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.
Addendum: May 2007. 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.
A  B 
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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. Marián Trenkler, Magicke Hviezdy (Magic stars), Obsory Matematiky, Fyziky a
Informatiky, 51(1998). 
AntiMagic 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) 
Associated Magic Cubes, Tesseracts, etc. 
Features are the same as those for the associated
magic square. There are 4 fundamental 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. W. S. Andrews, Magic squares & Cubes, 1917 
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). Benson & Jacoby, New Recreations in Magic Squares, 1976, pp
2633 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 4/m of
S. 
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 4/m of S. Many cubes have the 8 corners of all subcubes of one or several orders sum correctly. 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. 
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 M.S.  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. For a normal magic square, S = ½ m(m^{2}+1). 
Continuous M. Sqr.  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 Magic Square  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. Square  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 Magic Sqr. Triple Magic Sqr. 
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 to construct. 
A  B 
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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,
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. 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 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. 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. 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. Normally applied to figures of more then four dimensions. However, a square,
cube and tesseract are hypercubes of two, three and four
dimensions. 
Impure Magic Square  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 Magic Sqr.  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. 
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. 
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. 
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. 
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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, etc  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 
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. [1] 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 
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 cube (panmagic ratio). Magic cube ratio Panmagic cube ratio 
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 is researching this subject but he 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 Star, Normal  A normal magic star consists of a set of integers 1,
2, 3, ..., 2n which are placed at the 2n exterior points of intersection of
the lines which form a regular polygram, such that the sum of the four integers found in
any of the n lines is given by: S = 4n+2 where S is called the magic sum,
and n is the order of the star. Also called Pure. See my Magic Stars pages. 
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. J.R.Hendricks, Magic Squares to Tesseracts by Computer, 1998 
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 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. 
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. 
n  Traditionally used to indicate the order of a magic array.
Many hobbyists now use m for this purpose, reserving
n to indicate dimension. 
Nasik  Nasik is an unambiguous
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. 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. 
A  B 
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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. 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 . A. H. Frost, On the General Properties of Magic
Squares, Quarterly Journal of Mathematics, 15, 3449. 
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 
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. If all these sets sum correctly, the magic cube is pantriagonal. 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. 
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  This is a new definition! 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 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. See Benson & Jacoby Magic Cubes New
Recreations, 1981, for an Order8 of this type, first published in 1888. 
Perfect Magic Hypercube  A hypercube of dimension n is perfect if
all pannagonals sum correctly, and all lower dimension hypercubes
are perfect. For example: A perfect magic cube has all triagonals summing correctly and all magic squares contained in it are perfect (perfect is an old name for pandiagonal). Due to confusion over the term perfect the preferred term for this class is nasik. 
Perfect Magic Square  See Pandiagonal Magic Square.
(The term originated with La Hire.) This class was originally called
nasik by A. H. Frost.
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! 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. Due to confusion over the term perfect the preferred term for this class is nasik. J.R.Hendricks, Magic Squares to Tesseracts by Computer, 1998 
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 Squares  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. The author 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. 
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. 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. 
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. 
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.
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
It is therefore necessary to define which is the principle square from which the
others in the set are derived from. 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 travelling along the line causes only 1 coordinate to change). 
A  B 
C  D 
E  F  G 
H  I  J  L 
M 
N_O 
P 
Q  R 
S 
T  V  W 
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. See my Selfsimilar Magic Squares page. 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 
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). 
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 
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. 
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 
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.
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. 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.
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 
Symmetrical M.S.  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. Any magic square may be converted to another one by adding a constant to each number. 
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. 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. 
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March 30, 2009
Harvey Heinz harveyheinz@shaw.ca
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