# Glossary

### 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.

 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. ### T - V - W ### A - B

 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, 1998Marián 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. Anti-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) Associated Magic Cubes, Tesseracts, etc. Features are the same as those for the associated magic square. There are 4 fundamental 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

### E - F - G

 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. 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 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.

### N - O

 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 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. 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.

### S

 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.See my Self-similar Magic Squares page. Link to Mr. Suzuki ‘s Magic Squares page from my links 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 Order-4 and their Magic Square Loops, Journal of Recreational Mathematics, 29:4, page 274 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, pp241-244J.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 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. The lowest of any corner number must be in the upper left hand corner. 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. The diagram is oriented so only one point is at the top. The top point of the diagram has the lowest value of all the points. 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 M.S. See Associated Magic Square.

### T - V - W

 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. Exchange the left and right columns, then the top and bottom rows. 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. 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. 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. Last updated March 30, 2009
Harvey Heinz   harveyheinz@shaw.ca