Algorithms to make pantriagonal magic cubes

1. Algorithms for orders divisible by 4

   A pantriagonal magic cube can exist for any order divisible by 4. An associated pantriagonal magic cube can exist under the same condition. If the order is higher than 4, there exists a pantriagonal diagonal magic cube, which satisfies a stronger condition than pantriagonal. Furthermore, if the order is divisible by 8, there exists a Nasik magic cube, which satisfies a stronger condition than pantriagonal diagonal.

2.1 Non-associated pantriagonal magic cubes (m = 4x) (also complete and 2-compact)

2.2 Associated pantriagonal magic cubes (m = 4x)

2. Algorithms for odd orders

   A pantriagonal magic cube of odd order can exist only if the order is higher than 4. An associated pantriagonal magic cube of odd order can exist under the same condition. If the order is higher than 8, there exists a Nasik magic cube, which satisfies a stronger condition than pantriagonal.

2.1 When the order m is NOT divisible by 3 (m = 2x+1, m >= 5) (associated)

2.2 When the order m is divisible by 3 (m = 2x+1, m >= 5) (associated)

An associated pantriagonal magic cube a_ijk of order m, where i,j,k = 0,...,m-1, is given by the following equation:
a_ijk = S_m,3(b_ijk) m² + S_m,3(c_ijk) m + S_m,3(d_ijk) + 1,

where
b_ijk = 2i + j + k + (m+3)/2   (mod. m),   0 <= b_ijk < m,
c_ijk = 2i - j + k + (m+1)/2   (mod. m),   0 <= c_ijk < m,
d_ijk = 2i + j - k + (m+1)/2   (mod. m),   0 <= d_ijk < m.
S_m,3(x) = Q_m/3,3([x/3], x mod 3).

Q_p,q(x, y), where 0 <= x < p and 0 <= y < q, is given by the following equations (identical to Q_p,q(x, y) for Nasik magic cubes):
Q_p,q(x, y) = qx + y (if 0 < x < p-1 and x is even),
qx + (q-1-y) (if 0 < x < p-1 and x is odd),
y/2 + (q-1)/2 (if x = 0 and y is even),
(y-1)/2 (if x = 0 and y is odd),
y/2 + (p-1)q (if x = p-1 and y is even),
(y-1)/2 + pq - (q-1)/2 (if x = p-1 and y is odd).

See the page for Nasik magic cubes to see examples of Q_p,q(x, y) and S_m,q(x).

3. Algorithms for singly-even orders

   A pantriagonal magic cube exists for any singly-even (even but not divisible by 4) order higher than 4. An associated pantriagonal magic cube of singly-even order also exists under the same condition.

3.1 Non-associated pantriagonal magic cubes (m = 4x+2, m >= 6) (also complete)

3.2 Associated pantriagonal magic cubes (m = 4x+2, m >= 6)

An associated pantriagonal magic cube a_ijk of order m, where i,j,k = 0,...,m-1, is given by the following equations:
a_ijk = b_ijk (m/2)³ + c_ijk + 1,

where
b_ijk = v₁ (if t = (m-2)/4),
7-v₀ (if t < (m-2)/4 and t + (m-2)/4 is even),
v₀ (if t < (m-2)/4 and t + (m-2)/4 is odd),
c_ijk = k^* (m/2)² + i^* (m/2) + j^* (if i+j+k is even),
(m/2)³ - 1 - {k^* (m/2)² + i^* (m/2) + j^*} (if i+j+k is odd),
where
i^* = i/2 (if i is even),
(m-1-i)/2 (if i is odd),
j^* = j/2 (if j is even),
(m-1-j)/2 (if j is odd),
k^* = k/2 (if k is even),
(m-1-k)/2 (if k is odd),
t = min[{(i^* + j^* - 2k^*) mod (m/2)}, {(-i^* - j^* + 2k^*) mod (m/2)}],
v₀ = 4(k mod 2) + 2(j mod 2) + {(i+j+k) mod 2},
v₁ = 7 - v₀/2 (if v₀ is even),
3 - (v₀-1)/2 (if v₀ is odd).

By generalizing this method, we can construct an associated panmagic hypercube of any singly-even order and any odd dimension.

If m is 10 or higher singly-even integer, we can construct another associated pantriagonal magic cube of order m. Such a magic cube a_ijk, where i,j,k = 0,...,m-1, is given by the following equations:
when m is not divisible by 3:
a_ijk = b_ijk (m/2)³ + c_ijk (m/2)² + d_ijk (m/2) + e_ijk + 1,

when m is divisible by 3:
a_ijk = b_ijk (m/2)³ + S_m/2(c_ijk) (m/2)² + S_m/2(d_ijk) (m/2) + S_m/2(e_ijk) + 1,

where
S_m/3(x) = Q_m/6,3([x/3], x mod 3).

Q_p,q(x, y), where 0 <= x < p and 0 <= y < q, is given by the following equations (identical to Q_p,q(x, y) for Nasik magic cubes):
Q_p,q(x, y) = qx + y (if 0 < x < p-1 and x is even),
qx + (q-1-y) (if 0 < x < p-1 and x is odd),
y/2 + (q-1)/2 (if x = 0 and y is even),
(y-1)/2 (if x = 0 and y is odd),
y/2 + (p-1)q (if x = p-1 and y is even),
(y-1)/2 + pq - (q-1)/2 (if x = p-1 and y is odd).

See the page for Nasik magic cubes to see examples of Q_p,q(x, y) and S_m,q(x).

The cubes b_ijk, c_ijk, d_ijk, and e_ijk are defined as follows:
b_ijk = G^*_8,m/2(v, z),
c_ijk = i + j + k + 1   (mod. m/2),   0 <= c_ijk < m/2,
d_ijk = i - j + k   (mod. m/2),   0 <= d_ijk < m/2,
e_ijk = i + j - k   (mod. m/2),   0 <= e_ijk < m/2,

where
v = 4[2i/m] + 2[2j/m] + {([2i/m] + [2j/m] +[2k/m]) mod 2},   (0 <= v <= 7),
z = i + j + k - 3(m-2)/4   (mod. m/2),   -(m-2)/4 <= z <= (m-2)/4.

The function G^*_N,m/2(v, z) (where N is a power of 2, m/2 is odd, and 0 <= v < N) is defined as follows:
when m/2 = 3 (mod. 4):
G^*_N,m/2(v, z) = v (if z = 0),
v/2 (if |z| > 0, z is even, and v is even),
N/2 + (v-1)/2 (if |z| > 0, z is even, and v is odd),
N - 1 - v/2 (if |z| > 0, z is odd, and v is even),
N/2 - 1 - (v-1)/2 (if |z| > 0, z is odd, and v is odd).

when m/2 = 1 (mod. 4):
G^*_N,m/2(v, z) = v (if z = 0),
N - 1 - v (if |z| = 1),
v/2 (if |z| > 1, z is even, and v is even),
N/2 + (v-1)/2 (if |z| > 1, z is even, and v is odd),
N - 1 - v/2 (if |z| > 1, z is odd, and v is even),
N/2 - 1 - (v-1)/2 (if |z| > 1, z is odd, and v is odd).

The function G^*_N,m/2(v, z) satifies the following properties (only these properties are needed for G^*_N,m/2(v, z)).

(1) { G^*_N,m/2(v, z) | 0 <= v < N } = { 0, 1, ..., N-1 } for every integer z,
(2) The sum Sum_z=0,m/2-1{G^*_N,m/2(2v, z)} is independent of v,
(3) Sum_z=0,m/2-1{G^*_N,m/2(2v, z)} + Sum_z=0,m/2-1{G^*_N,m/2(2v+1, z)} = (N-1)m/2,
(4) G^*_N,m/2(v, z) = G^*_N,m/2(v, -z),
(5) G^*_N,m/2(v, z) + G^*_N,m/2(N-1-v, z) = N-1.

Here is an example for m = 10.

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