Algorithms to make pan and strictly magic tesseracts

1. Algorithms for orders divisible by 4

   A pan and strictly magic tesseract, namely, a pan-4-agonal and 2,3,4-agonal magic tesseract, of even order can exist only if the order is divisible by 4 and higher than 7. In this case, if the order is divisible by 16, there exist a Nasik magic tesseract, which satisfies a stronger condition than pan and strictly magic.

1.1 Non-associated pan and strictly magic tesseracts (m = 4x, m >= 8) (also complete)

A pan and strictly 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):
a_ijkh = b_ijkh m³ + b_jkhi m² + b_khij m + b_hijk + 1,

where
b_ijkh = T_m(h) (if B_m/2(i mod m/2, j mod m/2, k mod m/2) = 0),
m - 1 - T_m(h) (if B_m/2(i mod m/2, j mod m/2, k mod m/2) = 1),
T_m(x) = x (x < m/2), 3m/2-1-x (otherwise)    (identical to the definition of T_m(x) for pantriagonal magic cubes).

B_r(u, v, w), where r = m/2, u = i mod m/2, v = j mod m/2, and w = k mod m/2, is an order-r diagonal binary magic cube, that is, a diagonal non-normal magic cube which consists of zeroes and ones. The construction method of B_r(u, v, w) is as follows:

(1) Case r = 4x, r >= 4 (m = 8x, m >= 8)

B_r(u, v, w), where u, v, w = 0, ..., r-1, is given by
B_r(u, v, w) = [(T_r(u) + T_r(v) + T_r(w)) / (r/2)] mod 2.

The following is an example for r = 4 (m = 8):

B₄(u, v, w)

(2) Case r = 4x+2, r >= 6 (m = 8x+4, m >= 12)

B_r(u, v, w), where u, v, w = 0, ..., r-1, is given by
B_r(u, v, w) = B_r^*(u, v, w) xor D_r(u, v, w),

where
B_r^*(u, v, w) = V(C_r/2(u^*, v^*, w^*)) xor [u/(r/2)] xor [v/(r/2)] xor [w/(r/2)],
C_r/2(u^*, v^*, w^*) = (u^* + v^* - 2w^*) mod (r/2),
V(x) = x mod 2 (if x <= 1), (x+1) mod 2 (if x >= 2),
D_r(u, v, w) = 0 or 1 (define later),
u^* = u (if u < r/2), r-1-u (if u >= r/2),
v^* = v (if v < r/2), r-1-v (if v >= r/2),
w^* = w (if w < r/2), r-1-w (if w >= r/2).

D_r(u, v, w) is a function to adjust the diagonals of B_r^*(u, v, w).
D_r(u, v, w) = 1 if u, v, and w satisfy one of the following conditions:

• u^*-1 = v^*-1 = w^* (mod. r/2), u >= r/2,
• u^* = v^*-1 = w^* (mod. r/2), u < r/2,
• u^*-1 = v^* = w^* (mod. r/2), v >= r/2,
• u^*-1 = v^*+1 = w^* (mod. r/2), v >= r/2.

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