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**magic hypercube**- a magic hypercube of dimension
`n`and order`m`is defined as a`n`-dimensional hypercubical array of`m`^{n}numbers in which the sum for every orthogonal and every`n`-agonal is the same value (called the (magic) constant).

Magic hypercubes of dimensions 2, 3, and 4 are called**magic squares**,**magic cubes**, and**magic tesseracts**, respectively. **semimagic hypercube**- the same as magic hypercube except that the sum of each
`n`-agonal does not have to equal to the magic constant. Every associated**semimagic***hypercube*is a magic*hypercube*. **magic rectangle**- a magic rectangle of dimension
`n`and order (`m`_{1},`m`_{2}, ...,`m`_{n}) is defined as a`n`-dimensional rectangular array of`m`_{1}`m`_{2}...`m`_{n}numbers in which for each direction, every orthogonal sums to the same value depending on the direction. There is**no**condition for`r`-agonals.

A**magic rectangle**in which every element of the order is the same is a semimagic hypercube.

Some construction methods of magic hypercube use**magic rectangles**. **magic pattern**- A pattern of numbers in which the sum of every line of the pattern is the same value.
**Magic patterns**is categolized into**magic circles**,**magic stars**,*etc.*. **magic object**- the general term for (semi)magic hypercube, magic rectangle, magic pattern, and so on.
**Latin square**- a (2-dimentional)
**Latin square**is defined as an`m`by`m`array of`m`numerical or literal symbols in which each symbol appears exactly once in each row and each column of the array. It is**not**required that tha same condition apply to the diagonal. If they do, the square is called a**diagonal Latin square**. A**Latin square**can be regarded as a (non-normal) semimagic square if numerical symbols is used. A**Latin hypercube**of dimension 3 or higher is defined similarly.

**Latin squares**are frequently used for generating magic squares, and applied to other fields, for example, the**design of experiments**.

**normal**(magic object)- a magic object which consists of consecutive integers start from 1.

According to circumstances, we use consecutive integers start from 0 for a**normal**magic object for mathematical convenience.

Aale de Winkel calls the former range the "**regular number range**" and the latter range the "**analitic number range**".

This site adopts the**regular number range**in principle.

A**normal**(semi)magic hypercube of dimension`n`and order`m`consists of consecutive integers from 1 to`m`^{n}in the**regular number range**, and from 0 to`m`^{n}-1 in the**analitic number range**. **non-normal**(magic object)- a magic object which is
**not**normal. In this site, we assume that every cell of a (non-normal) magic object is integer. **prime**magic object- a magic object which consists of prime numbers. It is a kind of a non-normal magic object. The integer 1 is often used as an element of
**prime**magic object though 1 is not a prime number. **constant**magic object- a (trivial) magic object which consists of only a single number. It is a kind of a non-normal magic object. Of course, such a magic object is worthless.
**row**,**column**,**pillar**,**file**- directions of the first, the second, the third, and the fourth dimension, respectively, in a (semi)magic hypercube or a magic rectangle.
**r-agonal**(of a hypercube)- an r-dimensional space 'diagonal'. Specially, 2-agonal, 3-agonal, and 4-agonal are also called
**diagonal**,**triagonal**, and**quadragonal**, respectively.**1-agonal (monoagonal)**means**orthogonal**.

**pan-r-agonal**(of a hypercube)- a broken r-agonal parallel to some r-agonal of the hypercube. See the figure below.

Pan-2-agonal, pan-3-agonal, and pan-4-agonal are also called**pandiagonal**,**pantriagonal**, and**panquadragonal**, respectively.**Pan-1-agonal (panmonoagonal)**is a synonym of**1-agonal (monoagonal)**.

**example of pandiagonal****X****X****X****X****X**

**(magic) constant**- the sum which requires to be the same value in a (semi)magic hypercube or a magic pattern. Also called the
**magic sum**.

The**constant**of a normal magic hypercube of dimension`n`and order`m`equals to`m`(`m`^{n}+1)/2 in the regular number range.

In a magic rectangle, the**constant**is defined for each direction (the row constant, the column constant, etc.). **singly even**number- an even integer which is not divisible by 4. A singlly-even magic hypercube is difficult to construct.
**doubly even**number- an integer which is divisible by 4.

magic hypercube (cube, tesseract)`r`-agonal- a magic hypercube of dimension
`n`such that every`r`-agonal of the hypercube sums to the magic constant.

**2-agonal**,**3-agonal**, and**4-agonal**are also called**diagonal**,**triagonal**, and**quadragonal**, respectively. The description**1-agonal (monoagonal)**is usually left out.

We also use expressions like**2,3-agonal**(that is, both 2-agonal and 3-agonal).

A hypercube of dimension`n`is a magic hypercube if and only if the hypercube is**1,**.`n`-agonal **pan-**magic hypercube (square, cube, tesseract)`r`-agonal- a magic hypercube of dimension
`n`such that every pan-`r`-agonal of the hypercube sums to the magic constant.

**pan-2-agonal**,**pan-3-agonal**, and**pan-4-agonal**are also called**pandiagonal**,**pantriagonal**, and**panquadragonal**, respectively. The description**pan-1-agonal (panmonoagonal)**is usually left out.

We also use expressions like**pan-2,3-agonal**(that is, both pan-2-agonal and pan-3-agonal).

**panmagic**hypercube (square, cube, tesseract)- a pan-(1,)
`n`-agonal magic hypercube, where`n`is the dimension of the hypercube.

**Panmagic square**,**panmagic cube**, and**panmagic tesseract**are identical to pandiagonal magic square, pantriagonal magic cube, and panquadragonal magic tesseract, respectively. **strictly magic**hypercube (cube, tesseract)- a 1,2,3,...,
`n`-agonal magic hypercube, where`n`is the dimension of the hypercube. The term**strictly**was suggested by Walter Trump.

**Strictly magic cube**is identical to diagonal magic cube. Every magic square is strictly magic.

**No strictly magic cube**of order 3 or 4 can exist and**no strictly magic tesseract**of order 3, 4, or 5 can exist, either. **Nasik**magic hypercube (square, cube, tesseract)- a pan-1,2,3,...,
`n`-agonal magic hypercube, where`n`is the dimension of the hypercube. Also called**strictly panmagic**hypercube.

**Nasik magic square**,**Nasik magic cube**, and**Nasik magic tesseract**are identical to pandiagonal magic square (or, panmagic square), pan-2,3-agonal magic cube, and pan-2,3,4-agonal magic tesseract, respectively.

Note that the condition of**Nasik**is stronger than that of**panmagic and strictly magic**. **pantriagonal diagonal**magic hypercube (cube, tesseract)- a magic hypercube which is both pantriagonal and diagonal. Harvey D. Heitz calls this property
**PantriagDiag**.

There exists a**pantriagonal diagonal**magic cube which is**not**Nasik. **pan and strictly magic**hypercube (cube, tesseract)- a magic hypercube which is both panmagic and strictly magic, namely, a pan-
`n`-agonal and 1,2,3,...,`n`-agonal`n`-dimensional magic hypercube.

**Pan and strictly magic**cube is identical to PantriagDiag magic cube. **pseudo-pan-**magic hypercube (cube, tesseract)`r`-agonal- a magic hypercube of dimension
`n`such that every pan-`r`-agonal on its oblique (`n`-1)-dimensional slices is magic.

Pseudo-pan-3-agonal is also called**pseudo-pantriagonal**. **pseudo-panmagic**hypercube (cube, tesseract)- a pseudo-pan-
`n`-agonal magic hypercube, where`n`is the dimension of the hypercube.

Every panmagic hypercube is**pseudo-panmagic**. **horizontal-diagonal**magic hypercube (cube, tesseract)- a magic hypercube such that for some aspect of the hypercube, every square of horizontal slices of the hypercube is magic.

Every diagonal magic cube is**horizontal-diagonal**. **horizontal-pandiagonal**magic hypercube (cube, tesseract)- a magic hypercube such that for some aspect of the hypercube, every square of horizontal slices of the hypercube is pandiagonal magic square.

Every pandiagonal magic cube is**horizontal-pandiagonal**. **s-magic**cube- a magic cube such that each of six squares on the surface of the cube is magic. The prefix 's-' means 'surface'. This term is named by Walter Trump.

Every diagonal magic cube is**s-magic**. **simple**magic hypercube (square, cube, tesseract)- a magic hypercube of dimension
`n`which is**not**satisfy any condition of**(pan-)r-agonal**except n-agonal.

**Simple**magic squares are**not**pandiagonal,**simple**magic cubes are neither pantriagonal nor diagonal, and**simple**magic tesseracts are**not**panquadragonal, triagonal, or diagonal. **class**of magic hypercube- classification of magic hypercubes of dimension n by the magicness of
**(pan-)r-agonals**for each r such that 2 <= r <= n.

**Magic squares**are classified by the two classes, simple (2-agonal) and panmagic (pan-2-agonal).

**Magic cubes**are classified by the six classes, simple (3-agonal), pantriagonal (pan-3-agonal), diagonal (2,3-agonal), pantriagonal diagonal (2-agonal and pan-3-agonal), pandiagonal (pan-2-agonal and 3-agonal), and Nasik (pan-2,3-agonal).

**Magic tesseracts**are classified by the 18 classes.

**complement(ary) pair**(of a magic object)- a pair of cells whose sum equals to 2*
`c`, where`c`is the average of all the cells of the magic object. If a magic hypercube of dimension`n`and order`m`is normal in the regular number range, the sum of such a pair equals to`m`^{n}+1.

One number of a**complement pair**is called the**complement(ary) number**of the other number of the pair. **complement(ary) pair line**- the line that join the two numbers of a complement pair of a magic hypercube. The pattern of all
**complement pair lines**of a magic hypercube is called the**complement(ary) pair pattern**of the hypercube. Magic hypercubes are classified by their**complement pair patterns**. **associated (associative)**(magic hypercube (square, cube, tesseract), magic rectangle)- the property that every pair of cells diametrically equidistant from the center of the hypercube or the rectangle is a complement pair. If the order is odd, the double of the center cell need to equal to the sum of complement pairs. Also called a
**symmetrical**magic hypercube. **axis/plane symmetrical**(magic hypercube (square, cube, tesseract), magic rectangle)- the property that every pair of cells diametrically equidistant from some axis/plane of the hypercube or the rectangle is a complement pair.
**complete**magic hypercube (square, cube, tesseract)- a magic hypercube of dimension
`n`and**even**order`m`that every pan-`n`-agonal contains`m`/2 complement pairs spaced`m`/2 apart.**Complete**is**not**defined for an**odd**order magic hypercube.

Every**complete**magic*hypercube*is panmagic. magic hypercube (cube, tesseract)`r`D-compact- an
`n`-dimensional magic hypercube with the feature that the 2^{r}cells of every order-2 subhypercube of dimension`r`, where 2 <=`r`<=`n`, sum to the same value. The smaller`r`is, the stronger this condition is.

For example, in a**2D-compact**magic hypercube, the four cells of every 2x2 subsquare sum to the same value, and, in a**3D-compact**magic hypercube, the eight cells of every 2x2x2 subcube do so. Also called a 2^{r}-**uniform**magic hypercube. 2D-compact is simply called**compact**for magic squares.

Every**2D-compact**magic hypercube is panmagic. On the other hand,**not**everymagic hypercube is panmagic for`r`D-compact`r`> 2.

magic hypercube (square, cube, tesseract)`d`-ply-`r`-agonal- a magic hypercube in which every
`k`-th powered hypercube is`r`-agonal, where`k`= 1,2,...,`d`.

2-ply-`r`-agonal, 3-ply-`r`-agonal,*etc.*are also called**bi-**,`r`-agonal**tri-**,`r`-agonal*etc.*, respectively. hypercube (square, cube, tesseract)`d`-ply-magic`d`-ply-1,`n`-agonal magic hypercube, where`n`is the dimension of the hypercube. The number`d`is called the**degree**of the hypercube. Also called**multimagic**hypercube of**degree**`d`.

2-ply-magic, 3-ply-magic,*etc.*are also called**bimagic**,**trimagic**,*etc.*, respectively.

In ahypercube, every`d`-ply-magic`k`-th powered hypercube is also a magic hypercube, where`k`= 1,2,...,`d`.**strictly**hypercube (square, cube, tesseract)`d`-ply-magic`d`-ply-1,2,...,`n`-agonal magic hypercube, where`n`is the dimension of the*hypercube*. Also called**strictly multimagic**hypercube of**degree**`d`.

stricty 2-ply-magic, strictly-3-ply-magic,*etc.*is also called**strictly bimagic**,**strictly trimagic***etc.*, recpectively.

In a**strictly**hypercube, every`d`-ply-magic`k`-th powered hypercube is also a strictly magic hypercube, where`k`= 1,2,...,`d`.

**inlaid**magic hypercube (square, cube, tesseract)- a magic hypercube that contain another magic hypercube of lower order within it.

In an**inlaid**magic hypercube, we call the hypercube itself the**main**hypercube and a hypercube inlaid within it a**sub**hypercube. **concentric**magic hypercube (square, cube, tesseract)- an inlaid magic hypercube that contain sub magic hypercubes of dimension
`n`and orders`m`-2,`m`-4, ... concentrically, where`n`and`m`is the dimension and the order of the main hypercube, respectively. The least order of the sub magic hypercubes concentrically inlaid is 3 or 4. **bordered**magic hypercube (square, cube, tesseract)- a concentric magic hypercube in which every concentric sub hypercube consists of consecutive integers. Also called
**consecutively concentric**magic hypercube.

Every sub hypercube of a**bordered**magic hypercube can transform into a normal magic hypercube by subtracting some constant from all cells of the sub hypercube.

**Note**: The term**bordered**is often used as a synonym of concentric, but this site distingushes the two terms as above. **bordered diagonal magic cube**- a bordered magic cube which is diagonal and every subcube whose order is higher than 4 is also diagonal. (It is impossible to make the subcube of order 3 or 4 diagonal.)

This magic cube was first constructed by the author in 2004.

The author proved that a**bordered diagonal magic cube**exists for any**even**order higher than 5.

**XmlHypercube format**- a description format by Aale de Winkel to descript magic hypercubes. In the old version, it was called
*HyperCube Language*. **perfect**magic*hypercube*(*square*,*cube*,*tesseract*)- People use this term in plural senses. This site does not use the term
*perfect*except the case of quotation so as not to confuse the reader.

Harvey D. Heinz & John R. Hendricks,

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This page was last updated on August 8, 2016.

"Magic Cubes and Tesseracts" http://magcube.la.coocan.jp/magcube/en/

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