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A strictly magic tesseract (namely, a 2,3,4-agonal magic tesseracts) of **even** order exists if the order is higher than 7. In this case, an associated strictly magic tesseract exists if the order is divisible by 4. On the other hand, if the order is **singly-even**, only a **non-associated** strictly magic tesseract can exist.

A strictly magic tesseract **cannot** exist for order 4, and it is unknown whether a strictly magic tesseract exists or not for order 6. If the order is divisible by 16, there exists a Nasik magic tesseract, which satisfies a stronger condition than strictly magic (see this).

The following algorithm works for orders divisible by 4. (It is complecated to construct a strictly magic tesseract of **singly-even** order.)

An associated strictly magic tesseract a_{ijkh} of order m, where i,j,k,h = 0,...,m-1, is given by the following equation by using a **complete pan and strictly magic tesseract** C[i, j, k, h]:

a_{ijk} = C[**T**_{m}(i), **T**_{m}(j), **T**_{m}(k), **T**_{m}(h)]

where

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

If the order is divisible by 8, an associated strictly magic tesseract a_{ijkh} of order m, where i,j,k,h = 0,...,m-1, can be constructed directly by the following equation:

a_{ijk} = **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} = **T**_{m}(i) + (m/4)**T**_{m}(j) + (m/4)**T**_{m}(k) + (m/4)**T**_{m}(h) (mod. m), 0 <= b_{ijkh} < m,

c_{ijkh} = (m/4)**T**_{m}(i) + **T**_{m}(j) + (m/4)**T**_{m}(k) + (m/4)**T**_{m}(h) (mod. m), 0 <= c_{ijkh} < m,

d_{ijkh} = (m/4)**T**_{m}(i) + (m/4)**T**_{m}(j) + **T**_{m}(k) + (m/4)**T**_{m}(h) (mod. m), 0 <= d_{ijkh} < m,

e_{ijkh} = (m/4)**T**_{m}(i) + (m/4)**T**_{m}(j) + (m/4)**T**_{m}(k) + **T**_{m}(h) (mod. m), 0 <= e_{ijkh} < m,

the definition of **T**_{m}(x) is the same as above.

A strictly magic tesseract of **odd** order exists if the order is higher than 14, and an associated strictly magic tesseract of **odd** order can exist under the same condition.

A strictly magic tesseract **cannot** exist for orders 3 and 5, and it is unknown whether a strictly magic tesseract exists or not for orders 7, 9, 11, and 13. If the order is higher than 16, there exists a Nasik magic tesseract, which satisfies a stronger condition than strictly magic (see this).

Let L_{13} be **lcm**{y| y is odd and 3 <= y <= 13} = 3^{2}x5x7x11x13, where **lcm** means the least common multiple, and **gcd** means the greatest common divisor.

In this case, m is prime to 3, 5, 7, 11, and 13.

A 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 pan-2,3-agonal):

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

where

b_{ijkh} = 2i + 4j + k + 7h + (m+13)/2 (mod. m), 0 <= b_{ijkh} < m,

c_{ijkh} = 2i - 4j + k + 7h + (m+5)/2 (mod. m), 0 <= c_{ijkh} < m,

d_{ijkh} = 2i + 4j - k + 7h + (m+11)/2 (mod. m), 0 <= d_{ijkh} < m,

e_{ijkh} = 2i + 4j + 7k - h + (m+11)/2 (mod. m), 0 <= e_{ijkh} < m.

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

LP({2,4,1,7}+(m+13)/2,{2,-4,1,7}+(m+5)/2,{2,4,-1,7}+(m+11)/2,{2,4,7,-1}+(m+11)/2)

The value of the determinant

| 2 4 1 7|

| 2 -4 1 7|

| 2 4 -1 7|

| 2 4 7 -1|

equals to -256 = -2^{8}, which is prime to any odd integer, so this set of the formulae always generates a normal magic tesseract.

Let q be **gcd**(m, L_{13}) and p be m/q. In this case, p > 1 and q > 1.

A 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 pan-2,3-agonal):

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 + 4j + k + 7h + (m+13)/2 (mod. m), 0 <= b_{ijkh} < m,

c_{ijkh} = 2i - 4j + k + 7h + (m+5)/2 (mod. m), 0 <= c_{ijkh} < m,

d_{ijkh} = 2i + 4j - k + 7h + (m+11)/2 (mod. m), 0 <= d_{ijkh} < m,

e_{ijkh} = 2i + 4j + 7k - h + (m+11)/2 (mod. m), 0 <= e_{ijkh} < m.

**S**_{m}(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 cubes to see examples of **Q**_{p,q}(x, y) and **S**_{m,q}(x).

In this case, m is a divisor of L_{13} = 3^{2}x5x7x11x13.

A 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 pan-2,3-agonal):

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 + 4j + k + 7h + (m+13)/2 (mod. m), 0 <= b_{ijkh} < m,

c_{ijkh} = 2i - 4j + k + 7h + (m+5)/2 (mod. m), 0 <= c_{ijkh} < m,

d_{ijkh} = 2i + 4j - k + 7h + (m+11)/2 (mod. m), 0 <= d_{ijkh} < m,

e_{ijkh} = 2i + 4j + 7k - h + (m+11)/2 (mod. m), 0 <= e_{ijkh} < m.

**S**^{*}_{m}(x) is defined as follows:

**S**^{*}_{m}(x) = **R**_{q,m/q}(x mod q, x mod (m/q)) - 1,

where q is the greatest prime factor of m (q is equal to 5, 7, 11, or 13).

**R**_{q,m/q} is an associated magic rectangle of order-(q,m/q). See the page for magic rectangles.

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This page was last updated on August 8, 2016.

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