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A pantriagonal magic tesseract exists for any order divisible by 4. In this case, a pan-3,4-agonal magic tesseract exists, which satisfies a stronger condition than pantriagonal. See algorithms to make pan-3,4-agonal magic tesseracts.

**Note**

An order-4 associated pantriagonal magic tesseract exists though an order-4 associated pan-3,4-agonal magic tesseract **cannot** exist. Here is an example of such a pantriagonal magic tesseract.

A pantriagonal magic tesseract of **odd** order m can exist only if the order is higher than 4, and an associated pantriagonal magic tesseract of odd order can exist under the same condition.

A pantriagonal magic tesseract a_{ijkh} of order m, where i,j,k,h = 0,...,m-1, is given by the following equation:

a_{ijkh} = b_{ijkh} m^{3} + c_{ijkh} m^{2} + d_{ijkh} m + e_{ijkh} + 1,

where

b_{ijkh} = i + j + k + h + (m+3)/2 (mod. m), 0 <= b_{ijkh} < m,

c_{ijkh} = i - j + k + h + (m+1)/2 (mod. m), 0 <= c_{ijkh} < m,

d_{ijkh} = i + j - k + h + (m+1)/2 (mod. m), 0 <= d_{ijkh} < m,

e_{ijkh} = i + j + k - h + (m+1)/2 (mod. m), 0 <= e_{ijkh} < m.

This tesseract is expressed as follows by the **XmlHypercube format** by Aale de Winkel:

LP({1,1,1,1}+(m+3)/2,{1,-1,1,1}+(m+1)/2,{1,1,-1,1}+(m+1)/2,{1,1,1,-1}+(m+1)/2)

A pantriagonal magic tesseract a_{ijkh} of order m, where i,j,k,h = 0,...,m-1, is given by the following equation:

a_{ijkh} = **S**_{m,3}(b_{ijkh}) m^{3} + **S**_{m,3}(c_{ijkh}) m^{2} + **S**_{m,3}(d_{ijkh}) m + **S**_{m,3}(e_{ijkh}) + 1,

where

b_{ijkh} = i + j + k + h + (m+3)/2 (mod. m), 0 <= b_{ijkh} < m,

c_{ijkh} = i - j + k + h + (m+1)/2 (mod. m), 0 <= c_{ijkh} < m,

d_{ijkh} = i + j - k + h + (m+1)/2 (mod. m), 0 <= d_{ijkh} < m,

e_{ijkh} = i + j + k - h + (m+1)/2 (mod. m), 0 <= e_{ijkh} < m,

**S**_{m,3}(x) = **Q**_{m/3,3}([x/3], x mod 3),

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).

A pantriagonal magic tesseract of singly-even order can exist if the order is higher than 9. An associated magic tesseract **cannot** exist.

**Note** It is unknown wheter an order-6 pantriagonal magic tesseract can exist or not.

A pantriagonal magic tesseract a_{ijkh} of order m, where i,j,k,h = 0,...,m-1, is given by the following equation:

When m is **not** divisible by 3:

a_{ijkh} = b_{ijkh} (m/2)^{4} + c_{ijkh} (m/2)^{3} + d_{ijkh} (m/2)^{2} + e_{ijkh} (m/2) + f_{ijkh} + 1,

When m is divisible by 3:

a_{ijkh} = b_{ijkh} (m/2)^{4} + **S**_{m/2,3}(c_{ijkh}) (m/2)^{3} + **S**_{m/2,3}(d_{ijkh}) (m/2)^{2} + **S**_{m/2,3}(e_{ijkh}) (m/2) + **S**_{m/2,3}(f_{ijkh}) + 1,

where

**S**_{m/2,3}(x) = **Q**_{m/6,3}([x/3], x mod 3).

**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).

The tesseracts b_{ijkh}, c_{ijkh}, d_{ijkh}, e_{ijkh}, and f_{ijkh} are defined as follows:

b_{ijkh} = **G**_{16}(v, z),

c_{ijkh} = J(i) + J(j) + J(k) + J(h) (mod. m/2), 0 <= c_{ijk} < m/2,

d_{ijkh} = J(i) - J(j) + J(k) + J(h) (mod. m/2), 0 <= d_{ijk} < m/2,

e_{ijkh} = J(i) + J(j) - J(k) + J(h) (mod. m/2), 0 <= e_{ijk} < m/2,

f_{ijkh} = J(i) + J(j) + J(k) - J(h) (mod. m/2), 0 <= f_{ijk} < m/2,

where

J(x) = x/2 (if x is **even**), (m-1-x)/2 (if x is **odd**),

v = 8(i mod 2) + 4(j mod 2) + 2(k mod 2) + (h mod 2), (0 <= v < 16),

z = J(i) + J(j) - J(k) - J(h) (mod. m/2), 0 <= z < m/2.

The function **G**_{N}(v, z) (where N is a power of 2, 0 <= v < N, and z >= 0) is defined as follows (this definition is identical to that for diagonal magic cubes):

when v is even:

**G**_{N}(v, 0) = v,

**G**_{N}(v, 1) = v + 1,

**G**_{N}(v, 2) = N/2 - 1 - v/2,

**G**_{N}(v, 3) = N - 1 - v/2,

**G**_{N}(v, 2x+4) = N - 1 - v for x >= 0,

**G**_{N}(v, 2x+5) = v for x >= 0,

when v is odd:

**G**_{N}(v, 0) = v,

**G**_{N}(v, 1) = v - 1,

**G**_{N}(v, 2) = N/2 - 1 - (v-1)/2,

**G**_{N}(v, 3) = N - 1 - (v-1)/2,

**G**_{N}(v, 2x+4) = N - 1 - v for x >= 0,

**G**_{N}(v, 2x+5) = v for x >= 0.

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