Algorithms to make panmagic tesseracts

1. Algorithms for orders divisible by 4

   A panmagic tesseract (namely, a pan-4-agonal or panquadragonal magic tesseract) of even order can exist only if the order is divisible by 4. In this case, a pan-3,4-agonal magic tesseract exists, which satisfies a stronger condition than panmagic. See algorithms to make pan-3,4-agonal magic tesseracts.

Note
   You can construct a panmagic tessseract of order m = 4x which is not pan-3,4-agonal by the following XmlHypercube format by Aale de Winkel. The tesseract generated by this expression is also complete and 4-compact.

   LP({1,m/4,m/4,m/2},{m/2,1,m/4,m/4},{m/4,m/2,1,m/4},{m/4,m/4,m/2,1})= [0,..,m/2-1,m-1,...,m/2]

Note
   It can be proved that every order-4 panmagic tesseract is complete and 4-compact.

2. Algorithms for odd orders

   A panmagic tesseract of odd order m can exist only if the order is higher than 6, and an associated panmagic tesseract of odd order can exist under the same condition.
The function gcd means the greatest common divisor.

2.1 Case m = 2x+1, m >= 7, and gcd(m, 3x5) = 1 (associated)

In this case, m is prime to both 3 and 5.
A panmagic 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 = 2i + j + k + h + 2   (mod. m),   0 <= b_ijkh < m,
c_ijkh = 2i - j + k + h + 1   (mod. m),   0 <= c_ijkh < m,
d_ijkh = 2i + j - k + h + 1   (mod. m),   0 <= d_ijkh < m,
e_ijkh = 2i + j + k - h + 1   (mod. m),   0 <= e_ijkh < m.

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

2.2 Case m = 2x+1, m >= 7, and 1 < gcd(m, 3x5) < m (associated)

Let q be gcd(m, 3x5) and p be m/q. In this case, p > 1 and q > 1.
A panmagic tesseract a_ijkh of order m, where i,j,k,h = 0,...,m-1, is given by the following equation:
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 + j + k + h + 2   (mod. m),   0 <= b_ijkh < m,
c_ijkh = 2i - j + k + h + 1   (mod. m),   0 <= c_ijkh < m,
d_ijkh = 2i + j - k + h + 1   (mod. m),   0 <= d_ijkh < m,
e_ijkh = 2i + j + k - h + 1   (mod. m),   0 <= e_ijkh < m,
S_m,q(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 magic cubes to see examples of Q_p,q(x, y) and S_m,q(x).

2.3 Case m = 2x+1, m >= 7, and gcd(m, 3x5) = m (associated)

In this case, always m = 15.
A panmagic tesseract a_ijkh of order m, where i,j,k,h = 0,...,m-1, is given by the following equation:
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 + j + k + h + 2   (mod. m),   0 <= b_ijkh < m,
c_ijkh = 2i - j + k + h + 1   (mod. m),   0 <= c_ijkh < m,
d_ijkh = 2i + j - k + h + 1   (mod. m),   0 <= d_ijkh < m,
e_ijkh = 2i + j + k - h + 1   (mod. m),   0 <= e_ijkh < m,
S^*₁₅(x) = R_3,5(x mod 3, x mod 5) - 1.

R_3,5 is an order-(3,5) associated magic rectangle. For more information, see the page for Nasik magic cubes.

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