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A panmagic tesseract (namely, a pan-4-agonal or panquadragonal magic tesseract) of **even** order can exist only if the order is divisible by 4. In this case, a pan-3,4-agonal magic tesseract exists, which satisfies a stronger condition than panmagic. See algorithms to make pan-3,4-agonal magic tesseracts.

**Note**

You can construct a panmagic tessseract of order m = 4x which is **not** pan-3,4-agonal by the following **XmlHypercube format** by Aale de Winkel. The tesseract generated by this expression is also complete and 4D-compact.

LP({1,m/4,m/4,m/2},{m/2,1,m/4,m/4},{m/4,m/2,1,m/4},{m/4,m/4,m/2,1})= [0,..,m/2-1,m-1,...,m/2]

**Note**

It can be proved that every order-4 panmagic tesseract is complete and 4D-compact.

A panmagic tesseract of **odd** order m can exist only if the order is higher than 6, and an associated panmagic tesseract of **odd** order can exist under the same condition.

The function **gcd** means the greatest common divisor.

In this case, m is prime to both 3 and 5.

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

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

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

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

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

LP({2,1,1,1}+2,{2,-1,1,1}+1,{2,1,-1,1}+1,{2,1,1,-1}+1)

Let q be **gcd**(m, 3x5) and p be m/q. In this case, p > 1 and q > 1.

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

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

where

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

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

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

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

In this case, always m = 15.

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

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

where

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

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

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

e_{ijkh} = 2i + j + k - h + 1 (mod. m), 0 <= e_{ijkh} < 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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This page was last updated on August 8, 2016.

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