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A Nasik magic tesseract (namely, a pan-2,3,4-agonal magic tesseract) of **even** order can exist only if the order is divisible by 16. In this case, if the order is 32 or higher, an associated Nasik magic tesseract, which is 4D-compact or non-4D-compact, exists.

A Nasik 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 4D-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/8)j + (m/4)k + (m/2)h (mod. m), 0 <= b_{ijkh} < m,

c_{ijkh} = (m/2)i + j + (m/8)k + (m/4)h (mod. m), 0 <= c_{ijkh} < m,

d_{ijkh} = (m/4)i + (m/2)j + k + (m/8)h (mod. m), 0 <= d_{ijkh} < m,

e_{ijkh} = (m/8)i + (m/4)j + (m/2)k + h (mod. m), 0 <= e_{ijkh} < m,

**T**_{m}(x) = x (where x < m/2), 3m/2-1-x (otherwise) (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 = 16:

LP({1,2,4,8},{8,1,2,4},{4,8,1,2},{2,4,8,1})= [0,1,2,3,4,5,6,7,15,14,13,12,11,10,9,8]

In general:

LP({1,m/8,m/4,m/2},{m/2,1,m/8,m/4},{m/4,m/2,1,m/8},{m/8,m/4,m/2,1})= [0,..,m/2-1,m-1,...,m/2]

**Note**

It can be proved that every order-16 Nasik magic tesseract is complete and 4D-compact, so an order-16 associated Nasik magic tesseract **cannot** exist. On the other hand, if the order is higher than 16 and divisible by 16, there exists a Nasik magic tesseract which is **not** complete or **not** 4D-compact. See **1.2** and **1.3** for **non-complete** Nasik magic tesseracts. A Nasik magic tesseract which is complete but **not** 4D-compact is constructed by the following **XmlHypercube format**.

There **cannot** exist a Nasik magic tesseract which is 2D-compact or 3D-compact.

**m >= 32** and m is divisible by 32 (**Nasik**, **complete**, but **not 4D-compact**) :

LP({1,m/16,m/8,m/4},{m/4,1,m/16,m/8},{m/8,m/4,1,m/16},{m/16,m/8,m/4,1})= [0,..,m/2-1,m-1,...,m/2]

**m >= 32** and m is divisible by 16 but **not** by 32 (**Nasik**, **complete**, but **not 4D-compact**) :

LP({8,m/16,m/8,m/4},{m/4,8,m/16,m/8},{m/8,m/4,8,m/16},{m/16,m/8,m/4,8})= [0,..,m/2-1,m-1,...,m/2]

Compare with the case of Nasik magic cubes.

An associated Nasik magic tesseract (non-4D-compact) a_{ijkh} of order m, where i,j,k,h = 0,...,m-1, is given by the following equation:

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

where

b_{ijkh} = i + (m/16)j + (m/8)k + (m/4)h + 7m/32 (mod. m), 0 <= b_{ijkh} < m,

c_{ijkh} = (m/4)i + j + (m/16)k + (m/8)h + 7m/32 (mod. m), 0 <= c_{ijkh} < m,

d_{ijkh} = (m/8)i + (m/4)j + k + (m/16)h + 7m/32 (mod. m), 0 <= d_{ijkh} < m,

e_{ijkh} = (m/16)i + (m/8)j + (m/4)k + h + 7m/32 (mod. m), 0 <= e_{ijkh} < m,

**U**_{m}(x) = x (if x < m/4 or x >= 3m/4), m-1-x (otherwise).

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

for m = 32:

LP({1,2,4,8},{8,1,2,4},{4,8,1,2},{2,4,8,1})= [7,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,24,25,26,27,28,29,30,31,0,1,2,3,4,5,6]

An associated Nasik magic tesseract (non-4D-compact) a_{ijkh} of order m, where i,j,k,h = 0,...,m-1, is given by the following equation:

a_{ijkh} = **U**_{m,8}(b_{ijkh}) m^{3} + **U**_{m,8}(c_{ijkh}) m^{2} + **U**_{m,8}(d_{ijkh}) m + **U**_{m,8}(e_{ijkh}) + 1,

where

b_{ijkh} = 8i + (m/16)j + (m/8)k + (m/4)h + 7(m/16+1)/2 (mod. m), 0 <= b_{ijkh} < m,

c_{ijkh} = (m/4)i + 8j + (m/16)k + (m/8)h + 7(m/16+1)/2 (mod. m), 0 <= c_{ijkh} < m,

d_{ijkh} = (m/8)i + (m/4)j + 8k + (m/16)h + 7(m/16+1)/2 (mod. m), 0 <= d_{ijkh} < m,

e_{ijkh} = (m/16)i + (m/8)j + (m/4)k + 8h + 7(m/16+1)/2 (mod. m), 0 <= e_{ijkh} < m,.

**U**_{m,8}(x) = **P**_{m,16}(x mod 16, x mod (m/16)).

**P**_{m,16}(x, y) (where 0 <= x < 16, 0 <= y < m/16) is given by the following table, where e = m/16 and q = (m/16-3)/2.

When m = 48 (q = 0), only the center part colored by dark green is available.

0 | 16+0 | ... | 16(q-1)+0 | 16q+0 | 16q+30 | 16q+8 | 16(q-1)+15 | ... | 16+15 | 15 |

2 | 16+2 | ... | 16(q-1)+2 | 16q+2 | 16q+26 | 16q+10 | 16(q-1)+13 | ... | 16+13 | 13 |

12 | 16+12 | ... | 16(q-1)+12 | 16q+11 | 16q+24 | 16q+3 | 16(q-1)+3 | ... | 16+3 | 3 |

14 | 16+14 | ... | 16(q-1)+14 | 16q+9 | 16q+28 | 16q+1 | 16(q-1)+1 | ... | 16+1 | 1 |

5 | 16+5 | ... | 16(q-1)+5 | 16q+5 | 16q+20 | 16q+13 | 16(q-1)+10 | ... | 16+10 | 10 |

7 | 16+7 | ... | 16(q-1)+7 | 16q+7 | 16q+16 | 16q+15 | 16(q-1)+8 | ... | 16+8 | 8 |

9 | 16+9 | ... | 16(q-1)+9 | 16q+14 | 16q+18 | 16q+6 | 16(q-1)+6 | ... | 16+6 | 6 |

11 | 16+11 | ... | 16(q-1)+11 | 16q+12 | 16q+22 | 16q+4 | 16(q-1)+4 | ... | 16+4 | 4 |

16(e-1)+11 | 16(e-2)+11 | ... | 16(e-q)+11 | 16q+43 | 16q+25 | 16q+35 | 16(e-q)+4 | ... | 16(e-2)+4 | 16(e-1)+4 |

16(e-1)+9 | 16(e-2)+9 | ... | 16(e-q)+9 | 16q+41 | 16q+29 | 16q+33 | 16(e-q)+6 | ... | 16(e-2)+6 | 16(e-1)+6 |

16(e-1)+7 | 16(e-2)+7 | ... | 16(e-q)+7 | 16q+32 | 16q+31 | 16q+40 | 16(e-q)+8 | ... | 16(e-2)+8 | 16(e-1)+8 |

16(e-1)+5 | 16(e-2)+5 | ... | 16(e-q)+5 | 16q+34 | 16q+27 | 16q+42 | 16(e-q)+10 | ... | 16(e-2)+10 | 16(e-1)+10 |

16(e-1)+14 | 16(e-2)+14 | ... | 16(e-q)+14 | 16q+46 | 16q+19 | 16q+38 | 16(e-q)+1 | ... | 16(e-2)+1 | 16(e-1)+1 |

16(e-1)+12 | 16(e-2)+12 | ... | 16(e-q)+12 | 16q+44 | 16q+23 | 16q+36 | 16(e-q)+3 | ... | 16(e-2)+3 | 16(e-1)+3 |

16(e-1)+2 | 16(e-2)+2 | ... | 16(e-q)+2 | 16q+37 | 16q+21 | 16q+45 | 16(e-q)+13 | ... | 16(e-2)+13 | 16(e-1)+13 |

16(e-1)+0 | 16(e-2)+0 | ... | 16(e-q)+0 | 16q+39 | 16q+17 | 16q+47 | 16(e-q)+15 | ... | 16(e-2)+15 | 16(e-1)+15 |

Compare with the definition of

(1) 0 <=

(2) If

(3) Sum

(4) Sum

(5)

Back

An associated 4D-compact Nasik 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} = **U**_{m}(b^{*}_{ijkh}) (if i is even), m - 1 - **U**_{m}(b^{*}_{ijkh}) (if i is odd),

c_{ijkh} = **U**_{m}(c^{*}_{ijkh}) (if j is even), m - 1 - **U**_{m}(c^{*}_{ijkh}) (if j is odd),

d_{ijkh} = **U**_{m}(d^{*}_{ijkh}) (if k is even), m - 1 - **U**_{m}(d^{*}_{ijkh}) (if k is odd),

e_{ijkh} = **U**_{m}(e^{*}_{ijkh}) (if h is even), m - 1 - **U**_{m}(e^{*}_{ijkh}) (if h is odd),

b^{*}_{ijkh} = j^{*} + (m/16)k^{*} + (m/8)h^{*} (mod. m), 0 <= b^{*}_{ijkh} < m,

c^{*}_{ijkh} = k^{*} + (m/16)h^{*} + (m/8)i^{*} (mod. m), 0 <= c^{*}_{ijkh} < m,

d^{*}_{ijkh} = h^{*} + (m/16)i^{*} + (m/8)j^{*} (mod. m), 0 <= d^{*}_{ijkh} < m,

e^{*}_{ijkh} = i^{*} + (m/16)j^{*} + (m/8)k^{*} (mod. m), 0 <= e^{*}_{ijkh} < m,

i^{*} = i (if i is even), m - 1 - i (if i is odd),

j^{*} = j (if j is even), m - 1 - j (if j is odd),

k^{*} = k (if k is even), m - 1 - k (if k is odd),

h^{*} = h (if h is even), m - 1 - h (if h is odd),

**U**_{m}(x) = x (x < m/4 *or* x >= 3m/4), m-1-x (otherwise) (identical to the definition of **U**_{m}(x) in 1.2.1).

An associated 4D-compact Nasik 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} = **U**_{m,8}(b^{*}_{ijkh}) (if i is even), m - 1 - **U**_{m,4}(b^{*}_{ijkh}) (if i is odd),

c_{ijkh} = **U**_{m,8}(c^{*}_{ijkh}) (if j is even), m - 1 - **U**_{m,4}(c^{*}_{ijkh}) (if j is odd),

d_{ijkh} = **U**_{m,8}(d^{*}_{ijkh}) (if k is even), m - 1 - **U**_{m,4}(d^{*}_{ijkh}) (if k is odd),

e_{ijkh} = **U**_{m,8}(e^{*}_{ijkh}) (if h is even), m - 1 - **U**_{m,4}(d^{*}_{ijkh}) (if h is odd),

b^{*}_{ijkh} = 4j^{*} + (m/16)k^{*} + (m/8)h^{*} (mod. m), 0 <= b^{*}_{ijkh} < m,

c^{*}_{ijkh} = 4k^{*} + (m/16)h^{*} + (m/8)i^{*} (mod. m), 0 <= c^{*}_{ijkh} < m,

d^{*}_{ijkh} = 4h^{*} + (m/16)i^{*} + (m/8)j^{*} (mod. m), 0 <= d^{*}_{ijkh} < m,

e^{*}_{ijkh} = 4i^{*} + (m/16)j^{*} + (m/8)k^{*} (mod. m), 0 <= e^{*}_{ijkh} < m,

i^{*} = i (if i is even), m - 1 - i (if i is odd),

j^{*} = j (if j is even), m - 1 - j (if j is odd),

k^{*} = k (if k is even), m - 1 - k (if k is odd),

h^{*} = h (if h is even), m - 1 - h (if h is odd),

**U**_{m,8}(x) = **P**_{m,16}(x mod 16, x mod (m/16)). (identical to the definition of **U**_{m,8}(x) in 1.2.2).

A Nasik (namely, pan-2,3,4-agonal) magic tesseract of **odd** order can exist only if the order is higher than 16, and an associated Nasik magic tesseract of **odd** order can exist under the same condition.

Let L_{15} be **lcm**{y| y is odd and 3 <= y <= 15} = 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 Nasik 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 + 2j + 4k + 8h + 7 (mod. m), 0 <= b_{ijkh} < m,

c_{ijkh} = i - 2j + 4k + 8h + 5 (mod. m), 0 <= c_{ijkh} < m,

d_{ijkh} = i + 2j - 4k + 8h + 3 (mod. m), 0 <= d_{ijkh} < m,

e_{ijkh} = i + 2j + 4k - 8h - 1 (mod. m), 0 <= e_{ijkh} < m.

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

LP({1,2,4,8}+7,{1,-2,4,8}+5,{1,2,-4,8}+3,{1,2,4,-8}-1)

Let q be **gcd**(m, L_{15}) and p be m/q. In this case, p > 1 and q > 1.

A Nasik 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} = i + 2j + 4k + 8h + 7 (mod. m), 0 <= b_{ijkh} < m,

c_{ijkh} = i - 2j + 4k + 8h + 5 (mod. m), 0 <= c_{ijkh} < m,

d_{ijkh} = i + 2j - 4k + 8h + 3 (mod. m), 0 <= d_{ijkh} < m,

e_{ijkh} = i + 2j + 4k - 8h - 1 (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 magic cubes to see examples of **Q**_{p,q}(x, y) and **S**_{m,q}(x).

In this case, m is a divisor of L_{15} = 3^{2}x5x7x11x13.

A Nasik 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} = i + 2j + 4k + 8h + 7 (mod. m), 0 <= b_{ijkh} < m,

c_{ijkh} = i - 2j + 4k + 8h + 5 (mod. m), 0 <= c_{ijkh} < m,

d_{ijkh} = i + 2j - 4k + 8h + 3 (mod. m), 0 <= d_{ijkh} < m,

e_{ijkh} = i + 2j + 4k - 8h - 1 (mod. m), 0 <= e_{ijkh} < m,

where [x] means the integer part of x.

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

**(1) if m is NOT equal to 45**

**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 7, 11, or 13).

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

**(2) if m is equal to 45**

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

**R**_{3,3,5} is an associated 3-dimensional magic rectangle of order (3,3,5). Concretely, it is given by the following tables:

**a magic rectangle of order (3,5,7) (associated)** (shown as an order-(3,5,3) rectangle **R**_{3,5,3})

31 | 3 | 38 | 13 | 30 |

24 | 41 | 12 | 36 | 2 |

14 | 25 | 19 | 20 | 37 |

29 | 40 | 4 | 35 | 7 |

1 | 18 | 23 | 28 | 45 |

39 | 11 | 42 | 6 | 17 |

9 | 26 | 27 | 21 | 32 |

44 | 10 | 34 | 5 | 22 |

16 | 33 | 8 | 43 | 15 |

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