Algorithms to make pan-3,4-agonal magic tesseracts

1. Algorithms for orders divisible by 4

   A pan-3,4-agonal magic tesseract of even order can exist only if the order is divisible by 4. If the order is 8 or higher, an associated pan-3,4-agonal magic tesseract can exist. Moreover, a 2-compact pan-3,4-agonal magic tesseract can exist for any order divisible by 4. (Every 2-compact magic tesseract is pan-3,4-agonal and cannot be diagonal).

1.1 Non-associated pan-3,4-agonal magic tesseracts (m = 4x) (also complete and 2-compact)

A pan-3,4-agonal 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 complete and 2-compact):
a_ijkh = T_m(b_ijkh) m³ + T_m(c_ijkh) m² + T_m(d_ijkh) m + T_m(e_ijkh) + 1,

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

This definition of T_m(x) is identical to the definition of T_m(x) for pantriagonal magic cubes.

This tesseract is expressed as follows by the XmlHypercube format by Aale de Winkel:
for m = 4:
   LP({1,2,2,2},{2,1,2,2},{2,2,1,2},{2,2,2,1})= [0,1,3,2]
In general:
   LP({1,m/2,m/2,m/2},{m/2,1,m/2,m/2},{m/2,m/2,1,m/2},{m/2,m/2,m/2,1})= [0,..,m/2-1,m-1,...,m/2]

Note
   It can be proved that every order-4 pan-3,4-agonal magic tesseract is complete and 2-compact. On the other hand, if the order is 8 or higher and divisible by 4, a magic tesseract exists which is pan-3,4-agonal but not 2-compact. If the order m is prime to 7, such a tesseract is given by the following XmlHypercube format. The tesseract generated by this expression is not 4-compact, either.

for m >= 8 and primt to 7 (pan-3,4-agonal, complete, but not 4-compact):
   LP({1,2,2,2},{2,1,2,2},{2,2,1,2},{2,2,2,1})= [0,..,m/2-1,m-1,...,m/2]

As mentioned above, this expression gives an 2-compact magic tesseract for m = 4. There exists a similar expression by XmlHypercube format for m not prime to 7.

An associated pan-3,4-agonal 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 2-compact):
a_ijkh = b_ijkh m³ + c_ijkh m² + d_ijkh m + e_ijkh + 1,

where
b_ijkh = H_m(i)   (if j+k+h is even),    m - 1 - H_m(i)   (otherwise),
c_ijkh = H_m(j)   (if i+k+h is even),    m - 1 - H_m(j)   (otherwise),
d_ijkh = H_m(k)   (if i+j+h is even),    m - 1 - H_m(k)   (otherwise),
e_ijkh = H_m(h)   (if i+j+k is even),    m - 1 - H_m(h)   (otherwise),
H_m(x) = H'_m(x)   (if x < m/2),    H'_m(m-1-x)   (otherwise).

The function H'_m(x), where 0 <= x < m/2, is defined as follows:

when m is divisible by 8:
H'_m(x) = x (if x = 0 or 3 (mod. 4)),
m - 1 - x (otherwise),

when m is divisible by 4 but not by 8:
H'_m(x) = x (if x < m/2-5 and x = 0 or 3 (mod. 4), or x = m/2-4),
m - 1 - x (otherwise).

2. Algorithms for odd orders

   A pan-3,4-agonal magic tesseract of odd order m can exist only if the order is higher than 7, and an associated pan-3,4-agonal magic 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 >= 9, and gcd(m, 3x5x7) = 1 (associated)

In this case, m is prime to 3, 5, and 7.
A pan-3,4-agonal 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 = 4i + j + k + h + 3   (mod. m),   0 <= b_ijkh < m,
c_ijkh = 4i - j + k + h + 2   (mod. m),   0 <= c_ijkh < m,
d_ijkh = 4i + j - k + h + 2   (mod. m),   0 <= d_ijkh < m,
e_ijkh = 4i + j + k - h + 2   (mod. m),   0 <= e_ijkh < m.

This tesseract is expressed as follows by the XmlHypercube format by Aale de Winkel:
   LP({4,1,1,1}+3,{4,-1,1,1}+2,{4,1,-1,1}+2,{4,1,1,-1}+2)

2.2 Case m = 2x+1, m >= 9, and 1 < gcd(m, 3x5x7) < m (associated)

Let q be gcd(m, 3x5x7) and p be m/q. In this case, p > 1 and q > 1.
A pan-3,4-agonal 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,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 = 4i + j + k + h + 3   (mod. m),   0 <= b_ijkh < m,
c_ijkh = 4i - j + k + h + 2   (mod. m),   0 <= c_ijkh < m,
d_ijkh = 4i + j - k + h + 2   (mod. m),   0 <= d_ijkh < m,
e_ijkh = 4i + j + k - h + 2   (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 >= 9, and gcd(m, 3x5x7) = m (associated)

In this case, m is equal to 15(=3x5), 21(=3x7), 35(=5x7), or 105(=3x5x7).
A pan-3,4-agonal 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(b_ijkh) m³ + S^*_m(c_ijkh) m² + S^*_m(d_ijkh) m + S^*_m(e_ijkh) + 1,

where
b_ijkh = 4i + j + k + h + 3   (mod. m),   0 <= b_ijkh < m,
c_ijkh = 4i - j + k + h + 2   (mod. m),   0 <= c_ijkh < m,
d_ijkh = 4i + j - k + h + 2   (mod. m),   0 <= d_ijkh < m,
e_ijkh = 4i + j + k - h + 2   (mod. m),   0 <= e_ijkh < m,
S^*₁₅(x) = R_3,5(x mod 3, x mod 5) - 1,
S^*₂₁(x) = R_3,7(x mod 3, x mod 7) - 1,
S^*₃₅(x) = R_5,7(x mod 5, x mod 7) - 1,
S^*₁₀₅(x) = R_3,5,7(x mod 3, x mod 5, x mod 7) - 1.

R_3,5, R_3,7, R_5,7, and R_3,5,7 are associated magic rectangles of orders (3,5), (3,7), (5,7), and (3,5,7), respectively. For more information, see the page for Nasik magic cubes.

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