | 11 | 11 |

11 | 0 | 0 | 11 | 11 | 0 | 11 | 0 | 0 | 11 | 11 | 0 |

0 | 11 | 0 | 11 | 0 | 11 | 0 | 11 | 0 | 11 | 0 | 11 |

11 | 11 | 0 | 11 | 0 | 0 | 11 | 11 | 0 | 11 | 0 | 0 |

11 | 0 | 11 | 0 | 11 | 0 | 11 | 0 | 11 | 0 | 11 | 0 |

0 | 11 | 11 | 0 | 0 | 11 | 0 | 11 | 11 | 0 | 0 | 11 |

An associated 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 3D-compact):

a_{ijk} = 8^{3}(m/8)^{2} b_{ijk} + 8^{3}(m/8) c_{ijk} + 8^{3} d_{ijk} + e_{ijk} + 1.

where

b_{ijk} =
[i/8]
(if {(i + 4j + 2k + 1) mod 8} < 4 ),

(m/8) - 1 - [i/8]
(if {(i + 4j + 2k + 1) mod 8} >= 4 ),

c_{ijk} =
[j/8]
(if {(2i + j + 4k + 1) mod 8} < 4 ),

(m/8) - 1 - [j/8]
(if {(2i + j + 4k + 1) mod 8} >= 4 ),

d_{ijk} =
[k/8]
(if {(4i + 2j + k + 1) mod 8} < 4 ),

(m/8) - 1 - [k/8]
(if {(4i + 2j + k + 1) mod 8} >= 4 ),

e_{ijk} = 2^{8} B[8]_{ijk} + 2^{7} B[7]_{ijk} + 2^{6} B[6]_{ijk} + 2^{5} B[5]_{ijk} + 2^{4} B[4]_{ijk} + 2^{3} B[3]_{ijk} + 2^{2} B[2]_{ijk} + 2 B[1]_{ijk} + B[0]_{ijk}.

The binary magic cubes B[0]_{ijk} to B[8]_{ijk} is defined as follows:

B[8]_{ijk} = (i_{2} + j_{1} + k_{0}) mod 2,

B[7]_{ijk} = (i_{2} + i_{1} + j_{1} + j_{0} + k_{0}) mod 2,

B[6]_{ijk} = (i_{2} + i_{0} + j_{1} + j_{0} + k_{0}) mod 2,

B[5]_{ijk} = (i_{0} + j_{2} + k_{1}) mod 2,

B[4]_{ijk} = (i_{0} + j_{2} + j_{1} + k_{1} + k_{0}) mod 2,

B[3]_{ijk} = (i_{0} + j_{2} + j_{0} + k_{1} + k_{0}) mod 2,

B[2]_{ijk} = (i_{1} + j_{0} + k_{2}) mod 2,

B[1]_{ijk} = (i_{1} + i_{0} + j_{0} + k_{2} + k_{1}) mod 2,

B[0]_{ijk} = {i_{0} + j_{2} + j_{1} + k_{2} + k_{1} + (j_{1}+j_{0})(k_{1}+k_{0})} mod 2.

where

i_{2} = [i/4] mod 2, i_{1} = [i/2] mod 2, i_{0} = i mod 2,

j_{2} = [j/4] mod 2, j_{1} = [j/2] mod 2, j_{0} = j mod 2,

k_{2} = [k/4] mod 2, k_{1} = [k/2] mod 2, k_{0} = k mod 2.

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This page was last updated on August 8, 2016.

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