
Following is a list of some 125 definitions of terms
relating to magic squares, cubes, stars, etc. It puts in one location both
traditional and modern terminology along with explanations of its usage.
Where I felt it would be appropriate, I have
included a source reference.
In some cases I have included relevant facts.
In the definitions, bold type indicates a term that
has its own definition.
Unless I specifically indicate otherwise, all
references to magic squares mean normal (pure) magic squares composed of
the natural numbers from 1 to m^{2}. Likewise for cubes,
tesseracts, etc.
I welcome your comments, both constructive criticism
and suggestions for additional definitions or improvements in the wording
of a definition.
NOTE: Traditionally, in magic square circles, the letter
n has been used to
denote the order of the square. Studying magic rectilinear figures in
higher dimensions (hypercubes) has become increasingly popular.
In the 1990's, magic hypercube guru John Hendricks started using
the letter m for the order of a magic square, cube, etc., and
reserving the letter n for dimension. This convention is
gradually becoming more popular, so I have now changed all references
for n as order to m.
( I have maintained n as the order of magic stars as they are
only 2 dimensional.) 
.
Magic Square Lexicon: Illustrated is an expanded
version of this Web page. It contains 239 definitions, and about 200
illustrations. See details of this book at Book for
Sale. 
agonal 
Root word for a line that goes from one corner to an opposite corner of
a hypercube. Examples: diagonal, triagonal, quadragonal, ragonal, nagonal.
By extension, a 1agonal is an orthogonal line i.e. a row, column, etc.
Hendricks, John R., The Pan4agonal Magic Tesseract, The American
Mathematical Monthly, Vol. 75, No. 4, April 1968, p. 384.
John R. Hendricks, The Pan3Agonal Magic Cube, JRM 5:1:1972, pp
5154

Almostmagic Stars 
A magic pentagram (5pointed star), we now know, must have 5
lines summing to an equal value.
However, such a figure cannot be constructed using consecutive integers.
Charles Trigg calls a pentagram with only 4 lines with equal sums but
constructed with the consecutive numbers from 1 to 10, an almostmagic
pentagram.
Charles W. Trigg, JRM29:1, p. 811, 1998
Marián Trenkler (Safarik
University, Slovakia) has independently coined the phrase almostmagic, but
generalizes it for all orders of stars.
His definition: If there are numbers 1, 2, …, 2n located in a star Sn
( or Tn) so that the sum on n – 2 lines is 4n + 2, on
the others 4n + 1 and 4n + 3, we call it an almostmagic star.
NOTE that by his definition, the order5 almostmagic star has only 3 lines
summing correctly.
Trigg’s order5 (the only order he defines) requires 4 lines summing the
same.
Marián Trenkler, Magicke Hviezdy (Magic stars), Obsory
Matematiky, Fyziky a Informatiky, 51(1998).
See my pages on Almostmagic Stars.

Almostmagic Squares 
An array of consecutive numbers, from 1 to m^{2},
where the rows, columns and two main diagonals sum to a set of 2(m +
1) consecutive integers. Antimagic squares are a subset of
heterosquares.
Joseph S. Madachy, Mathemaics On Vacation, pp 101110.
(Also JRM 15:4, p.302)

Antimagic
Squares 
An antimagic
square is a subset of the
heterosquare in which the 3m+2. line sums
are distinct numbers in consecutive order.
Joseph S. Madachy, Mathemaics On Vacation, 1966, pp
101110. (Also JRM 15:4, p.302)

Aspect 
An aspect is an apparently
different but in reality only a disguised version of the magic
square, cube, tesseract, etc. It is obtained by rotations and/or
reflections of the basic figure.
See Normalizing and Standard position.

Associated Magic Cubes, Tesseracts, etc. 
Features are the same as those for the associated
magic square.
There are 4 fundamental (essentially different) order3
magic cubes. Each of these can appear in 48 aspects due to rotations and
reflections.
There are 58 essentially different order3 magic tesseracts (4th dimension).
Each of these can appear in 384 aspects due to rotations and reflections.
Just as the 1 order3 magic square is associated, so to are the 4 order3
magic cubes and the 58 order3 magic tesseracts.
All of these figures can be converted to another aspect by complimenting
each number (the selfsimilar feature).

Associated Magic Squares 
A magic square where all pairs of cells diametrically
equidistant from the center of the square equal the sum of the first and
last terms of the series, or m^{2} + 1. Also called
Symmetrical or centersymmetric. The center cell of odd
order associated magic squares is always equal to the middle number of the
series. Therefore the sum of each pair is equal to 2 times the center cell.
In an order5 magic square, the sum of the 2 symmetrical pairs plus the
center cell is equal to the constant, and any two symmetrical pairs
plus the center cell sum to the constant. i.e. the two pairs do not have to
be symmetrical to each other.
In an even order magic square the sum of any 2 symmetrical pairs will
equal the constant (the sum of the 2 members of a symmetrical pair is equal
to the sum of the first and last terms of the series).
As with any magic square, each associated magic square has 8 aspects due to
rotations and reflections. any such magic square can be converted to another
aspect by complimenting each number (the selfsimilar
feature).
There are NO singlyeven associated magic squares.
All even order associated magic squares are semipandiagonal.
The one order3 magic square is associative.
There are 48 order4 associative magic squares.
Order5 is the smallest that has associated, pandiagonal magic squares, and
only 400 of the 3600 pandiagonal magic squares are also associated. None of
the 36 essentially different magic squares of this order are
associated.
W. S. Andrews, Magic squares & Cubes, 1917
Benson & Jacoby, New Recreations with Magic Squares, 1976

Basic Magic Square

See Fundamental Magic Square.

Bent diagonals 
Diagonals that proceed only to the center of the magic
square and then change direction by 90 degrees. For example, with an order8
magic square, starting from the top left corner, one bent diagonal would
consist of the first 4 cells down to the right, then the next 4 cells would
go up to the right, ending in the top right corner. Another bent diagonal
would consist of the same first 4 cells down to the right, then the next 4
cells would go down to the left, ending in the bottom left corner.
Bent
diagonals are the prominent feature of Franklin magic squares.

Bimagic Square 
If a certain magic square is still magic when each integer
is raised to the second power, it is called bimagic. If (in addition
to being bimagic) the integers in the square can be raised to the third
power and the resulting square is still magic, the square is then called a
trimagic square. These squares are also referred to as doublemagic
and triplemagic. To date the smallest bimagic square seems to be
order 8, and the smallest trimagic square is order 12.
See my multimagic page.
Aale de Winkel reports, based on John Hendricks
digital equations, that there are 68,016 order9 bimagic squares.
email of May 14, 2000

Bordered Magic Square 
It is possible to form a magic square (of any odd or even
order) and then put a border of cells around it so that you get a new magic
square of order m + 2 (and in fact keep doing this indefinitely). The
center magic square is always an associated magic square but is never a
normal magic square because it must contain the middle numbers in the
series. i.e. There must be (m^{2} 1)/2 lowest numbers and
their complements (the highest numbers) in the border where m^{2}
is the order of the square the border surrounds. This applies to each
border. The outside border is called the first border and the borders are
numbered from the outside in.
When a border (or borders) is removed from a
Bordered magic square, the square is still magic (although no longer
normal).
The Bordered Magic Square is often called a Concentric Magic
Square but modern usage considers them different.
Benson & Jacoby, New Recreations in Magic Squares,
1976, pp 2633
W. S. Andrews, Magic squares & Cubes, 1917
There are 174,240 border squares out of the 549,504 order 5 magic squares
and already 567,705,600 order 6 magic squares constructed.
J.L.Fults, Magic Squares, 1974

Broken diagonal pair 
Two short diagonals that are parallel to but on opposite
sides of a main diagonal and together contain the same number of cells as
are contained in each row, column and main diagonal (i.e. the order).
These are sometimes referred to as pandiagonals, and are the
prominent feature of Pandiagonal magic squares.
J. L. Fults, Magic Squares, 1974

Cell 
The basic element of a magic square, magic cube, magic
star, etc. Each cell contains one number, usually an integer.
There are m^{2} cells in a magic square of order m,
m^{3} cells in a magic cube, m^{4} cells
in a magic tesseract, 2n cells in a magic star, etc.
RouseBall & Coxeter, Mathematical Recreations and Essays,
1892, 13 Edition, p.194

Column 
Each vertical sequence of numbers. There are m
columns of height m in an orderm magic square.

Compact 
Gakuho Abe used this term for a magic square where the four
cells of all 2x2 squares contained within it summed to S.
Note that this is a requirement for Ollerenshaw’s mostperfect magic
squares.
Gakuho
Abe, Fifty Problems of Magic Squares, Self published 1950. Later
republished in Discrete Math, 127, 1994, pp 313.
Addendum; In April 2007 Aale de Winkel proved that corners of all
rectangular shapes in a compact magic square are panmagic.
Addendum; In April 2008, Dwane Campbell independently also proved the above.
And in May/08 this editor extended the concept to odd
order pandiagonal magic squares by including the center cell in the pattern.
See Pandiag.htm.

Compactplus 
Refers to a magic cube when the eight corners of all
orders of subcubes contained within a cube, including wraparound, sum to
S. I have adapted this term from Gakuho Abe’s [1] term ‘compact’
which he used to indicate that all 2x2 squares sum to S.
Many cubes have the 8 corners of all subcubes of one
or several orders sum to 8S/m where m is the order of the
parent cube. All subcubes from orders 2 to 8 sum
correctly in an order 8 ‘perfect’ magic cube. This includes wraparound, so
in effect there are 64 subcubes of each order.
Kanji Setsuda uses the term ‘composite’ for this
feature in magic cubes but I feel that this can cause confusion with
‘composite’ magic squares.
Kanji Setsuda’s Compact (composite) and Complete magic Cubes Web pages may
be accessed from here.
http://homepage2.nifty.com/KanjiSetsuda/pages/EnglishP1.html

Complete 
This definition also applies to magic cubes. Every
pantriagonal contains m/2 complement pairs, spaced m/2 apart.
Note that this is a requirement for Ollerenshaw’s mostperfect magic
squares. Coined by Kanji Setsuda.
Obviously, this feature appears only in
even order cubes.
Years before, McClintock had defined
‘complete’ squares as pandiagonal magic squares with two additional
properties: all 2x2 subsquares sum to the same value which is 2m^{2}+2,
where m is the order of the magic square, and the integers
come in complementary pairs distanced ½m along the diagonals.
(See my definition for mostperfect’.
McClintock, E.
1897, On the Most Perfect Forms of Magic Squares, with Methods for their
Production. , A., J. Math. 19, 99120 (see p. 116)

Complementary Numbers 
In a normal magic square, the first and last numbers in the
series are complementary numbers. Their sum forms the next number in the
series (m^{2} + 1). All other pairs of numbers which also sum
to m^{2} + 1 are also complementary.

Composition
Magic Square 
It is simple to construct magic squares of order mn (m times n)
where m and n are themselves magic squares.
For a normal magic
square of this type, the series used is from 1 to (mn)^{2}.
An order 9 composite magic square would consist of 9 order 3 magic squares
themselves arranged as an order 3 magic square and using the series from 1
to 81.
An order 12 composite magic square could be made from 9 order 4 magic
squares by arranging the order 4 squares themselves as an order 3 square (or
12 order 3 magic squares arranged as an order 4 magic square). In either
case, the series used would be from 1 to 144.

Concentric
Magic Square 
The center square (or squares) consist of nonconsecutive numbers in a
concentric magic square. In a bordered magic square, these central squares
contain consecutive numbers. See Bordered Magic Square.

Constant (S) 
The sum produced by each row, column, and main diagonal (and possibly
other arrangements). Also called the magic sum.
The constant (S) of a normal magic square is (m^{3}+m)/2
If the magic square consists of consecutive numbers, but not starting at
1, the constant is (m^{3}+m)/2+m(a1) where
a equals the starting number.
If the magic square consists of numbers with a fixed increment, then
S = am + b(m/2)(m^{2}1) where a
= starting number and b = increment.
See Series.
For a normal magic square, S = ½ m(m^{2}+1).
For a normal magic cube, S = ½ m(m^{3}+1).
For a tesseract S = ½ m(m^{4}+1).
In general; for a ddimensional hypercube S = ½ m(m^{n}+1)
For a normal magic star, when n is the order, S = 4n +
2. See Magic
Star Definitions.

Continuous M. S. 
Seldom used now. See Pandiagonal Magic Square.

Cyclical Permutations 
A pandiagonal magic square may be converted to another by simply moving
one row or column to the opposite side of the square. For example, an
order5 pandiagonal magic square may be converted to 24 other pandiagonal
magic squares. Any of the 25 numbers in the square may be brought to the top
left corner (or any other position) by this method. See also
Transformations and Transposition.

Diabolic M. S. 
Seldom used now. See Pandiagonal Magic Square.

Diagonal 
The line that goes through the middle of a magic square, from a corner to
the opposite corner.
The basic requirement for a square to be magic, is that these two lines sum
correctly, along with the n rows and n columns.
See also Broken, Leading, Main, Right, Opposite Short,
Short.

Diagonal
Magic Cube 
One of the main classes of magic cubes (as defined by John Hendricks). A
diagonal magic cube is one where both main diagonals are correct in all
planar arrays. This means that there are 3m orthogonal simple magic
squares in the magic cube. The Myers cube is a well known example of
this type.
By the older (still quite common) definition, these cubes were called
'perfect'. Of course, so were Hendricks 'pandiagonal magic cube and
perfect magic cube!

Diametrically
Equidistant 
A pair of cells the same distance from, but on opposite sides of the
center, of the magic square. Other terms meaning the same thing are skew
related and symmetrical cells.

Disguised M. S. 
See Fundamental magic square.

Division
Magic Square 
Construct the same as the multiply magic square, then interchange
diagonal opposite corners. Now, by multiplying the outside numbers of each
line, and dividing by the middle number, the constant is obtained.

Double M. S.
Triple M. S.

See Bimagic magic square.

Doubly Even 
The order (side) of the magic square is evenly divisible by 4. i.e. 4,
8, 12, etc.
Probably the easiest type of magic square to construct.

Essentially Different 
There are 36 essentially different order5 pandiagonal magic
squares each of which have 99 variations (total of 100 aspects) by
permutations of the rows, columns and diagonals. These 3600 magic squares
are all Fundamental because each one still has it’s 3 rotations
and 4 reflections. A magic square is essentially different when;
The number in the top lefthand corner is 1,
The number in the cell next to the 1 in the top row is less then any
other number in the top row, in the left hand column or in the diagonal
containing the 1, and
The number in the lefthand column of the second row is less then the
number in the lefthand column of the last row.
Benson & Jacoby, New Recreations with Magic Squares,
1976, p 129.

Eulerian square 
See GraecoLatin square.

Even Order 
The order (side) of the magic square is evenly divisible by two.

Expansion Band 
See Framed Magic Square.
If used in a magic cube, Hendricks refers to the expansion band as an
expansion shell.
J.R.Hendricks, Inlaid Magic Squares and Cubes, 1999

Files 
The fourth dimension lines of numbers in a tesseract, or
higher order hypercube. Analogous to rows and columns, the x
and y direction lines of numbers in a magic square or cube and pillars,
the z direction in a magic cube.
J.R.Hendricks, Magic Squares to Tesseracts by Computer,
1998

Framed
Magic Square 
A subset of Inlaid magic square where an expansion band
of numbers is placed around the inlaid magic square. Or the frame may be
designed first, leaving room for the inlaid squares. The frame may be
one, two, or even more rows and columns thick.
Unlike a Bordered magic square, the interior square may be a
Normal magic square. Of course the total of all the cells in each row,
column, and main diagonal, including the cells in the frame, must sum
correctly to the constant.
J.R.Hendricks, Inlaid Magic Squares and Cubes, 1999

Franklin Magic Square 
A type of magic square designed by Benjamin Franklin in which there are
many combinations that sum to the constant, the most prominent being bent
diagonals. However, they are only semimagical, as the main
diagonals do not sum correctly.
The neverbefore published order 16
Franklin square discovered by Paul Pasles does have correct main diagonals
and so is a magic square. It is on my Franklin page.

Fundamental Magic cube, tesseract, etc 
There are 4 fundamental (basic) magic cubes of order3.
Each may be disguised to make 48 other (apparently) different
magic cubes by means of rotations and reflections. These variations are NOT
considered new magic cubes for purposes of enumeration, but are referred to
as aspects.
There are 58 fundamental (basic) magic tesseracts of
order3. Each may be disguised to make 384 other (apparently)
different magic tesseracts by means of rotations and reflections. 
Fundamental Magic Square 
There is 1 fundamental (basic) magic square of order3 and 880 of
order4, each with 7 variations (aspects) due to rotations and
reflections.
In fact, any magic square may be disguised to make 7 other
(apparently) different magic squares by means of rotations and
reflections. These variations are NOT considered new magic squares for
purposes of enumeration. Also known as Basic Magic Square.
Any of the eight variations may be considered the fundamental one.
However, see Standard Position, magic square and Index.

Fundamental Magic Star 
A magic star may be disguised to make 2n1 apparently different
magic stars where n is the order (number of points) of the magic
star.
These variations are NOT considered new magic stars for purposes of
enumeration. Also known as Basic Magic Star.
Any of these 2n variations may be considered the fundamental one.
However, see Standard Position, magic star and Index.

Geometric
Magic Square 
Instead of using numbers in arithmetic progression as in a Normal
Magic Square , a geometric progression is used. These progressions may
be exponential or ratio.
In the exponent type the numbers in the cells consist of a base value and an
exponent. The base value is the same in each cell. The exponents are the
numbers in a regular magic square.
The ratio type uses a ratio for the horizontal step and a ratio for
the vertical step.
The constant is obtained by multiplying the cell contents.
W.S.Andrews, Magic Squares and Cubes, 1917, pp283294
discusses this type of magic square.

GraecoLatin Square 
When two Latin squares are constructed, one with Latin letters
and one with Greek letters, in such a way that when superposed, each Latin
letter appears once and only once with each Greek letter, the resulting
square is called a GraecoLatin square. This type of square is sometimes
referred to as a Eulerian square.
This type is often used to generate magic squares by assigning suitable
integers to the letters.
For convenience, upper case letters are often used for the one square and
lower case letters for the other one.
See Regular & Irregular.
Martin Gardner, New Mathematical Diversions from
Scientific American, 1966, Euler’s Spoilers: The Discovery of an Order10
GraecoLatin Square.

Heterosquare 
Similar to a magic square except all rows, columns, and
main diagonals sum to different (not necessarily consecutive) integers.
If the 2m+2 line sums are distinct from the
interior numbers of the square, it is called a 'real' heterosquare, a term
coined by Peter Bartsch.
Two
simple methods of generating an order 3 heterosquare is to write the natural
numbers from 1 to 9 in a spiral, starting from a corner and moving inward,
or starting from the center and moving out.
A special form of heterosquare (a subset) is the antimagic
square. For this type of
square, the line sums must be consecutive numbers.
Joseph S. Madachy, Mathemaics On Vacation, 1966, pp
101110. (Also JRM 15:4, p.302)

Horizontal step 
The difference between adjacent numbers in each series.
It is not a reference to the columns of the magic square.
In a normal magic square, the horizontal step and
vertical step are both 1.
J. L. Fults, Magic Squares, 1974

Horizontally paired

Two cells in the same row and the same distance from the
center of the magic square.

Hypercube 
A geometric figure consisting of all angles right and all
sides equal.
A square, cube and tesseract are
hypercubes of two, three and four dimensions.

Impure M. S. 
The numbers composing the magic square are not integers or
are not in the range from 1 to m^{2}.i.e. are not consecutive
or the series does not start at 1.
It may contain n series of n numbers where the
horizontal and/or vertical steps are not 1, or it may contain
numbers with random spacing between them.

Indian M. S. 
See Pandiagonal Magic Square.

Index 
The position in a list of magic squares of a given order
where a given magic square fits, after it has been converted to the
standard position. The correct placement or index of magic squares is
determined by comparing each cell of two magic square of
the same order starting with the top leftmost cell and proceeding across
the top row, then across the second row, etc. until the two corresponding
cells differ. The magic square with the smallest value in this cell is then
given the lower index number. See also Fundamental and
Standard position.
The index was designed by Bernard Frénicle de Bessy and published
posthumously in 1693 with the 880 basic solutions for the order4
magic square.
Magic stars may be indexed in a similar fashion.
Obviously, only normal magic squares and magic stars may be indexed.
Benson & Jacoby New Recreations with Magic Squares, 1976,
p.123124.

Inlaid
Magic Square 
A magic square that contains within it other magic squares.
However, unlike a bordered magic square, where the border must
contain the lowest and highest numbers in the series, there is no such
restriction on the inlaid magic square. The inlaid square may even be
a normal magic square. Inlays are often placed in the
quadrants of a magic square, and the inlays may themselves contain
inlays.
Overlapping magic squares are a form of Inlaid
and Patchwork magic squares.
J.R.Hendricks, Inlaid Magic Squares and Cubes, 1999

Irregular 
See Regular & Irregular

Isolike Magic Stars 
An order8B type magic star can be
constructed by a systematic transformation of magic squares of certain
orders. This is a broad term that covers cases where all the numbers are not
used or some numbers are duplicated. The resulting star has either 8, 10 or
12 lines of n numbers that sum correctly.
They may be constructed from diamagic or plusmagic,
quadrant magic squares of odd orders greater then 5 (orders 4 and 5
produce isomorphic magic squares). Because the magic square contains more
numbers then can appear in the star, not all numbers are used. Their
discovery was a direct result of Aale de Winkel’s work with panmagic
stars which use all the numbers but require the use of duplicate
numbers.
Actually, such a star, but without two of the diagonal lines (only
10 lines) can be constructed from a suitable order9 magic square.
See my page on
Isolike Magic Stars for samples and more information.
Go to Aale de Winkel’s Magic Object pages from my
Links page.

Isomorphic
Magic Stars 
An order8B type magic star then can be constructed by a
systematic transformation using all the numbers of a magic square.
If the magic square is order4 then the resulting star has 8 lines
of 4 numbers that sum correctly.
See one at
Unusual Magic Squares.
If the originating magic square is order5, it must be a plusmagic
quadrant magic square and the resulting star has 12 lines of 5
numbers summing correctly.
In both cases all the numbers in the magic square are used to form
the star.

Jaina
Magic Square 
Named for the first type of this square found as a Jaina
inscription in the City of Khajuraho, India. This term is seldom used now.
See Pandiagonal Magic Squares.

Latin Square 
An m x m array of m symbols in which
each symbol appears exactly once in each row and each column of the array. A
set of two Latin squares are frequently used for generating magic squares.
See GraecoLatin square.

Leading Diagonal 
Also called left diagonal. The line of numbers from
the upper left corner of the magic square to the lower right corner. See
Main Diagonals.

Lines of Numbers 
In a magic square, cube, tesseract or
hypercube these are normally referred to as rows, columns,
diagonals, pillars, files, triagonals,
quadragonals, etc. Each line contains n numbers where n is
the order of the magic array.
In a magic star they are the set of numbers forming a line between
two points.
In a normal magic star there is always four of these numbers,
regardless of the order of the star. An ornamental magic star may
have a set of any size.

Lozenge
Magic Square 
An odd order magic square where all the odd numbers are
arranged sequentially to occupy a 45 degree rotated square in the center of
the complete magic square. The (n^{2}1)/8 cells in each of
the corner areas contain the even numbers. 
m 
Used to indicate the order of a magic
hypercube.
Traditionally this function was performed by n.
However, with the recent popularity of higher dimension hypercubes, some
writers (notably J. R. Hendricks) have started using m for
this purpose, thus making n available for indicating
dimension.

Magic Circle, Hexagon, Cross 
Various arrangements of numbers, usually the
first n integers, where all lines or points add up to the same
constant value.

Magic Cube, Normal 
Similar to a magic square but 3 dimensional
instead of two. It contains the integers from 1 to m^{3}.
There are 3m^{2 }+ 4 lines that sum correctly. All rows,
columns, pillars, and the four triagonals must sum to 1/2m(m^{3}+1)
(the constant). The minor diagonals do not sum correctly although it
is possible that those in only one plane do.
There are 4 basic magic cubes of order3, each of which can be
shown in 48 aspects due to rotations and/or reflections.
J.R.Hendricks, Inlaid Magic Squares and Cubes, 1999
J.R.Hendricks, Magic Squares to Tesseracts by Computer, 1998
Benson & Jacoby, Magic Cubes:New Recreations, 1981

Magic Hypercube 
A magic square, cube, tesseract, or higher
dimension rectilinear object where all orthogonal lines and all
nagonals sum to a constant (n = dimension).
There are 2 main classes of magic squares,
6 main classes of magic cubes, and
18 main
classes of magic tesseracts.
See Mitsutoshi Nakamurs's site at
http://homepage2.nifty.com/googol/magcube/en/classes.htm

Magic Lines 
Lines connecting the centers of cells of a
Pure Magic Square. The line diagrams produced may be used for purposes
of classification.
If the areas between the lines are filled with contrasting colors,
interesting abstract patterns result. These are called sequence patterns.
Jim Moran, Magic Squares, 1981
Another type of line pattern is used for classification. It was first used
by H.E. Dudeney to classify the 880 order 4 magic squares into 12 groups. In
this method, each pair of complementary numbers are joined by a line.
The resulting combination of lines forms a distinct pattern
H.E.Dudeney, Amusements in Mathematics, 1917, p 120
Jim Moran Magic Squares, 1981, 0394747984 (lots of material)

Magic Cube Ratios 
These two terms were defined by Walter
Trump in January, 2004. Their value is mainly for cubes that are almost
magic. They are also of value for cubes that are simple magic but not quite
diagonal magic (magic ratio). Also for measuring magic cubes against a
perfect (nasik) cube (panmagic ratio).
Magic cube ratio
The magic ratio is the number of correct monagonals, diagonals, and
triagonals
divided by the highest possible, which is 3m^{2} monagonals +
6m diagonals + 4 triagonals.
Panmagic cube ratio
The panmagic ratio is the number of correct monagonals, pandiagonals, and
pantriagonals divided by the highest possible, which is 3m^{2}
monagonals + 6m^{2} pandiagonals + 4m^{2}
pantriagonals.

Magic Rectangle 
A rectangular array of cells numbered from 1
to m. All rows sum to the value which is the mean of all cell values
times the number of cells in the row. Likewise, all columns sum to the value
which is the mean of all cell values times the number of cells in the
column. Neither Andrews, Collison, Hendricks, Moran or Trenkler require that
the diagonals be magic.
However, Shineman, in a letter dated March 27, 2000, provided a 4 x
16 magic rectangle in which 4 equally spaced leading and 4 equally spaced
right diagonals each summed correctly.
Aale de Winkel has researched this subject and refers to them as
Magic Beams (usually in a multidimensional context). Go to his Magic Object
pages from my links page.

Magic Square 
An m x m array of cells
with each cell containing a number. These numbers are arranged so that the
sum for each row, each column, and the two main diagonals
are all the same. 
Magic Square, Normal

A magic square composed of the natural
numbers from 1 to m^{2}. Also called pure, or
traditional. 
Magic Sum 
The value each row, column, etc., sums to. It
is denoted by S. See constant
For a magic star, S is the sum of the numbers in each line.

Magic Tesseract 
A magic tesseract is a fourdimensional array,
equivalent to the magic cube and magic square of lower
dimensions, containing the numbers 1, 2, 3, …, m^{4} arranged
in such a way that the sum of the numbers in each of the m^{3}
rows, m^{3} columns, m^{3}
pillars, m^{3} files and in the eight major
quadragonals passing through the center and joining opposite corners is
a constant sum S, called the magic sum, which is
given by: S = ½ m(m^{4}+1) and where n is
called the order of the tesseract.
There are 58 basic magic tesseracts of order3. Each may be shown
in 384 aspects due to rotations and/or reflections.
There is 1 basic hypercube of dimension2, order3,
with 8 aspects.
There are 4 basic hypercubes of dimension3, order3, each with 48
aspects.
There are 58 basic hypercubes of dimension4, order3, each with 384
aspects.
There are 2992 basic hypercubes of dimension5, order3, each with 3840
aspects.
There are 543328 basic hypercubes of dimension6, order3, each with 46080
aspects.J.R.Hendricks, Magic Squares to Tesseracts by
Computer, 1998
J.R.Hendricks, All thirdorder Magic Tesseracts, 1999
C.Planck (W.S.Andrews, Magic Squares and
Cubes,1917, pp 363375) refers to these as
octahedroids, and their space diagonals as hyperdiagonals.

Main Diagonals 
The two diagonal series of cells that go from
corner to corner of the magic square.
Each must sum to the constant in order for the array to be magic.
The leading (or left) diagonal is the one from upper left to
lower right.
The right diagonal is the one from lower left to upper right.

MostPerfect Magic Square 
A normal pandiagonal magic square of
doublyeven order with two added properties. Any twobytwo block
of adjacent cells (including wraparound) sum to the same value which
is 2m^{2}+2, where m is the order of the magic
square, and the integers come in complementary pairs distanced ½m
along the diagonals.
K. Ollerenshaw and D. Brée, MostPerfect
Pandiagonal Magic Squares, 1998
Ian Stewart, Mathematical Recreations,
Scientific American, November 1999
Note that both these authors use the series from 0 to m^{2}2
for mathematical convenience. The sum of each 2 by 2 square array is then 2m^{2}2.
See also Reversible Square.

Multiplication Magic Square 
A magic square where the constant is
obtained by multiplying the values in the cells. Also called a geometric
magic square.

Myers Cube 
A magic cube were all 3m squares are
simple magic. All six oblique squares are also simple magic, or one may be
pandiagonal magic. This type of cube is now referred to as a Diagonal
magic cube.
C. Boyer and W. Trump refer to this type of cube as perfect.

n 
Traditionally used to indicate the order
of a magic array. Many hobbyists now use m for this purpose,
reserving n to indicate dimension. Continue to use n
for order of magic stars.

Nasik 
Nasik is an unambiguous
and preferred alternative to Hendricks term perfect for magic
squares, cubes, tesseracts, etc., where all possible lines sum to a
constant. It is a refinement to Frost's use which applied to all classes
of cubes with pandiagonallike features. For more information see my
Theory of Paths Nasik.
C. Planck,
The Theory of Path Nasiks, Printed
for private circulation by A. J. Lawrence, Printer, Rugby (England),1905
(Available from The University Library, Cambridge).

Nasik
Magic Square 
The term is seldom used now in relation to m.s.
(but see nasik). See Pandiagonal Magic Square.
This term was coined by Rev. A. H. Frost for the town in India where he
served as a missionary.
A.H.Frost, On the General Properties of Nasik Squares,
Quarterly Journal of Mathematics, 15, 1878, pp 3449.

Normal 
When used in reference to a magic square,
magic cube, magic star, etc, it indicates the magic array uses
consecutive positive integers starting with 1. An equally popular term for
this condition is pure.

Normalized position

See Standard position. 
Normalizing 
Rotating and /or reflecting a
magic square or magic star to achieve the standard position
so the figure may be assigned an index number.

Octants 
The eight parts of a doublyeven order magic
cube if you split the cube in half in each dimension. i.e. if you divide an
order8 cube in this fashion, the octants are the eight order4 cubes
positioned at each of the eight corners of the original cube.
J.R.Hendricks, Inlaid Magic Squares and Cubes, 1999

Opposite short diagonal pairs 
Two short diagonals that are parallel to but
on opposite sides of a main diagonal and each containing the same number of
cells. See SemiPandiagonal.
J. L. Fults, Magic Squares, 1974

Order m 
Indicates the number of cells per side of the
magic square, cube, tesseract, etc. (But see order n.)

Order n 
n traditionally indicated the
number of cells per side of the magic square, cube, tesseract, etc. m
is now used increasingly for this purpose.
For a magic star, n indicates the number of points in the
star pattern.

Order,
Doublyeven

The order is evenly divisible by 4. i.e. 4, 8,
12, etc. Probably the easiest to construct.

Order, Odd 
The order is not divisible by 2, i.e. 3 (the
smallest possible magic square), 5, 7, etc.

Order,
Singlyeven 
The order is evenly divisible by 2 but not by
4. i.e. 6, 10, 14, etc. This order is by far the hardest to construct.

Ornamental Magic Square

A general term for magic squares containing
unusual features. Some examples are; Bordered, Composition, Inlaid,
Lozenge, Overlapping, Reversible, Serrated. 
Ornamental Magic Star 
Any Magic Star containing unusual features. It
may have one star embedded in another, more then four numbers to a line,
consist of prime numbers (or any unusual number series), etc.

Overlapping Magic Square 
A special type of inlaid magic square
where 1 square partially (or completely) overlaps another magic square
(probably of a different order). See Andrews, Magic
Squares & Cubes, 1917, p.276 for a combination of 4 m.s. & p.240 for a 13
square combination.

Pandiagonals 
See Broken diagonal pairs

Pandiagonal Magic Square 
Also known as Diabolic, Nasic, Continuous,
Indian, Jaina or Perfect. To be pandiagonal, the broken
diagonal pairs must also sum to the constant. This is considered
the top class of magic squares.
Some pandiagonal magic squares are also associative
(order 5 & higher) . Also some are Mostperfect (doublyeven
orders only). There are 4n lines that sum
correctly (n rows, n columns and 2n
diagonals).
There is only 1 basic order 3 magic square and it is not
pandiagonal.
Of the 880 basic order 4 magic squares, only 48 are pandiagonal and
none of these are associative.
Order5 has 3600 basic pandiagonal magic squares (Only 36 essentially
different).
Order7 has 678,222,720 basic pandiagonal magic squares.
Order8 has more then 6,500,000,000 pandiagonal magic squares.
There are NO singlyeven normal pandiagonal magic
squares This was proved in 1878 by A. H. Frost , and more elegantly by C.
Planck in 1919 .
It was thought that there are no order 9 normal pandiagonal magic squares,
but in 1998 Gahuka Abe discovered a whole class of such squares.
All the above assume we are considering only normal,
Fundamental magic squares.
Another term for this type of square is nasik, which implies
perfect (all possible lines sum to S).
A. H. Frost, On the General Properties of
Nasik Squares, Quarterly Journal of Mathematics, 15, 1878, 3449.
C. Planck in 1919 The Monest 29, 307316.

Pandiagonal Magic Cube 
A Pandiagonal Magic Cube has the normal requirements of a
magic cube plus the additional one that all the squares (planes) also
be pandiagonal. Remember that an ordinary magic cube does not require
even the main diagonals of these squares to be correct.
There are 9m^{2} + 4 lines that sum correctly (m^{2}
rows, m^{2} columns, m^{2} pillars, 4 main
triagonals and 6m^{2} Diagonals). Order7 is the smallest
possible order pandiagonal magic cube.
This is one of the original definitions of a
Perfect Magic Cube.
Rev. A. H. Frost published an order 7 pandiagonal magic cube in 1866!
J.R.Hendricks, Magic Squares to Tesseracts by
Computer, 1999
A. H. Frost, Invention of Magic Cubes, Quarterly
Journal of Mathematics, 7, 1866, 92102.

Panmagic Stars 
An order8B type magic star then can be constructed by a
systematic transformation of oddorder pandiagonal magic squares greater
then order5.
Aale de Winkel investigated this type of magic star in the spring
of 1999 which later resulted in his and my joint investigation of Isolike
magic stars.
Unlike isomagic stars which cannot use all the numbers,
panmagic stars usually use all the numbers in the originating magic
square but require the use of duplicate numbers to complete the pattern.
A variation is what Aale calls the butterfly. See my
Isolike Magic Stars for more information.
Go to his page on Panmagic Stars from my links page.

Panquadragonals 
Broken quadragonal sets that are parallel to a
quadragonal and that sum to the magic constant. A set may consist of 2,
3, or 4 segments that together contain m cells. If all these sets sum
correctly, the magic tesseract is panquadragonal. It is
analogous to a pandiagonal magic square but instead of moving a row
or column from one side to the other and maintaining the magic properties,
you move any cube from one side to the other.
See also, Pantriagonals.
J.R.Hendricks, Magic Squares to Tesseracts by
Computer, 1999

Pantriagonals 
Broken triagonal sets of lines of a magic cube that
are parallel to a triagonal and that sum to the magic constant. Such
a set may consist of 2 or 3 segments that together contain m cells.
There are m^{2}  1 such sets parallel to each of the four
triagonals.
J.R.Hendricks, Magic Squares to Tesseracts by
Computer, 1999

Pantriagonal Magic Cube 
If all triagonal sets (Pantriagonals) sum
correctly, the magic cube is pantriagonal. It is analogous to
a pandiagonal magic square but instead of moving a row or column from
one side to the other and maintaining the magic properties, you may move any
plane from one side to the other.
There are 7m^{2} lines that sum correctly (m^{2}
rows, m^{2} columns, m^{2} pillars, and 4m^{2}
triagonals). There may be some correct diagonals in the cube but they are
not required.
Order4 is the smallest possible order pantriagonal magic cube.
See also, Pandiagonal Magic Cube.
J.R.Hendricks, Magic Squares to Tesseracts by Computer, 1999

Patchwork
Magic Square 
An Inlaid magic square that has magic squares or odd
magic shapes within it. The most common shape is a magic rectangle,
but diamonds, crosses, tees and L shapes are also possible. These shapes are
magic if the constant in each direction is proportional to the number of
cells.
For example, a 4 x 6 rectangle may have the constant of 100 in the
short direction and 150 in the long direction. Diagonals (of the magic
shapes) are not required to be magical. An example by David Collison is an
order 14 magic square, containing 4 order 4 magic squares in the
quadrants, a magic cross in the center, 4 magic tees, and 4 magic elbows
in the corners.
J. R. Hendricks, Magic Square Course, 1992, page
312 (now outofprint)

Perfect
Magic Cube 
A perfect magic cube is pantriagonal and all of its
orthogonal planes are pandiagonal magic squares. There are 13m^{2}
lines that sum correctly (m^{2} rows, m^{2}
columns, m^{2} pillars, 4m^{2} triagonals and
6m^{2} diagonals). There are 3m +6 one segment and 6m6
twosegment pandiagonal magic squares. Order8 is the smallest possible
order perfect magic cube.
So a perfect magic cube is a combination pantriagonal and
pandiagonal magic cube!
Due to confusion over the term perfect the preferred term now for
this class is nasik.
J.R.Hendricks, Magic Squares to
Tesseracts by Computer, 1999.
One (of several) older definitions of a Perfect Magic Cube
is now called a Diagonal Cube (of which the Myers Cube is an
example). See Benson & Jacoby Magic Cubes New Recreations,
1981, for an Order8 of this type, first published in 1888.
See Nasik.

Perfect Magic Hypercube 
A hypercube of dimension n is perfect if all pannagonals
sum correctly, and all lower dimension hypercubes are perfect (nasik).
For example: A perfect magic cube has all triagonals summing
correctly and all magic squares contained in it are perfect (perfect and
nasik are old names for a pandiagonal magic square).
Due to confusion over the term perfect
the preferred term for this class is
nasik. See Nasik.

Perfect
Magic Square 
See Pandiagonal Magic Square. (The term originated
with La Hire.) This class was originally called nasik by A. H. Frost.
See Nasik.
Emory McClintock, On the Most Perfect Forms
of Magic Squares, with Methods for their Production, American Journal of
Mathematics, 1897, 19, pp 99120. (2^{nd} page)

Perfect Magic Tesseract 
A tesseract is perfect if all panquadragonals
are correct, and all the magic squares and magic cubes within it are
perfect. i.e. the magic squares are all pandiagonal and the
magic cubes are all pantriagonal and pandiagonal. There
are 40m^{3} lines that sum correctly. They are m^{3}
rows, m^{3} columns, m^{3}
pillars, m^{3} files, 8m^{3}
quadragonals, 16m^{3} triagonals, and 12m^{3}
diagonals. The smallest order perfect tesseract is order16.
This is a new definition!
Please also review the revised definition for the Perfect Magic Cube.
These new definitions are more compatible with that of a perfect
(pandiagonal) magic square.
By extension, this definition is consistent for all dimensions of
hypercubes! John R. Hendricks constructed the first perfect
magic tesseract in 1998. It was confirmed correct after an independent
computer check by Clifford Pickover in 1999.
It is order16 and uses the numbers from 1 to 65536. It sums to 524,296 in
163,840 ways and contains 64 perfect magic cubes and 1536 pandiagonal magic
squares.
Due to confusion over the term
perfect the preferred term for this class is nasik.
See Nasik.
J.R.Hendricks, Magic Squares to Tesseracts
by Computer, 1998
J.R.Hendricks, Perfect nDimensional Magic Hypercubes of Order2^{n},
1999
Private correspondence with Hendricks and Pickover.

Pillars 
The Z dimension in a coordinate system of addressing the
cells in a magic cube.
(x = rows and y = columns.)
J.R.Hendricks, Magic Squares to Tesseracts by Computer,
1998

Prime Number Magic Square 
A magic square consisting only of prime numbers. They are
not too difficult to construct. The difficulty is in constructing ones
consisting of consecutive primes. The first order 3 magic squares of this
type was only published in 1988 and consists of nine, 10 digit primes. H. L.
Nelson proved there are only two such squares with prime numbers under 2^{31}.
Harry L. Nelson, JRM No. 20:3, 1988, p.214216.
In 1913 it was proved (?) (Scientific American
vol.210, no.3 pp. 1267) that it is impossible to construct a
consecutive prime number magic square of order smaller then 12. The order 12
magic square shown by the author, however, contained the digit 1 and missed
out the digit 2. (Of course the number 1 is no longer considered a prime,
and the number 2 can never appear in a prime number magic square, because it
is the only even number, and parity would be destroyed.)
The minimum consecutive prime number magic square of order3 starts with
1480028129.
The minimum consecutive prime number magic square of order4 starts with 31.
The minimum consecutive prime number magic square of order5 starts with
269.
The minimum consecutive prime number magic square of order6 starts with 67.
The minimum consecutive prime number magic square of order9 starts with 43.

Pure
Magic Square

See Magic Square, Normal.

Pure Magic Star 
See Magic Star, Normal. 
Quadragonal 
A 4dimensional version of the 2dimensional diagonal
and the 3dimensional triagonal. A Magic Tesseract requires
eight of these lines of m numbers summing correctly that go from one
corner to the opposite corner through the center of the tesseract. Also
called a 4agonal.
J.R.Hendricks, Magic Squares to Tesseracts by
Computer, 1998

Quadrant 
A quarter of a magic square. The four quadrants are;
upperleft, upperright, lowerleft and lowerright. If the magic square is
even, the size of each quadrant is m/2 square. If the magic
square is odd, the center row or center column is common to two
orthogonally adjacent quadrants.

Quadrant
Magic Square 
Some magic squares of orders m equal to 4x +
1, have patterns of m cells appearing in each quadrant
that sum to the magic constant.
If a magic square contains 4 of these patterns in the 4 quadrants, and if
they are all the same type, I call it a quadrantmagic square.
Odd order quadrant magic squares were studied by Aale de Winkel and this
editor in 1999.
An order13 such square was found with 14 patterns appearing in all 4
quadrants and another 18 patterns that appear in at least 1 quadrant. Only 6
possible patterns do not appear at all in this particular square!
See my Quadrant Magic Squares page for more information.

Regular
Magic Square 
See Associative magic squares
Also a major classification of Pandiagonal Magic Squares. See
Regular & Irregular

Regular & Irregular 
A common method of constructing Pandiagonal magic squares
makes use of 2 subsidiary squares where letters are used to represent
various constants. The values in the two squares are then combined to obtain
the value for the corresponding cell of the magic square.
If each letter appears an even number of times in each row and
column in both squares, the resulting pandiagonal is considered regular.
If they do not appear an equal number of times in the rows, columns
and diagonals of one or both squares, then the resulting pandiagonal is
irregular.
All pandiagonal magic squares of orders 4 and 5 are regular.
There are 38,102,400 regular pandiagonals of order 7 and 640,120,320
irregular.
Benson & Jacoby, New Recreations with Magic Squares,1976 ,
p93)

Reflection 
A transformation of a magic square by exchanging the
contents of cells on the right and left sides (or the top and bottom) as
though the matrix was reflected in a mirror. See Fundamental magic
square and aspects
. 
Reversible
Magic Square 
Because certain digits are the same when viewed in a
mirror, or upside down; it is possible to form integers that change to other
integers when viewed in a mirror or upside down. From these integers,
construction of magic squares are possible.
The best known example of this is the order 4 magic square called
the IXOHOXI (pronounced ixohooksie). This square uses the digits 1 & 8 to
form sixteen unique 4 digit integers and presents 4 different arrangements
of these integers when rotated 180 degrees, flipped horizontally, and
flipped vertically.
The digit 0 can be used for such a magic square if a leading 0 for
integers is permitted.
The digits 6 and 9 may also be used in a magic square for 180 degree
rotation but not reflection.

Reversible Square 
This type of square was defined and used by K. Ollerenshaw
in her work with MostPerfect Magic Squares. While not magic, they
are important because
a. there is a onetoone relationship between mostperfect
and reversible squares
b. the number of reversible squares of a given order may be
readily determined.
Thus by simply calculating the number of reversible squares for a given
order, the number of mostperfect magic squares for that order is
immediately known.Reversible squares are m x m arrays of
the numbers from 1 to m^{2} (Ollerenshaw uses the series from
0 to m^{2} – 1). They have these additional properties.
1. The sum of the two numbers at diagonally opposite corners of any
rectangle or subsquare within the reversible square will equal the sum of
the two numbers of the other pair of diagonally opposite corners.
2. The sum or the first and last numbers in each row or column
equal the sum of the next and the next to last number in each row or column,
etc.
3. Diametrically opposed number pairs sum to m^{2}
+ 1.
K. Ollerenshaw and D. Brée, MostPerfect Pandiagonal Magic
Squares, 1998

Reversible Square, Principle 
Reversible squares may be assembled in sets whose
members may be transformed from one to another by
1. Interchanging a pair of complementary rows and/or columns.
2. Interchanging two rows/columns in one half of the square
together with interchanging the complementary rows/columns in the other half
of the square.
It is therefore necessary to define which is the
principle square from which the others in the set are derived from. The
principle reversible square is defined as that one containing 1 and 2 as the
first two numbers in the first row and all its rows and columns in,
respectively, sequences of integers in ascending order.
There are three principle reversible squares for order4, each may
be transformed to 15 other reversible squares, making three sets of 16, for
a total of 48 for order4. Because each of these may be mapped to a
mostperfect magic square there are 48 mostperfect magic squares for
order4. i.e. all the order4 pandiagonal magic squares are
mostperfect.
K. Ollerenshaw and D. Brée,
MostPerfect Pandiagonal Magic Squares, 1998

Right Diagonal 
The diagonal line of numbers from the lower left to upper
right corners of the magic square.

Rotation 
A transformation of a magic square by rotating the
magic square clockwise or counterclockwise. See Fundamental magic
square.

Row 
Each horizontal sequence of numbers. There are m
rows of length m in an order m magic square. Rows, columns,
pillars, etc. (i.e. orthogonal lines) are sometimes called irows, or
1agonals (because traveling along the line causes only 1 coordinate to
change).

S 
Indicates the magic sum or constant. See
constant for equations.

Selfsimilar 
A magic square which after each number is converted to its
complement, is a rotated and/or reflected copy of the original magic square.
Any magic square in which the complementary pairs are symmetric
across either the horizontal or the vertical center line of the square is
selfsimilar. The resulting copy is either horizontally or vertically
reflected.
Because associated magic squares are symmetric across both these
lines all such magic squares are selfsimilar and the copy is horizontally
and vertically reflected from the original.
Mutsumi Suzuki discovered magic squares with this feature and named
it selfsimilar. He has listed 16 order5 magic squares and 352 order4
magic squares of this type.
The process of complementing each number of a magic object is also
known as ‘complementary pair interchange’ (CPI). See an
excellent paper on this subject in Robert S. Sery, Magic Squares of
Order4 and their Magic Square Loops, Journal of Recreational Mathematics,
29:4, page 274
See my Selfsimilar Magic Squares page. Link to Mr. Suzuki ‘s Magic Squares
page from my links page.

SemiDiabolic 
See SemiPandiagonal magic square.

Semimagic square 
The rows & columns sum correctly but one or both main
diagonals do not. 
Semipandiagonal
magic square 
Also known as Semi–Diabolic They have the
property that the sum of the cells in the opposite short
diagonals are equal to the magic constant.
In an odd order square, these two opposite short diagonals,
which together contain m1 cells, will, when added to the center cell
equal the square’s constant. The two opposite short diagonals, which
together contain m+1 cells, will sum to the constant if the center
cell is subtracted from their total.
In an even order square, the two opposite short diagonals
which together consist of m cells will sum to the square's constant.
Of the 880 fundamental magic squares of order 4, 384 are
semipan ( 48 of these are also associative).
All semipan magic squares are NOT associated, but all associated (that
is centersymmetric) magic squares are semipan magic

Semipantriagonal
magic cube 
The magic cube equivalent of the semipandiagonal magic
square. Simply replace references to semipandiagonal in the above
definition with semipantriagonal . Also, instead of two short diagonal
pairs for the square case, there are four short triagonal pairs for the
cube.
This is just one more example of how magic square principles are
simply extended to magic cubes.

Sequence patterns 
The center of the cells containing consecutive numbers are
joined by lines. See magic lines.

Series 
A magic square usually contains n series of
n numbers. The horizontal step within each series is a
constant. The vertical step between corresponding numbers of each
series is also a constant. This step can be but need not be the same as the
horizontal step.
A normal magic square has the starting number, the
horizontal step and the vertical step all equal to 1.
After the N initial series are established, the magic square is constructed
using any appropriate method. If N = the squares order, a = starting
number, d = the horizontal step D = the vertical step, and K = sum of
numbers in the first series; then S = (N^{3} + N) / 2 + N (a
 1 ) + ( K  N ) [ N ( d  1 ) + ( D  1 )]
W.S.Andrews, Magic Squares and Cubes,1917, pp
5463
J.L.Fults, Magic Squares, 1974, pp 3739

Serrated
Magic Square 
A magic square rotated 45 degrees.
W.S.Andrews, Magic Squares and Cubes, 1917, pp241244
J.R.Hendricks, Ed Shineman, Jr. (and others) refer to these as
Magic Diamonds.
J.R.Hendricks, Inlaid Magic Squares and
Cubes, 1999

Short Diagonal 
One which runs parallel to a main diagonal from 1 side of
the square to an adjacent side.
These are usually considered in pairs (magic Squares), trios (magic cubes),
etc., in which case they are called broken diagonals or
pandiagonals.

Simple
Magic Square 
A square array of numbers, usually integers, in which all
the rows, columns, and the two main diagonals have the same sum. As these
are the minimum specifications to qualify as a magic square this term
signifies it has no special features. The one order 3 magic square is not
simple (it is associative). Of the 880 order 4 magic squares, 448 are
classified as simple.

Singlyeven order 
The side of the square is divisible by two but not by four.
This is the most difficult order to construct.

Skew related 
See Symmetrical cells
RouseBall & Coxeter, Mathematical Recreations and
Essays, 1892, (13 Edition, p.194)

Space diagonals 
See triagonals, quadragonals

Standard Position Magic Squares 
Any magic square may be disguised to make 7 other
(apparently) different magic squares by means of rotations and
reflections. These variations are NOT considered as new magic squares for
purposes of enumeration. For the purpose of listing and indexing magic
squares, a standard position must be defined. The magic square is then
rotated and/or reflected until it is in this position. This position was
defined by Frénicle in 1693 and consists of only two requirements.
1. The lowest of any corner number must be in the upper left
hand corner.
2. The cell in the top row adjacent to the top left corner
must be lower then the leftmost position of the second row (also adjacent to
the top left corner).
This process is called Normalizing. Achieving the first
condition may require rotation. The second may require rotation and
reflection. Once the magic square is in this position, it may be put in the
correct index position in a list of magic squares of a given order.
This definition has meaning (and relevance) for a normal magic
square.
Benson & Jacoby, New Recreations with Magic Squares, 1976,
p 123.

Standard Position Magic Stars 
A magic star may be disguised to make 2n1
apparently different magic stars where n is the order (number of
points) of the magic star.
Three characteristics determine the Standard position.
1. The diagram is oriented so only one point is at the top.
2. The top point of the diagram has the lowest value of all the
points.
3. The valley to the right of the top point has a lower value then
that of the valley to the left.
This process is called Normalizing. Achieving the first and second
conditions may require rotation. The third may require reflection. Once the
magic star is in this position, it may be put in the correct index
position in a list of magic stars of a given order.
This definition has meaning (and relevance) for a normal magic
star.
See my Magic Stars Definitions page.

Subtraction Magic Square 
Interchange the contents of diagonal opposite corners of an
order3 magic square. Now, if you add the two outside numbers and subtract
the center one from the sum, you get the constant 5.

Symmetrical cells 
Two cells that are the same distance and on opposite sides
of the center of the cell are called symmetrical cells. In an odd order
square the center is itself a cell. In an even order square the
center is the intersection of 4 cells. Other definitions for these pairs are
skew related and diametrically equidistant.
J. L. Fults, Magic Squares, 1974
RouseBall & Coxeter, Mathematical Recreations and Essays,1892 (13
Edition, p.194,202)

Symmetrical Magic Square 
See Associated Magic Square. 
Talisman
Magic Square 
A Talisman square is an m x m array of the
integers from 1 to m^{2} so that the difference between any
integer and its neighbors, horizontally, vertically, of diagonally, is
greater then some given constant. The rows, columns and diagonals will NOT
sum to the same value so the square is not magic in the normal sense of the
word. This type of square was discovered and named by Sidney Kravitz.
Joseph S. Madachy, Mathemaics On Vacation,
1966, pp 110112.

Transformation 
Any order5 pandiagonal magic square may be converted to
another magic square by permuting the rows and columns in the order
13524. Each of these two magic squares can be transformed to another by
exchanging the rows and columns with the diagonals. Finally, each of these
four squares may be converted to 24 other magic squares by cyclical
permutations.
Benson & Jacoby, Magic squares & Cubes, 1976,
pp.128131.
Another type of transformation converts any
normal magic square to its complement by subtracting each integer
in the magic square from m^{2} + 1. In some cases this
results in a copy of the original magic square. See my
Selfsimilar page.
Any order5 magic square can also be transposed to another one by
either of the following two transformations.
1. Exchange the left and right columns, then the top and bottom
rows.
2. Exchange columns 1 and 2 and columns 4 and 5. Then exchange rows
1 and 2, and rows
4 and 5.
These two methods, of course, also work for all odd orders greater then
order5.
Any magic square may be converted to another one by adding a
constant to each number.
My Transformations pages lists over 40
transformations of various types!

Transposition 
The permutation of the rows and columns of a
pandiagonal magic square in order to change it into another
pandiagonal magic square.
For order5 this is cyclical 13524. For order7 there are two
noncyclical permutations, 1357246 and 1473625.
The other transposition method for pandiagonals is to exchange the
rows and columns with the diagonals.
Benson & Jacoby, Magic squares & Cubes, 1976,
pp.146154.
The above authors devote a chapter in their book to
transposition, but freely use the term transformation
elsewhere in the same book. Other authors seem to prefer the term
transformation.
A third method of converting a pandiagonal magic
square to a different one is to simply move a row or column from one side of
the square to the other.
In general, either term may be considered any method of
converting one magic square into another one.

Traditional M. S. 
See Magic Square, Normal

Triagonal 
A space diagonal that goes from 1 corner of a magic
cube to the opposite corner, passing through the center of the cube. There
are 4 of these in a magic cube and all must sum correctly (as well as
the rows, columns and pillars) for the cube to be
magic. As you go from cell to cell along the line, all three coordinates
change.
In tesseracts or higher order hypercubes, this is
called an nagonal or space diagonal. Of course, with
these higher dimensions there are more coordinates. See also quadragonals.
J.R.Hendricks, Inlaid Magic Squares and
Cubes, 1999.

Trimagic Square 
See Bimagic Square.

Vertical step 
The difference between corresponding numbers of the n
series. It is not a reference to the rows of the
magic square.
In a normal magic square, the horizontal step and
vertical step are both 1.
J. L. Fults, Magic Squares, 1974
W.S.Andrews, Magic Squares and Cubes,1917

Vertically paired 
Two cells in the same column and the same distance from the
center of the square.

Wraparound 
Used in pandiagonal magic squares to indicate that
lines are actually loops. Each edge may be considered to be joined to the
opposite edge. If you move from left to right along a row, when you
reach the right edge of the magic square, you wraparound to the
first cell on the left of the same row.
Or consider that the pandiagonal magic square is repeated in all
four directions. Any n x n section of this array may be
considered as a pandiagonal magic square. This results from the fact the
broken diagonal pairs form complete lines.

