Algorithm to make strictly magic tesseracts

1. Algorithm for orders divisible by 8

   A strictly magic tesseract (namely, a 2,3,4-agonal magic tesseracts) can exist for orders divisible by 8. In this case, an associated strictly magic tesseract can exist. If the order is divisible by 16, there exists a Nasik magic tesseract, which satisfies a stronger condition than strictly magic. It is unknown what condition is required and sufficient for the existence of strictly magic tesseracts.

1.1 Associated strictly magic tesseracts (m = 8x)

An associated strictly magic tesseract a_ijkh of order m, where i,j,k,h = 0,...,m-1, is given by the following equation:
a_ijk = T_m(b_ijkh) m³ + T_m(c_ijkh) m² + T_m(d_ijkh) m + T_m(e_ijkh) + 1,

where
b_ijkh = T_m(i) + (m/4)T_m(j) + (m/4)T_m(k) + (m/4)T_m(h)   (mod. m),   0 <= b_ijkh < m,
c_ijkh = (m/4)T_m(i) + T_m(j) + (m/4)T_m(k) + (m/4)T_m(h)   (mod. m),   0 <= c_ijkh < m,
d_ijkh = (m/4)T_m(i) + (m/4)T_m(j) + T_m(k) + (m/4)T_m(h)   (mod. m),   0 <= d_ijkh < m,
e_ijkh = (m/4)T_m(i) + (m/4)T_m(j) + (m/4)T_m(k) + T_m(h)   (mod. m),   0 <= e_ijkh < m,
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).

2. Algorithms for odd orders

   A strictly magic tesseract of odd order can exist if the order is higher than 14, and an associated strictly magic tesseract of odd order can exist under the same condition.
   Let L₁₃ be lcm{y| y is odd and 3 <= y <= 13} = 3²x5x7x11x13, where lcm means the least common multiple, and gcd means the greatest common divisor.

2.1 Case m = 2x+1, m >= 15, and gcd(m, L₁₃) = 1 (associated, pan-2,3-agonal)

In this case, m is prime to 3, 5, 7, 11, and 13.
A strictly magic tesseract a_ijkh of order m, where i,j,k,h = 0,...,m-1, is given by the following equation (a_ijkh is also pan-2,3-agonal):
a_ijkh = b_ijkh m³ + c_ijkh m² + d_ijkh m + e_ijkh + 1,

where
b_ijkh = 2i + 4j + k + 7h + (m+13)/2   (mod. m),   0 <= b_ijkh < m,
c_ijkh = 2i - 4j + k + 7h + (m+5)/2   (mod. m),   0 <= c_ijkh < m,
d_ijkh = 2i + 4j - k + 7h + (m+11)/2   (mod. m),   0 <= d_ijkh < m,
e_ijkh = 2i + 4j + 7k - h + (m+11)/2   (mod. m),   0 <= e_ijkh < m.

This tesseract is expressed as follows by the XmlHypercube format by Aale de Winkel:
   LP({2,4,1,7}+(m+13)/2,{2,-4,1,7}+(m+5)/2,{2,4,-1,7}+(m+11)/2,{2,4,7,-1}+(m+11)/2)

The value of the determinant
| 2 4 1 7|
| 2 -4 1 7|
| 2 4 -1 7|
| 2 4 7 -1|
equals to -256 = -2⁸, which is prime to any odd integer, so this set of the formulae always generates a normal magic tesseract.

2.2 Case m = 2x+1, m >= 15, and 1 < gcd(m, L₁₃) < m (associated, pan-2,3-agonal)

Let q be gcd(m, L₁₃) and p be m/q. In this case, p > 1 and q > 1.
A strictly magic tesseract a_ijkh of order m, where i,j,k,h = 0,...,m-1, is given by the following equation (a_ijkh is also pan-2,3-agonal):
a_ijkh = S_m,q(b_ijkh) m³ + S_m,q(c_ijkh) m² + S_m,q(d_ijkh) m + S_m,q(e_ijkh) + 1,

where
b_ijkh = 2i + 4j + k + 7h + (m+13)/2   (mod. m),   0 <= b_ijkh < m,
c_ijkh = 2i - 4j + k + 7h + (m+5)/2   (mod. m),   0 <= c_ijkh < m,
d_ijkh = 2i + 4j - k + 7h + (m+11)/2   (mod. m),   0 <= d_ijkh < m,
e_ijkh = 2i + 4j + 7k - h + (m+11)/2   (mod. m),   0 <= e_ijkh < m.
S_m(x) = Q_p,q([x/q], x mod q),

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 cubes to see examples of Q_p,q(x, y) and S_m,q(x).

2.3 Case m = 2x+1, m >= 15, and gcd(m, L₁₃) = m (associated)

In this case, m is a divisor of L₁₃ = 3²x5x7x11x13.
A strictly magic tesseract a_ijkh of order m, where i,j,k,h = 0,...,m-1, is given by the following equation (a_ijkh is also pan-2,3-agonal):
a_ijkh = S^*_m(b_ijkh) m³ + S^*_m(c_ijkh) m² + S^*_m(d_ijkh) m + S^*_m(e_ijkh) + 1,

where
b_ijkh = 2i + 4j + k + 7h + (m+13)/2   (mod. m),   0 <= b_ijkh < m,
c_ijkh = 2i - 4j + k + 7h + (m+5)/2   (mod. m),   0 <= c_ijkh < m,
d_ijkh = 2i + 4j - k + 7h + (m+11)/2   (mod. m),   0 <= d_ijkh < m,
e_ijkh = 2i + 4j + 7k - h + (m+11)/2   (mod. m),   0 <= e_ijkh < m.

S^*_m(x) is defined as follows:
S^*_m(x) = R_q,m/q(x mod q, x mod (m/q)) - 1,

where q is the greatest prime factor of m (q is equal to 5, 7, 11, or 13).
R_q,m/q is an associated magic rectangle of order-(q,m/q). See the page for magic rectangles.

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