Hypercube Glossary

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Introduction

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 m2. 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.)

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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.

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agonal

Root word for a line that goes from one corner to an opposite corner of a hypercube. Examples: diagonal, triagonal, quadragonal, r-agonal, n-agonal. By extension, a 1-agonal is an orthogonal line i.e. a row, column, etc.
Hendricks, John R., The Pan-4-agonal Magic Tesseract, The American Mathematical Monthly, Vol. 75, No. 4, April 1968, p. 384.
John R. Hendricks, The Pan-3-Agonal Magic Cube, JRM 5:1:1972, pp 51-54
 

Almost-magic Stars

   A magic pentagram (5-pointed 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 almost-magic pentagram. 
Charles W. Trigg, JRM29:1, p. 8-11, 1998

   Marián Trenkler (Safarik University, Slovakia) has independently coined the phrase almost-magic, 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 almost-magic star.
   NOTE that by his definition, the order-5 almost-magic star has only 3 lines summing correctly.
Trigg’s order-5 (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 Almost-magic Stars.
 

Almost-magic Squares

   An array of consecutive numbers, from 1 to m2, where the rows, columns and two main diagonals sum  to a set of 2(m + 1) consecutive integers. Anti-magic squares are a sub-set of heterosquares.
Joseph S. Madachy, Mathemaics On Vacation, pp 101-110. (Also JRM 15:4, p.302)
 

Anti-magic 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 101-110. (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) order-3 magic cubes. Each of these can appear in 48 aspects due to rotations and reflections.
   There are 58 essentially different order-3 magic tesseracts (4th dimension). Each of these can appear in 384 aspects due to rotations and reflections.
Just as the 1 order-3 magic square is associated, so to are the 4 order-3 magic cubes and the 58 order-3 magic tesseracts.
   All of these figures can be converted to another aspect by complimenting each number (the self-similar 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 m2 + 1. Also called Symmetrical or center-symmetric. 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 order-5 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 self-similar feature).
   There are NO singly-even associated magic squares.
All even order associated magic squares are semi-pandiagonal.
The one order-3 magic square is associative.
There are 48 order-4 associative magic squares.
Order-5 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 order-8 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 order-9 bimagic squares. e-mail 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 (m2 -1)/2 lowest numbers and their complements (the highest numbers) in the border where m2 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 26-33  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 pan-diagonals, and are the prominent feature of Pandiagonal magic squares.
J. L. Fults, Magic Squares, 1974
 

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Cell

   The basic element of a magic square, magic cube, magic star, etc. Each cell contains one number, usually an integer. There are m2 cells in a magic square of order m, m3 cells in a magic cube, m4 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 order-m 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 most-perfect magic squares.
 Gakuho Abe, Fifty Problems of Magic Squares, Self published 1950. Later republished in Discrete Math, 127, 1994, pp 3-13.

   Addendum; In April 2007 Aale de Winkel proved that corners of all rectangular shapes in a compact magic square are pan-magic.
   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 sub-cubes contained within a cube, including wrap-around, 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 sub-cubes of one or several orders sum to 8S/m where m is the order of the parent cube. All sub-cubes from orders 2 to 8 sum correctly in an order 8 ‘perfect’ magic cube. This includes wrap-around, so in effect there are 64 sub-cubes 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 most-perfect 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 sub-squares sum to the same value which is 2m2+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 most-perfect’.
McClintock, E. 1897, On the Most Perfect Forms of Magic Squares, with Methods for their Production. , A., J. Math. 19, 99-120 (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 (m2 + 1). All other pairs of numbers which also sum to m2 + 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 non-consecutive 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 (m3+m)/2
If the magic square consists of consecutive numbers, but not starting at 1, the constant is (m3+m)/2+m(a-1) where a equals the starting number.
   If the magic square consists of numbers with a fixed increment, then S = am + b(m/2)(m2-1) where a = starting number and b = increment.  See Series.
For a normal magic square, S = ½ m(m2+1).
For a normal magic cube, S = ½ m(m3+1).
For a tesseract S = ½ m(m4+1).
In general; for a d-dimensional hypercube S = ½ m(mn+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 order-5 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.
 

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Essentially Different

   There are 36 essentially different order-5 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 left-hand 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 left-hand column of the second row is less then the number in the left-hand column of the last row.
Benson & Jacoby, New Recreations with Magic Squares, 1976, p 129.
 

Eulerian square

See Graeco-Latin 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 semi-magical, as the main diagonals do not sum correctly.
   The never-before 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 order-3. 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 order-3. 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 order-3 and 880 of order-4, 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 2n-1 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, pp283-294 discusses this type of magic square.
 

Graeco-Latin 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 Graeco-Latin 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 Order-10 Graeco-Latin Square.
 

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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 sub-set) is the antimagic square. For this type of square, the line sums must be consecutive numbers.
Joseph S. Madachy, Mathemaics On Vacation, 1966, pp 101-110. (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 m2.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 order-4 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.123-124.
 
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
 
Iso-like Magic Stars    An order-8B 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 pan-magic 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 order-9 magic square.
  
See my page on Iso-like Magic Stars for samples and more information.
Go to Aale de Winkel’s Magic Object pages from my Links page.
 
Isomorphic
Magic Stars
   An order-8B type magic star then can be constructed by a systematic transformation using all the numbers of a magic square.
   If the magic square is order-4 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 order-5, 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 Graeco-Latin 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 (n2-1)/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    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 m3. There are 3m2 + 4 lines that sum correctly. All rows, columns, pillars, and the four triagonals must sum to 1/2m(m3+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 order-3, 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
 n-agonals 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, 0-394-74798-4 (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 3m2 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 3m2 monagonals + 6m2 pandiagonals + 4m2 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 multi-dimensional 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 m2. 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 four-dimensional array, equivalent to the magic cube and magic square of lower dimensions, containing the numbers 1, 2, 3, …, m4 arranged in such a way that the sum of the numbers in each of the m3 rows, m3 columns, m3 pillars, m3 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(m4+1) and where n is called the order of the tesseract.
   There are 58 basic magic tesseracts of order-3. Each may be shown in 384 aspects due to rotations and/or reflections.
There is             1 basic hypercube   of dimension-2, order-3,          with 8 aspects.
There are           4 basic hypercubes of dimension-3, order-3, each with 48 aspects.
There are         58 basic hypercubes of dimension-4, order-3, each with 384 aspects.
There are     2992 basic hypercubes of dimension-5, order-3, each with 3840 aspects.
There are 543328 basic hypercubes of dimension-6, order-3, each with 46080 aspects.

J.R.Hendricks, Magic Squares to Tesseracts by Computer, 1998
J.R.Hendricks, All third-order Magic Tesseracts, 1999
C.Planck (W.S.Andrews, Magic Squares and Cubes,1917, pp 363-375) 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.
 
Most-Perfect Magic Square    A normal pandiagonal magic square of doubly-even order with two added properties. Any two-by-two block of adjacent cells (including wrap-around) sum to the same value which is 2m2+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, Most-Perfect Pandiagonal Magic Squares, 1998
  
Ian Stewart, Mathematical Recreations, Scientific American, November 1999

Note that both these authors use the series from 0 to m2-2 for mathematical convenience. The sum of each 2 by 2 square array is then 2m2-2. 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.
 

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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 pandiagonal-like 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 34-49.
 
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 doubly-even order magic cube if you split the cube in half in each dimension. i.e. if you divide an order-8 cube in this fashion, the octants are the eight order-4 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 Semi-Pandiagonal.
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,
Doubly-even
 
   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,
Singly-even
   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.
 

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Pan-diagonals 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 Most-perfect (doubly-even 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.
Order-5 has 3600 basic pandiagonal magic squares (Only 36 essentially different).
Order-7 has 678,222,720 basic pandiagonal magic squares.
Order-8 has more then 6,500,000,000 pandiagonal magic squares.
   There are NO singly-even 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, 34-49.
   C. Planck in 1919 The Monest 29, 307-316.
 
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 9m2 + 4 lines that sum correctly (m2 rows, m2 columns, m2 pillars, 4 main triagonals and 6m2 Diagonals). Order-7 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, 92-102.
 
Pan-magic Stars    An order-8B type magic star then can be constructed by a systematic transformation of odd-order pandiagonal magic squares greater then order-5.
   Aale de Winkel investigated this type of magic star in the spring of 1999 which later resulted in his and my joint investigation of Iso-like magic stars.
   Unlike iso-magic stars which cannot use all the numbers, pan-magic 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 Iso-like Magic Stars for more information.
Go to his page on Pan-magic Stars from my links page.
 
Pan-quadragonals    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 pan-quadragonal. 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, Pan-triagonals.
   J.R.Hendricks, Magic Squares to Tesseracts by Computer, 1999
 
Pan-triagonals    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 m2 - 1 such sets parallel to each of the four triagonals.
   J.R.Hendricks, Magic Squares to Tesseracts by Computer, 1999
 
Pan-triagonal Magic Cube    If all triagonal sets (Pan-triagonals) 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 7m2 lines that sum correctly (m2 rows, m2 columns, m2 pillars, and 4m2 triagonals). There may be some correct diagonals in the cube but they are not required.
Order-4 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 out-of-print)
 
Perfect
Magic Cube
   A perfect magic cube is pantriagonal and all of its orthogonal planes  are pandiagonal magic squares. There are 13m2 lines that sum correctly (m2 rows, m2 columns, m2 pillars, 4m2 triagonals and 6m2 diagonals). There are 3m +6 one- segment and 6m-6 two-segment pandiagonal magic squares. Order-8 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 Order-8 of this type, first published in 1888.  See Nasik.
 
Perfect Magic Hypercube    A hypercube of dimension n is perfect if all pan-n-agonals 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 99-120. (2nd page)
 
Perfect Magic Tesseract    A tesseract is perfect if all pan-quadragonals 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 40m3 lines that sum correctly. They are m3 rows, m3 columns, m3 pillars, m3 files, 8m3 quadragonals, 16m3 triagonals, and 12m3 diagonals. The smallest order perfect tesseract is order-16.
   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 order-16 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 n-Dimensional Magic Hypercubes of Order2n, 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 231.
Harry L. Nelson, JRM No. 20:3, 1988, p.214-216.

   In 1913 it was proved (?) (Scientific American vol.210, no.3 pp. 126-7) 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 order-3 starts with 1480028129.
The minimum consecutive prime number magic square of order-4 starts with 31.
The minimum consecutive prime number magic square of order-5 starts with 269.
The minimum consecutive prime number magic square of order-6 starts with 67.
The minimum consecutive prime number magic square of order-9 starts with 43.
 

Pure
Magic Square
 
See Magic Square, Normal.
 
Pure Magic Star See Magic Star, Normal.

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Quadragonal    A 4-dimensional version of the 2-dimensional diagonal and the 3-dimensional 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 4-agonal.
   J.R.Hendricks, Magic Squares to Tesseracts by Computer, 1998
 
Quadrant    A quarter of a magic square. The four quadrants are; upper-left, upper-right, lower-left and lower-right. 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 quadrant-magic square.

   Odd order quadrant magic squares were studied by Aale de Winkel and this editor in 1999.
An order-13 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 ixo-hooksie). 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 Most-Perfect Magic Squares. While not magic, they are important because
    a. there is a one-to-one relationship between most-perfect 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 most-perfect magic squares for that order is immediately known.

Reversible squares are m x m arrays of the numbers from 1 to m2 (Ollerenshaw uses the series from
0 to m2 – 1). They have these additional properties.
   1. The sum of the two numbers at diagonally opposite corners of any rectangle or sub-square 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 m2 + 1.
K. Ollerenshaw and D. Brée, Most-Perfect 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 order-4, each may be transformed to 15 other reversible squares, making three sets of 16, for a total of 48 for order-4. Because each of these may be mapped to a most-perfect magic square there are 48 most-perfect magic squares for order-4. i.e. all the order-4 pandiagonal magic squares are most-perfect.
     K. Ollerenshaw and D. Brée, Most-Perfect 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 i-rows, or 1-agonals (because traveling along the line causes only 1 co-ordinate to change).
 

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S    Indicates the magic sum or constant. See constant for equations.
 
Self-similar    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 self-similar. The resulting copy is either horizontally or vertically reflected.
   Because associated magic squares are symmetric across both these lines all such magic squares are self-similar and the copy is horizontally and vertically reflected from the original.
   Mutsumi Suzuki discovered magic squares with this feature and named it self-similar. He has listed 16 order-5 magic squares and 352 order-4 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 Order-4 and their Magic Square Loops, Journal of Recreational Mathematics, 29:4, page 274
See my Self-similar Magic Squares page. Link to Mr. Suzuki ‘s Magic Squares page from my links page.
 
Semi-Diabolic    See Semi-Pandiagonal magic square.
 
Semi-magic square    The rows & columns sum correctly but one or both main diagonals do not.

 

Semi-pandiagonal
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 m-1 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 semi-pan ( 48 of these are also associative).
All semi-pan magic squares are NOT associated, but all associated (that is center-symmetric) magic squares are semi-pan magic
 
Semi-pantriagonal
magic cube
   The magic cube equivalent of the semi-pandiagonal magic square. Simply replace references to semi-pandiagonal in the above definition with semi-pantriagonal . 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 = (N3 + N) / 2 + N (a - 1 ) + ( K - N ) [ N ( d - 1 ) + ( D - 1 )]
   W.S.Andrews, Magic Squares and Cubes,1917, pp 54-63
  
J.L.Fults, Magic Squares, 1974, pp 37-39
 
Serrated
Magic Square
   A magic square rotated 45 degrees. W.S.Andrews, Magic Squares and Cubes, 1917, pp241-244
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.
 
Singly-even 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 2n-1 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 order-3 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.

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Talisman
Magic Square
   A Talisman square is an m x m array of the integers from 1 to m2 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 110-112.
 
Transformation    Any order-5 pandiagonal magic square may be converted to another magic square by permuting the rows and columns in the order 1-3-5-2-4. 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.128-131.
  
Another type of transformation converts any normal magic square to its complement by subtracting each integer in the magic square from m2 + 1. In some cases this results in a copy of the original magic square. See my Self-similar page.
   Any order-5 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 order-5.
   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 order-5 this is cyclical 1-3-5-2-4. For order-7 there are two non-cyclical permutations, 1-3-5-7-2-4-6 and 1-4-7-3-6-2-5.
   The other transposition method for pandiagonals is to exchange the rows and columns with the diagonals.
Benson & Jacoby, Magic squares & Cubes, 1976, pp.146-154.
   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 n-agonal 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.
 
Wrap-around    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 wrap-around 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.
 

This page was originally posted April 2000
It was last updated April 11, 2011
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz