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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).

A pan-3,4-agonal 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 and 2D-compact):

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

where

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

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

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

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

**T**_{m}(x) = x (if x < m/2), 3m/2-1-x (otherwise).

This definition of **T**_{m}(x) is identical to the definition of **T**_{m}(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.

An associated pan-3,4-agonal 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 2D-compact):

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

where

b_{ijkh} = **H**_{m}(i) (if j+k+h is even), m - 1 - **H**_{m}(i) (otherwise),

c_{ijkh} = **H**_{m}(j) (if i+k+h is even), m - 1 - **H**_{m}(j) (otherwise),

d_{ijkh} = **H**_{m}(k) (if i+j+h is even), m - 1 - **H**_{m}(k) (otherwise),

e_{ijkh} = **H**_{m}(h) (if i+j+k is even), m - 1 - **H**_{m}(h) (otherwise),

**H**_{m}(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).

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.

In this case, m is prime to 3, 5, and 7.

A pan-3,4-agonal 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} = 4i + j + k + h + 3 (mod. m), 0 <= b_{ijkh} < m,

c_{ijkh} = 4i - j + k + h + 2 (mod. m), 0 <= c_{ijkh} < m,

d_{ijkh} = 4i + j - k + h + 2 (mod. m), 0 <= d_{ijkh} < m,

e_{ijkh} = 4i + j + k - h + 2 (mod. m), 0 <= e_{ijkh} < 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)

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 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} = 4i + j + k + h + 3 (mod. m), 0 <= b_{ijkh} < m,

c_{ijkh} = 4i - j + k + h + 2 (mod. m), 0 <= c_{ijkh} < m,

d_{ijkh} = 4i + j - k + h + 2 (mod. m), 0 <= d_{ijkh} < m,

e_{ijkh} = 4i + j + k - h + 2 (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, m is equal to 15(=3x5), 21(=3x7), 35(=5x7), or 105(=3x5x7).

A pan-3,4-agonal 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}(b_{ijkh}) m^{3} + **S**^{*}_{m}(c_{ijkh}) m^{2} + **S**^{*}_{m}(d_{ijkh}) m + **S**^{*}_{m}(e_{ijkh}) + 1,

where

b_{ijkh} = 4i + j + k + h + 3 (mod. m), 0 <= b_{ijkh} < m,

c_{ijkh} = 4i - j + k + h + 2 (mod. m), 0 <= c_{ijkh} < m,

d_{ijkh} = 4i + j - k + h + 2 (mod. m), 0 <= d_{ijkh} < m,

e_{ijkh} = 4i + j + k - h + 2 (mod. m), 0 <= e_{ijkh} < m,

**S**^{*}_{15}(x) = **R**_{3,5}(x mod 3, x mod 5) - 1,

**S**^{*}_{21}(x) = **R**_{3,7}(x mod 3, x mod 7) - 1,

**S**^{*}_{35}(x) = **R**_{5,7}(x mod 5, x mod 7) - 1,

**S**^{*}_{105}(x) = **R**_{3,5,7}(x mod 3, x mod 5, x mod 7) - 1.

**R**_{3,5}, **R**_{3,7}, **R**_{5,7}, and **R**_{3,5,7} 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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This page was last updated on August 8, 2016.

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