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

[Order 4x] [Order 2x+1]

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 2D-compact pan-3,4-agonal magic tesseract can exist for any order divisible by 4. (Every 2D-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 2D-compact)

A pan-3,4-agonal magic tesseract aijkh of order m, where i,j,k,h = 0,...,m-1, is given by the following equation (aijkh is also complete and 2D-compact):
aijkh = Tm(bijkh) m3 + Tm(cijkh) m2 + Tm(dijkh) m + Tm(eijkh) + 1,

where
bijkh = i + (m/2)j + (m/2)k + (m/2)h   (mod. m),   0 <= bijkh < m,
cijkh = (m/2)i + j + (m/2)k + (m/2)h   (mod. m),   0 <= cijkh < m,
dijkh = (m/2)i + (m/2)j + k + (m/2)h   (mod. m),   0 <= dijkh < m,
eijkh = (m/2)i + (m/2)j + (m/2)k + h   (mod. m),   0 <= eijkh < m,
Tm(x) = x (if x < m/2),   3m/2-1-x (otherwise).

This definition of Tm(x) is identical to the definition of Tm(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 2D-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 2D-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 4D-compact, either.

for m >= 8 and primt to 7 (pan-3,4-agonal, complete, but not 4D-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 2D-compact magic tesseract for m = 4. There exists a similar expression by XmlHypercube format for m not prime to 7.

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1.2 Associated panmagic tesseracts (m = 4x, x >= 8) (also 2D-compact)

An associated pan-3,4-agonal magic tesseract aijkh of order m, where i,j,k,h = 0,...,m-1, is given by the following equation (aijkh is also 2D-compact):
aijkh = bijkh m3 + cijkh m2 + dijkh m + eijkh + 1,

where
bijkh = Hm(i)   (if j+k+h is even),    m - 1 - Hm(i)   (otherwise),
cijkh = Hm(j)   (if i+k+h is even),    m - 1 - Hm(j)   (otherwise),
dijkh = Hm(k)   (if i+j+h is even),    m - 1 - Hm(k)   (otherwise),
eijkh = Hm(h)   (if i+j+k is even),    m - 1 - Hm(h)   (otherwise),
Hm(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).

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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 aijkh of order m, where i,j,k,h = 0,...,m-1, is given by the following equation:
aijkh = bijkh m3 + cijkh m2 + dijkh m + eijkh + 1,

where
bijkh = 4i + j + k + h + 3   (mod. m),   0 <= bijkh < m,
cijkh = 4i - j + k + h + 2   (mod. m),   0 <= cijkh < m,
dijkh = 4i + j - k + h + 2   (mod. m),   0 <= dijkh < m,
eijkh = 4i + j + k - h + 2   (mod. m),   0 <= eijkh < 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 aijkh of order m, where i,j,k,h = 0,...,m-1, is given by the following equation:
aijkh = Sm,q(bijkh) m3 + Sm,q(cijkh) m2 + Sm,q(dijkh) m + Sm,q(eijkh) + 1,

where
bijkh = 4i + j + k + h + 3   (mod. m),   0 <= bijkh < m,
cijkh = 4i - j + k + h + 2   (mod. m),   0 <= cijkh < m,
dijkh = 4i + j - k + h + 2   (mod. m),   0 <= dijkh < m,
eijkh = 4i + j + k - h + 2   (mod. m),   0 <= eijkh < m,
Sm,q(x) = Qp,q([x/q], x mod q),

where [x] means the integer part of x.

Qp,q(x, y), where 0 <= x < p and 0 <= y < q, is given by the following equations (identical to Qp,q(x, y) for Nasik magic cubes):
Qp,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 Qp,q(x, y) and Sm,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 aijkh of order m, where i,j,k,h = 0,...,m-1, is given by the following equation:
aijkh = S*m(bijkh) m3 + S*m(cijkh) m2 + S*m(dijkh) m + S*m(eijkh) + 1,

where
bijkh = 4i + j + k + h + 3   (mod. m),   0 <= bijkh < m,
cijkh = 4i - j + k + h + 2   (mod. m),   0 <= cijkh < m,
dijkh = 4i + j - k + h + 2   (mod. m),   0 <= dijkh < m,
eijkh = 4i + j + k - h + 2   (mod. m),   0 <= eijkh < m,
S*15(x) = R3,5(x mod 3, x mod 5) - 1,
S*21(x) = R3,7(x mod 3, x mod 7) - 1,
S*35(x) = R5,7(x mod 5, x mod 7) - 1,
S*105(x) = R3,5,7(x mod 3, x mod 5, x mod 7) - 1.

R3,5, R3,7, R5,7, and R3,5,7 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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