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# 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 aijkh of order m, where i,j,k,h = 0,...,m-1, is given by the following equation (aijkh is also complete):
aijkh = bijkh m3 + bjkhi m2 + bkhij m + bhijk + 1,

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

Br(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 Br(u, v, w) is as follows:

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

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

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

B4(u, v, w)

 0 0 1 1 0 1 0 1 1 0 1 0 1 1 0 0
 0 1 0 1 1 1 0 0 0 0 1 1 1 0 1 0
 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1
 1 1 0 0 1 0 1 0 0 1 0 1 0 0 1 1

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

Br(u, v, w), where u, v, w = 0, ..., r-1, is given by
Br(u, v, w) = Br*(u, v, w) xor Dr(u, v, w),

where
Br*(u, v, w) = V(Cr/2(u*, v*, w*)) xor [u/(r/2)] xor [v/(r/2)] xor [w/(r/2)],
Cr/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),
Dr(u, v, w) = 0 or 1 (the condition is defined later),

where
u* = min{ u, r-1-u }, v* = min{ v, r-1-v }, w* = min{ w, r-1-w }.

Dr(u, v, w) is a function to adjust the diagonals of Br*(u, v, w).
Dr(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.
The following is an example for r = 6 (m = 12):

C3(u*, v*, w*)
 0 1 2 1 2 0 2 0 1
 1 2 0 2 0 1 0 1 2
 0 1 2 1 2 0 2 0 1
V(C3(u*, v*, w*))
 0 1 1 1 1 0 1 0 1
 1 1 0 1 0 1 0 1 1
 1 0 1 0 1 1 1 1 0
B6*(u, v, w)
 0 1 1 0 0 1 1 1 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 1 1 1 0 0 1 1 0
 1 1 0 1 0 0 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 0 1 0 1 1
 1 0 1 0 1 0 0 1 1 0 0 1 1 1 0 1 0 0 0 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 0 1

 0 1 0 1 0 1 1 0 0 1 1 0 0 0 1 0 1 1 1 1 0 1 0 0 0 1 1 0 0 1 1 0 1 0 1 0
 0 0 1 0 1 1 0 1 0 1 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0
 1 0 0 1 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1 0 1 0 1 0 1 1 0 1 0 0 0 1 1 0 0 1
D6(u, v, w)
 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0
 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1

 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1
 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0
B6(u, v, w)
 0 0 1 0 1 1 1 1 0 0 0 1 1 0 1 0 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 0 0 1 1 0
 1 1 0 1 0 0 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 1
 1 0 1 1 0 0 0 1 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 1 1 0 0 1 1 0 1 1 0 0 1 0

 0 1 0 0 1 1 1 0 0 1 1 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 1 0 0 1 0 0 1 1 0 1
 0 0 1 0 1 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 1 0 1 0 0
 1 1 0 1 0 0 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 1 0 1 0 0 0 1 1 0 1 1 0 0 1

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