Algorithms to make pantriagonal magic tesseracts

1. Algorithms for orders divisible by 4

   A pantriagonal magic tesseract exists for any order divisible by 4. In this case, a pan-3,4-agonal magic tesseract exists, which satisfies a stronger condition than pantriagonal. See algorithms to make pan-3,4-agonal magic tesseracts.

Note
   An order-4 associated pantriagonal magic tesseract exists though an order-4 associated pan-3,4-agonal magic tesseract cannot exist. Here is an example of such a pantriagonal magic tesseract.

2. Algorithms for odd orders

   A pantriagonal magic tesseract of odd order m can exist only if the order is higher than 4, and an associated pantriagonal magic tesseract of odd order can exist under the same condition.

2.1 Case m = 2x+1, m >= 5, and m is NOT divisible by 3 (associated)

A pantriagonal magic tesseract a_ijkh of order m, where i,j,k,h = 0,...,m-1, is given by the following equation:
a_ijkh = b_ijkh m³ + c_ijkh m² + d_ijkh m + e_ijkh + 1,

where
b_ijkh = i + j + k + h + (m+3)/2   (mod. m),   0 <= b_ijkh < m,
c_ijkh = i - j + k + h + (m+1)/2   (mod. m),   0 <= c_ijkh < m,
d_ijkh = i + j - k + h + (m+1)/2   (mod. m),   0 <= d_ijkh < m,
e_ijkh = i + j + k - h + (m+1)/2   (mod. m),   0 <= e_ijkh < m.

This tesseract is expressed as follows by the XmlHypercube format by Aale de Winkel:
   LP({1,1,1,1}+(m+3)/2,{1,-1,1,1}+(m+1)/2,{1,1,-1,1}+(m+1)/2,{1,1,1,-1}+(m+1)/2)

2.2 Case m = 2x+1, m >= 5, and m is divisible by 3 (associated)

A pantriagonal magic tesseract a_ijkh of order m, where i,j,k,h = 0,...,m-1, is given by the following equation:
a_ijkh = S_m,3(b_ijkh) m³ + S_m,3(c_ijkh) m² + S_m,3(d_ijkh) m + S_m,3(e_ijkh) + 1,

where
b_ijkh = i + j + k + h + (m+3)/2   (mod. m),   0 <= b_ijkh < m,
c_ijkh = i - j + k + h + (m+1)/2   (mod. m),   0 <= c_ijkh < m,
d_ijkh = i + j - k + h + (m+1)/2   (mod. m),   0 <= d_ijkh < m,
e_ijkh = i + j + k - h + (m+1)/2   (mod. m),   0 <= e_ijkh < m,
S_m,3(x) = Q_m/3,3([x/3], x mod 3),

where [x] means the integer part of x.

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 tesseract of singly-even order can exist if the order is higher than 9. An associated magic tesseract cannot exist.

Note It is unknown wheter an order-6 pantriagonal magic tesseract can exist or not.

3.1 Case m = 4x+2 and m >= 10 (non-associated)

A pantriagonal magic tesseract a_ijkh of order m, where i,j,k,h = 0,...,m-1, is given by the following equation:
When m is not divisible by 3:
a_ijkh = b_ijkh (m/2)⁴ + c_ijkh (m/2)³ + d_ijkh (m/2)² + e_ijkh (m/2) + f_ijkh + 1,

When m is divisible by 3:
a_ijkh = b_ijkh (m/2)⁴ + S_m/2,3(c_ijkh) (m/2)³ + S_m/2,3(d_ijkh) (m/2)² + S_m/2,3(e_ijkh) (m/2) + S_m/2,3(f_ijkh) + 1,

where
S_m/2,3(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 tesseracts b_ijkh, c_ijkh, d_ijkh, e_ijkh, and f_ijkh are defined as follows:
b_ijkh = G₁₆(v, z),
c_ijkh = J(i) + J(j) + J(k) + J(h)   (mod. m/2),   0 <= c_ijk < m/2,
d_ijkh = J(i) - J(j) + J(k) + J(h)   (mod. m/2),   0 <= d_ijk < m/2,
e_ijkh = J(i) + J(j) - J(k) + J(h)   (mod. m/2),   0 <= e_ijk < m/2,
f_ijkh = J(i) + J(j) + J(k) - J(h)   (mod. m/2),   0 <= f_ijk < m/2,

where
J(x) = x/2   (if x is even),    (m-1-x)/2   (if x is odd),
v = 8(i mod 2) + 4(j mod 2) + 2(k mod 2) + (h mod 2),   (0 <= v < 16),
z = J(i) + J(j) - J(k) - J(h)   (mod. m/2),   0 <= z < m/2.

The function G_N(v, z) (where N is a power of 2, 0 <= v < N, and z >= 0) is defined as follows (this definition is identical to that for diagonal magic cubes):
when v is even:
G_N(v, 0) = v,
G_N(v, 1) = v + 1,
G_N(v, 2) = N/2 - 1 - v/2,
G_N(v, 3) = N - 1 - v/2,
G_N(v, 2x+4) = N - 1 - v   for x >= 0,
G_N(v, 2x+5) = v   for x >= 0,

when v is odd:
G_N(v, 0) = v,
G_N(v, 1) = v - 1,
G_N(v, 2) = N/2 - 1 - (v-1)/2,
G_N(v, 3) = N - 1 - (v-1)/2,
G_N(v, 2x+4) = N - 1 - v   for x >= 0,
G_N(v, 2x+5) = v   for x >= 0.

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