Algorithms to make Nasik magic cubes

1. Algorithms for orders divisible by 8

   A Nasik (namely, pan-2,3-agonal) magic cube of even order m can exist only if the order is divisible by 8. In this case, if the order is 16 or higher, an associated Nasik magic cube, which is 3-compact or non-3-compact, exists.

1.1 Non-associated Nasik magic cubes (m = 8x) (complete and 3-compact)

A Nasik magic cube a_ijk of order m, where i,j,k = 0,...,m-1, is given by the following equation (a_ijk is also complete and 3-compact):
a_ijk = T_m(b_ijk) m² + T_m(c_ijk) m + T_m(d_ijk) + 1,

where
b_ijk = i + (m/4)j + (m/2)k   (mod. m),   0 <= b_ijk < m,
c_ijk = (m/2)i + j + (m/4)k   (mod. m),   0 <= c_ijk < m,
d_ijk = (m/4)i + (m/2)j + k   (mod. m),   0 <= d_ijk < 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 cube is expressed as follows by the XmlHypercube format by Aale de Winkel:
for m = 8:
   LP({1,2,4},{4,1,2},{2,4,1})= [0,1,2,3,7,6,5,4]
for m = 16:
   LP({1,4,8},{8,1,4},{4,8,1})= [0,1,2,3,4,5,6,7,15,14,13,12,11,10,9,8]
In general:
   LP({1,m/4,m/2},{m/2,1,m/4},{m/4,m/2,1})= [0,..,m/2-1,m-1,...,m/2]
The F. A. P. Barnard order-8 Nasik magic cube was constructed by a similar way to this method.

Note
   It can be proved that every order-8 Nasik magic cube is complete and 3-compact, so an order-8 associated Nasik magic cube cannot exist. On the other hand, if the order is higher than 8 and divisible by 8, there exists a Nasik magic cube which is not complete or not 3-compact. See 1.2 and 1.3 for non-complete Nasik magic cubes. A Nasik magic cube which is complete but not 3-compact is constructed by the following XmlHypercube format. There cannot exist a 2-compact Nasik magic cube.

m >= 16 and m is divisible by 16 (Nasik, complete, but not 3-compact) :
   LP({1,m/8,m/4},{m/4,1,m/8},{m/8,m/4,1})= [0,..,m/2-1,m-1,...,m/2]
m >= 16 and m is divisible by 8 but not by 16 (Nasik, complete, but not 3-compact) :
   LP({4,m/8,m/4},{m/4,4,m/8},{m/8,m/4,4})= [0,..,m/2-1,m-1,...,m/2]

Compare with the case of Nasik magic tesseracts.

Addendum
   If the order is a power of 2, you can construct another Nasik magic cube by using binary digits. (The Gakuho Abe order-8 Nasik magic cube seems to have been constructed by this method.) For more information, see Dwane H. Campbell's site.

1.2 Associated Nasik magic cubes (m = 8x, m >= 16) (not 3-compact)

1.2.1 Case m = 16x (associated)

An associated Nasik magic cube (non-3-compact) a_ijk of order m, where i,j,k = 0,...,m-1, is given by the following equation:
a_ijk = U_m(b_ijk) m² + U_m(c_ijk) m + U_m(d_ijk) + 1,

where
b_ijk = i + (m/8)j + (m/4)k + 3m/16   (mod. m),   0 <= b_ijk < m,
c_ijk = (m/4)i + j + (m/8)k + 3m/16   (mod. m),   0 <= c_ijk < m,
d_ijk = (m/8)i + (m/4)j + k + 3m/16   (mod. m),   0 <= d_ijk < m,
U(x) = x (if x < m/4 or x >= 3m/4), m-1-x (otherwise)    (identical to the definition of T_m(x) for pantriagonal magic cubes).

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

1.2.2 Case m = 16x+8 and m >= 24 (associated)

An associated Nasik magic cube (non-3-compact) a_ijk of order m, where i,j,k = 0,...,m-1, is given by the following equation:
a_ijk = U_m,4(b_ijk) m² + U_m,4(c_ijk) m + U_m,4(d_ijk) + 1,

where
b_ijk = 4i + (m/8)j + (m/4)k + 3(m/8+1)/2   (mod. m),   0 <= b_ijk < m,
c_ijk = (m/4)i + 4j + (m/8)k + 3(m/8+1)/2   (mod. m),   0 <= c_ijk < m,
d_ijk = (m/8)i + (m/4)j + 4k + 3(m/8+1)/2   (mod. m),   0 <= d_ijk < m,
U_m,4(x) = P_m,8(x mod 8, x mod (m/8)).

P_m,8(x, y) (where 0 <= x < 8, 0 <= y < m/8) is given by the following table, where e = m/8 and q = (m/8-3)/2.
When m = 24 (q = 0), only the center part colored by dark green is available.

1.3 Associated 3-compact Nasik magic cubes (m = 8x, m >= 16)

1.3.1 Case m = 16x (associated and 3-compact)

An associated 3-compact Nasik magic cube a_ijk of order m, where i,j,k = 0,...,m-1, is given by the following equation:
a_ijk = b_ijk m² + c_ijk m + d_ijk + 1,

where
b_ijk = U_m(b^*_ijk) (if the index i is even),   m - 1 - U_m(b^*_ijk) (if i is odd),
c_ijk = U_m(c^*_ijk) (if the index j is even),   m - 1 - U_m(c^*_ijk) (if j is odd),
d_ijk = U_m(d^*_ijk) (if the index k is even),   m - 1 - U_m(d^*_ijk) (if k is odd),

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

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

1.2.2 Case m = 16x+8 and m >= 24 (associated and 3-compact)

An associated 3-compact Nasik magic cube a_ijk of order m, where i,j,k = 0,...,m-1, is given by the following equation:
a_ijk = b_ijk m² + c_ijk m + d_ijk + 1,

where
a_ijk = b_ijk m² + c_ijk m + d_ijk + 1,

where
b_ijk = U_m,4(b^*_ijk) (if the index i is even),   m - 1 - U_m,4(b^*_ijk) (if i is odd),
c_ijk = U_m,4(c^*_ijk) (if the index j is even),   m - 1 - U_m,4(c^*_ijk) (if j is odd),
d_ijk = U_m,4(d^*_ijk) (if the index k is even),   m - 1 - U_m,4(d^*_ijk) (if k is odd),

b^*_ijk = 2j^* + (m/8)k^*   (mod. m),   0 <= b^*_ijk < m,
c^*_ijk = 2k^* + (m/8)i^*   (mod. m),   0 <= c^*_ijk < m,
d^*_ijk = 2i^* + (m/8)j^*   (mod. m),   0 <= d^*_ijk < 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),

U_m,4(x) = P_m,8(x mod 8, x mod (m/8))    (the same as the definition of U_m,4(x) in 1.2.2).

2. Algorithms for odd orders

   A Nasik (namely, pan-2,3-agonal) magic cube of odd order m can exist only if the order is higher by 8, and an associated Nasik magic cube 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 Nasik magic cube a_ijk of order m, where i,j,k = 0,...,m-1, is given by the following equation:
a_ijk = b_ijk m² + c_ijk m + d_ijk + 1,

where
b_ijk = i + 2j + 4k + 3   (mod. m),   0 <= b_ijk < m,
c_ijk = i - 2j + 4k + 1   (mod. m),   0 <= c_ijk < m,
d_ijk = i + 2j - 4k - 1   (mod. m),   0 <= d_ijk < m.

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

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 Nasik magic cube a_ijk, where i,j,k = 0,...,m-1, is given by the following equation:
a_ijk = S_m,q(b_ijk) m² + S_m,q(c_ijk) m + S_m,q(d_ijk) + 1,

where
b_ijk = i + 2j + 4k + 3   (mod. m),   0 <= b_ijk < m,
c_ijk = i - 2j + 4k + 1   (mod. m),   0 <= c_ijk < m,
d_ijk = i + 2j - 4k - 1   (mod. m),   0 <= d_ijk < 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:
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).

Q_p,q(x, y) satifies the following properties (only these properties are needed for Q_p,q(x, y)).
(1) 0 <= Q_p,q(x, y) < pq (= m),
(2) If Q_p,q(x, y) = Q_p,q(x', y') then x = x' and y = y'    (normal),
(3) Sum_v=0,p-1 {Q_p,q(v, y)} = q(pq-1)/2    (column magicness),
(4) Q_p,q(x, y) + Q_p,q(p-1-x, q-1-y) = pq-1    (associated).

The following are examples of Q_p,q(x, y) and S_m,q(x) for m = 9 (p = 3, q = 3), m = 25 (p = 5, q = 5), and m = 45 (p = 3, q = 15).

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 Nasik magic cube a_ijk of order m, where i,j,k = 0,...,m-1, is given by the following equation:
a_ijk = S^*_m(b_ijk) m² + S^*_m(c_ijk) m + S^*_m(d_ijk) + 1,

where
b_ijk = i + 2j + 4k + 3   (mod. m),   0 <= b_ijk < m,
c_ijk = i - 2j + 4k + 1   (mod. m),   0 <= c_ijk < m,
d_ijk = i + 2j - 4k - 1   (mod. m),   0 <= d_ijk < 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. Concretely, they are given by the following tables:

magic rectangles of orders (3,5), (3,7), and (5,7) (associated)

Addendum:
   For m = 105, we can use a magic rectangle of order (3,35), (5,21), or (7,15) instead of that of order (3,5,7). (Aale de Winkel pointed out that.) We can construct these magic rectangles by using the order-(3,5,7) magic rectangle, or without using it. For more information, see this page for magic rectangles.

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