Algorithms to make Nasik magic tesseracts

1. Algorithms for orders divisible by 16

   A Nasik magic tesseract (namely, a pan-2,3,4-agonal magic tesseract) of even order can exist only if the order is divisible by 16. In this case, if the order is 32 or higher, an associated Nasik magic tesseract, which is 4-compact or non-4-compact, exists.

1.1 Non-associated Nasik magic tesseracts (m = 16x) (also complete and 4-compact)

A Nasik 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 4-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/8)j + (m/4)k + (m/2)h   (mod. m),   0 <= b_ijkh < m,
c_ijkh = (m/2)i + j + (m/8)k + (m/4)h   (mod. m),   0 <= c_ijkh < m,
d_ijkh = (m/4)i + (m/2)j + k + (m/8)h   (mod. m),   0 <= d_ijkh < m,
e_ijkh = (m/8)i + (m/4)j + (m/2)k + 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).

This tesseract is expressed as follows by the XmlHypercube format by Aale de Winkel:
for m = 16:
   LP({1,2,4,8},{8,1,2,4},{4,8,1,2},{2,4,8,1})= [0,1,2,3,4,5,6,7,15,14,13,12,11,10,9,8]
In general:
   LP({1,m/8,m/4,m/2},{m/2,1,m/8,m/4},{m/4,m/2,1,m/8},{m/8,m/4,m/2,1})= [0,..,m/2-1,m-1,...,m/2]

Note
   It can be proved that every order-16 Nasik magic tesseract is complete and 4-compact, so an order-16 associated Nasik magic tesseract cannot exist. On the other hand, if the order is higher than 16 and divisible by 16, there exists a Nasik magic tesseract which is not complete or not 4-compact. See 1.2 and 1.3 for non-complete Nasik magic tesseracts. A Nasik magic tesseract which is complete but not 4-compact is constructed by the following XmlHypercube format.
   There cannot exist a Nasik magic tesseract which is 2-compact or 3-compact.

m >= 32 and m is divisible by 32 (Nasik, complete, but not 4-compact) :
   LP({1,m/16,m/8,m/4},{m/4,1,m/16,m/8},{m/8,m/4,1,m/16},{m/16,m/8,m/4,1})= [0,..,m/2-1,m-1,...,m/2]
m >= 32 and m is divisible by 16 but not by 32 (Nasik, complete, but not 4-compact) :
   LP({8,m/16,m/8,m/4},{m/4,8,m/16,m/8},{m/8,m/4,8,m/16},{m/16,m/8,m/4,8})= [0,..,m/2-1,m-1,...,m/2]

Compare with the case of Nasik magic cubes.

1.2 Assciated Nasik magic tesseracts (m = 16x, m >= 32) (non-4-compact)

1.2.1 Case m = 32x

An associated Nasik magic tesseract (non-4-compact) a_ijkh of order m, where i,j,k,h = 0,...,m-1, is given by the following equation:
a_ijkh = U_m(b_ijkh) m³ + U_m(c_ijkh) m² + U_m(d_ijkh) m + U_m(e_ijkh) + 1,

where
b_ijkh = i + (m/16)j + (m/8)k + (m/4)h + 7m/32   (mod. m),   0 <= b_ijkh < m,
c_ijkh = (m/4)i + j + (m/16)k + (m/8)h + 7m/32   (mod. m),   0 <= c_ijkh < m,
d_ijkh = (m/8)i + (m/4)j + k + (m/16)h + 7m/32   (mod. m),   0 <= d_ijkh < m,
e_ijkh = (m/16)i + (m/8)j + (m/4)k + h + 7m/32   (mod. m),   0 <= e_ijkh < m,
U_m(x) = x (if x < m/4 or x >= 3m/4),   m-1-x (otherwise).

This tesseract is expressed as follows by the XmlHypercube format by Aale de Winkel:
for m = 32:
   LP({1,2,4,8},{8,1,2,4},{4,8,1,2},{2,4,8,1})= [7,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6]

1.2.2 Case m = 32x+16 and m >= 48

An associated Nasik magic tesseract (non-4-compact) a_ijkh of order m, where i,j,k,h = 0,...,m-1, is given by the following equation:
a_ijkh = U_m,8(b_ijkh) m³ + U_m,8(c_ijkh) m² + U_m,8(d_ijkh) m + U_m,8(e_ijkh) + 1,

where
b_ijkh = 8i + (m/16)j + (m/8)k + (m/4)h + 7(m/16+1)/2   (mod. m),   0 <= b_ijkh < m,
c_ijkh = (m/4)i + 8j + (m/16)k + (m/8)h + 7(m/16+1)/2   (mod. m),   0 <= c_ijkh < m,
d_ijkh = (m/8)i + (m/4)j + 8k + (m/16)h + 7(m/16+1)/2   (mod. m),   0 <= d_ijkh < m,
e_ijkh = (m/16)i + (m/8)j + (m/4)k + 8h + 7(m/16+1)/2   (mod. m),   0 <= e_ijkh < m,.
U_m,8(x) = P_m,16(x mod 16, x mod (m/16)).

P_m,16(x, y) (where 0 <= x < 16, 0 <= y < m/16) is given by the following table, where e = m/16 and q = (m/16-3)/2.
When m = 48 (q = 0), only the center part colored by dark green is available.

1.3 Assciated Nasik magic tesseracts (m = 16x, m >= 32) (also 4-compact)

1.3.1 Case m = 32x

An associated 4-compact Nasik 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 = U_m(b^*_ijkh) (if i is even), m - 1 - U_m(b^*_ijkh) (if i is odd),
c_ijkh = U_m(c^*_ijkh) (if j is even), m - 1 - U_m(c^*_ijkh) (if j is odd),
d_ijkh = U_m(d^*_ijkh) (if k is even), m - 1 - U_m(d^*_ijkh) (if k is odd),
e_ijkh = U_m(e^*_ijkh) (if h is even), m - 1 - U_m(e^*_ijkh) (if h is odd),

b^*_ijkh = j^* + (m/16)k^* + (m/8)h^*   (mod. m),   0 <= b^*_ijkh < m,
c^*_ijkh = k^* + (m/16)h^* + (m/8)i^*   (mod. m),   0 <= c^*_ijkh < m,
d^*_ijkh = h^* + (m/16)i^* + (m/8)j^*   (mod. m),   0 <= d^*_ijkh < m,
e^*_ijkh = i^* + (m/16)j^* + (m/8)k^*   (mod. m),   0 <= e^*_ijkh < m,

i^* = i (if i is even), m - 1 - i (if i is odd),
j^* = j (if j is even), m - 1 - j (if j is odd),
k^* = k (if k is even), m - 1 - k (if k is odd),
h^* = h (if h is even), m - 1 - h (if h is odd),

U_m(x) = x (x < m/4 or x >= 3m/4), m-1-x (otherwise)    (identical to the definition of U_m(x) in 1.2.1).

1.3.2 Case m = 32x+16 and m >= 48

An associated 4-compact Nasik 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 = U_m,8(b^*_ijkh) (if i is even), m - 1 - U_m,4(b^*_ijkh) (if i is odd),
c_ijkh = U_m,8(c^*_ijkh) (if j is even), m - 1 - U_m,4(c^*_ijkh) (if j is odd),
d_ijkh = U_m,8(d^*_ijkh) (if k is even), m - 1 - U_m,4(d^*_ijkh) (if k is odd),
e_ijkh = U_m,8(e^*_ijkh) (if h is even), m - 1 - U_m,4(d^*_ijkh) (if h is odd),

b^*_ijkh = 4j^* + (m/16)k^* + (m/8)h^*   (mod. m),   0 <= b^*_ijkh < m,
c^*_ijkh = 4k^* + (m/16)h^* + (m/8)i^*   (mod. m),   0 <= c^*_ijkh < m,
d^*_ijkh = 4h^* + (m/16)i^* + (m/8)j^*   (mod. m),   0 <= d^*_ijkh < m,
e^*_ijkh = 4i^* + (m/16)j^* + (m/8)k^*   (mod. m),   0 <= e^*_ijkh < m,

i^* = i (if i is even), m - 1 - i (if i is odd),
j^* = j (if j is even), m - 1 - j (if j is odd),
k^* = k (if k is even), m - 1 - k (if k is odd),
h^* = h (if h is even), m - 1 - h (if h is odd),

U_m,8(x) = P_m,16(x mod 16, x mod (m/16)).    (identical to the definition of U_m,8(x) in 1.2.2).

2. Algorithms for odd orders

   A Nasik (namely, pan-2,3,4-agonal) magic tesseract of odd order can exist only if the order is higher than 16, and an associated Nasik magic tesseract of odd order can exist under the same condition.
   Let L₁₅ be lcm{y| y is odd and 3 <= y <= 15} = 3²x5x7x11x13, where lcm means the least common multiple, and gcd means the greatest common divisor.

2.1 Case m = 2x+1, m >= 17, and gcd(m, L₁₅) = 1 (associated)

In this case, m is prime to 3, 5, 7, 11, and 13.
A Nasik 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 + 2j + 4k + 8h + 7   (mod. m),   0 <= b_ijkh < m,
c_ijkh = i - 2j + 4k + 8h + 5   (mod. m),   0 <= c_ijkh < m,
d_ijkh = i + 2j - 4k + 8h + 3   (mod. m),   0 <= d_ijkh < m,
e_ijkh = i + 2j + 4k - 8h - 1   (mod. m),   0 <= e_ijkh < m.

This tesseract is expressed as follows by the XmlHypercube format by Aale de Winkel:
   LP({1,2,4,8}+7,{1,-2,4,8}+5,{1,2,-4,8}+3,{1,2,4,-8}-1)

2.2 Case m = 2x+1, m >= 17, and 1 < gcd(m, L₁₅) < m (associated)

Let q be gcd(m, L₁₅) and p be m/q. In this case, p > 1 and q > 1.
A Nasik 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 = i + 2j + 4k + 8h + 7   (mod. m),   0 <= b_ijkh < m,
c_ijkh = i - 2j + 4k + 8h + 5   (mod. m),   0 <= c_ijkh < m,
d_ijkh = i + 2j - 4k + 8h + 3   (mod. m),   0 <= d_ijkh < m,
e_ijkh = i + 2j + 4k - 8h - 1   (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 magic cubes to see examples of Q_p,q(x, y) and S_m,q(x).

2.3 Case m = 2x+1, m >= 17, and gcd(m, L₁₅) = m (associated)

In this case, m is a divisor of L₁₅ = 3²x5x7x11x13.
A Nasik 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 = i + 2j + 4k + 8h + 7   (mod. m),   0 <= b_ijkh < m,
c_ijkh = i - 2j + 4k + 8h + 5   (mod. m),   0 <= c_ijkh < m,
d_ijkh = i + 2j - 4k + 8h + 3   (mod. m),   0 <= d_ijkh < m,
e_ijkh = i + 2j + 4k - 8h - 1   (mod. m),   0 <= e_ijkh < m,

where [x] means the integer part of x.
S^**_m(x) is defined as follows:

(1) if m is NOT equal to 45
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 7, 11, or 13).
R_q,m/q is an associated magic rectangle of order-(q,m/q). See the page for magic rectangles.

(2) if m is equal to 45
S^**_m(x) = R_3,3,5((x/3) mod 3, x mod 3, x mod 5) - 1.

R_3,3,5 is an associated 3-dimensional magic rectangle of order (3,3,5). Concretely, it is given by the following tables:

a magic rectangle of order (3,5,7) (associated) (shown as an order-(3,5,3) rectangle R_3,5,3)

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