Algorithms to make pandiagonal magic cubes

1. Algorithms for orders divisible by 8

   A pandiagonal magic cube of even order m can exist only if the order is divisible by 8. In this case, if the order is higher than 15, an associated pandiagonal magic cube can exist. Also refer the page for Nasik magic cubes, which class is higher than pandiagonal.

1.1 Non-associated pandiagonal magic cubes (m = 8x)

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

where
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).

The definitions of b_ijk, c_ijk, and d_ijk are as follows:
if i+j+k is even:
b_ijk = i + (m/4)j - (m/4+s)k   (mod. m),   0 <= b_ijk < m,
c_ijk = (m/4)i + j - (m/4+s)k   (mod. m),   0 <= c_ijk < m,
d_ijk = (m/4)i + (m/2)j + k   (mod. m),   0 <= d_ijk < m,

if i+j+k is odd:
b_ijk = -i - (m/4)j + (m/4+s)k + m/2   (mod. m),   0 <= b_ijk < m,
c_ijk = -(m/4)i - j + (m/4+s)k + m/2   (mod. m),   0 <= c_ijk < m,
d_ijk = (m/4)i + (m/2)j + k   (mod. m),   0 <= d_ijk < m,

where
s = -1 (if m-24 is divisible by 32), 1 (otherwise).

1.2 Associated pandiagonal magic cubes (m = 8x, m >= 16)

1.2.1 Case m = 16x

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

where
T_m(x) = x (where x < m/2), 3m/2-1-x (otherwise),
U_m(x) = x (if x < m/4 or x >= 3m/4), m-1-x (otherwise).
(identical to the definitions of T_m(x) and U_m(x) for pantriagonal magic cubes)

The definitions of b_ijk, c_ijk, and d_ijk are as follows:
if i+j+k is even:
b_ijk = i + (m/4)j - (m/4+s)k   (mod. m),   0 <= b_ijk < m,
c_ijk = (m/4)i + j - (m/4+s)k   (mod. m),   0 <= c_ijk < m,
d_ijk = (m/4)i + (m/8)j + k + 3m/16   (mod. m),   0 <= d_ijk < m,

if i+j+k is odd:
b_ijk = -i - (m/4)j + (m/4+s)k + m/2   (mod. m),   0 <= b_ijk < m,
c_ijk = -(m/4)i - j + (m/4+s)k + m/2   (mod. m),   0 <= c_ijk < m,
d_ijk = (m/4)i + (m/8)j + k + 3m/16   (mod. m),   0 <= d_ijk < m,

where
s = -1 (if m-24 is divisible by 32), 1 (otherwise).

1.2.2 Case m = 16x+8, m >= 24

An associated pandiagonal 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):
a_ijk = T_m(b_ijk) m² + T_m(c_ijk) m + U_m,4(d_ijk) + 1.

T_m(x) is already defined in the case of m = 16x. U_m,4(x) is defined in the page for Nasik magic cubes.
The definitions of b_ijk, c_ijk, and d_ijk are as follows:
if i+j+k is even:
b_ijk = i + (m/4)j - (m/4+s)k   (mod. m),   0 <= b_ijk < m,
c_ijk = (m/4)i + j - (m/4+s)k   (mod. m),   0 <= c_ijk < m,
d_ijk = (m/4)i + (m/8)j + 4k + 3(m/8+1)/2   (mod. m),   0 <= d_ijk < m,

if i+j+k is odd:
b_ijk = -i - (m/4)j + (m/4+s)k + m/2   (mod. m),   0 <= b_ijk < m,
c_ijk = -(m/4)i - j + (m/4+s)k + m/2   (mod. m),   0 <= c_ijk < m,
d_ijk = (m/4)i + (m/8)j + 4k + 3(m/8+1)/2   (mod. m),   0 <= d_ijk < m,

where
s = -1 (if m-24 is divisible by 32), 1 (otherwise).

2. Algorithms for odd orders

   A pandiagonal magic cube of odd order m can exist only if the order is higher by 6, and an associated pandiagonal 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 >= 7, and gcd(m, 3x5) = 1 (associated)

In this case, m is prime to both 3 and 5.
A pandiagonal 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 + 3k + {(m-1)/2+3}   (mod. m),   0 <= b_ijk < m,
c_ijk = i + 3j + 2k + {(m-1)/2+3}   (mod. m),   0 <= c_ijk < m,
d_ijk = 2i + 3j - k + {(m-1)/2+2}   (mod. m),   0 <= d_ijk < m.

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 pandiagonal magic cube a_ijk of order m, 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 + 3k + {(m-1)/2+3}   (mod. m),   0 <= b_ijk < m,
c_ijk = i + 3j + 2k + {(m-1)/2+3}   (mod. m),   0 <= c_ijk < m,
d_ijk = 2i + 3j - k + {(m-1)/2+2}   (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 (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 pandiagonal magic cube a_ijk of order m, where i,j,k = 0,...,m-1, is given by the following equation:
a_ijk = S^*(b_ijk) m² + S^*_m(c_ijk) m + S^*_m(d_ijk) + 1,

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