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A magic tesseract is a four-dimensional generalization of the two-dimensional magic square and the three-dimensional magic cube. A magic tesseract has magic constantso for , 2, ..., the magic tesseract constants are 1, 17, 123, 514, 1565, 3891, ... (OEIS A021003).Berlekamp et al. (1982, p. 783) give a magic tesseract. J. Hendricks has constructed magic tesseracts of orders three, four, five (Hendricks 1999a, pp. 128-129), and six (Heinz). M. Houlton has used Hendricks' techniques to construct magic tesseracts of orders 5, 7, and 9.There are 58 distinct magic tesseracts of order three, modulo rotations and reflections (Heinz, Hendricks 1999), one of which is illustrated above. Each of the 27 rows (e.g., 1-72-50), columns (e.g., 1-80-42), pillars (e.g., 1-54-68), and files (e.g., 1-78-44) sum to the magic constant 123.Hendricks (1968) has constructed a pan-4-agonal magic tesseract of order 4. No pan-4-agonal..

A magic square is a square array of numbers consisting of the distinct positive integers 1, 2, ..., arranged such that the sum of the numbers in any horizontal, vertical, or main diagonal line is always the same number (Kraitchik 1942, p. 142; Andrews 1960, p. 1; Gardner 1961, p. 130; Madachy 1979, p. 84; Benson and Jacoby 1981, p. 3; Ball and Coxeter 1987, p. 193), known as the magic constantIf every number in a magic square is subtracted from , another magic square is obtained called the complementary magic square. A square consisting of consecutive numbers starting with 1 is sometimes known as a "normal" magic square.The unique normal square of order three was known to the ancient Chinese, who called it the Lo Shu. A version of the order-4 magic square with the numbers 15 and 14 in adjacent middle columns in the bottom row is called Dürer's magic square. Magic squares of order 3 through 8 are shown..

A perfect magic cube is a magic cube for which the rows, columns, pillars, space diagonals, and diagonals of each orthogonal slice sum to the same number (i.e., the magic constant ). While this terminology is standard in the published literature (Gardner 1976, Benson and Jacoby 1981, Gardner 1988, Pickover 2002), it has been suggested at various times that such cubes instead be termed Myers cubes, Myers diagonal cubes, or diagonal magic cube (Heinz).There is a trivial perfect magic cube of order one, but no perfect cubes exist for orders 2-4 (Schroeppel 1972; Benson and Jacoby 1981, pp. 23-25; Gardner 1988). While normal perfect magic cubes of orders 7 and 9 have been known since the late 1800s, it was long not known if perfect magic cubes of orders 5 or 6 could exist (Wells 1986, p. 72), although Schroeppel (1972) and Gardner (1988) note that any such cube must have a central value of 63. (Confusingly, Benson and Jacoby (1981, p. 5)..

In 1750, Benjamin Franklin constructed the above semimagic square having magic constant 260. Any half-row or half-column in this square totals 130, and the four corners plus the middle total 260. In addition, bent diagonals (such as 52-3-5-54-10-57-63-16) also total 260 (Madachy 1979, p. 87).Describing his invention in 1771, Franklin stated, "I was at length tired with sitting there to hear debates, in which, as clerk, I could take no part, and which were often so unentertaining that I was induc'd to amuse myself with making magic squares or circles" (Franklin 1793).

Dürer's magic square is a magic square with magic constant 34 used in an engraving entitled Melancholia I by Albrecht Dürer (The British Museum, Burton 1989, Gellert et al. 1989). The engraving shows a disorganized jumble of scientific equipment lying unused while an intellectual sits absorbed in thought. Dürer's magic square is located in the upper right-hand corner of the engraving. The numbers 15 and 14 appear in the middle of the bottom row, indicating the date of the engraving, 1514.Dürer's magic square has the additional property that the sums in any of the four quadrants, as well as the sum of the middle four numbers, are all 34 (Hunter and Madachy 1975, p. 24). It is thus a gnomon magic square. In addition, any pair of numbers symmetrically placed about the center of the square sums to 17, a property making the square even more magical...

If replacing each number by its square or cube in a magic square produces another magic square, the square is said to be a trimagic square. Trimagic squares are also called trebly magic squares, and are 3-multimagic squares.Trimagic squares of order 12, 32, and larger are known. Tarry (1906) gave a method for constructing a trimagic square of order 128, Cazalas a method for trimagic squares of orders 64 and 81, R. V. Heath a method for constructing an order 64 trimagic square which is different from Cazalas's (Kraitchik 1942), and Benson (Benson and Jacoby 1976) a method for constructing an order 32 trimagic square.Walter Trump constructed the first trimagic square of order 12 in June 2002. This square, illustrated above, is the smallest possible trimagic square, since Boyer and Trump subsequently proved that a trimagic square of order less than 12 cannot exist (Boyer)...

A square which is magic under multiplication instead of addition (the operation used to define a conventional magic square) is called a multiplication magic square. Unlike (normal) magic squares, the entries for an th order multiplicative magic square are not required to be consecutive.The above multiplication magic square has a multiplicative magic constant of 4096 and was found by Antoine Arnauld in Nouveaux Eléments de Géométrie, Paris in 1667 (Boyer).The smallest possible magic constants for , , ... are 216, 5040, 302400, 25945920, ... (OEIS A114060). The solution (left) was found by Sayles in 1913 and also published by Dudeney (1917). Sayles also found the solution (right), which was subsequently proved to be minimal by Borkovitz and Hwang (1983). The series of best known smallest largest element for an multiplication magic square with , 4, ... begins 36, 28, 45, 66, 91, 160, 225, ... (Boyer)...

A magic square is said to be -multimagic if the square formed by replacing each element by its th power for , 2, ..., is also magic. A 2-multimagic square is called bimagic, a 3-multimagic square is called trimagic, a 4-multimagic square is called tetramagic, a 5-multimagic square is called pentamagic, and so on.The first known bimagic square had order eight and was constructed by Pfefferman (1891). Tetramagic and pentamagic squares were constructed by Christian Boyer and André Viricel in 2001 (Boyer 2001).

A magic square that remains magic when its border is removed. A nested magic square remains magic after the border is successively removed one ring at a time. An example of a nested magic square is the order 7 square illustrated above (i.e., the order 7, 5, and 3 squares obtained from it are all magic).The amazing square above (Madachy 1979, pp. 93-94) is a prime magic border square, so that the , , ..., and subsquares are all also prime magic squares.

A semimagic square is a square that fails to be a magic square only because one or both of the main diagonal sums do not equal the magic constant (Kraitchik 1942, p. 143). Note some care with terminology is necessary. For example, Jelliss terms a semimagic square a "magic square" and a magic square a "diagonally magic square."The number of distinct semimagic squares (treating squares differing by rotations and reflections as identical) of orders 1, 2, ... are 1, 0, 8, .... The eight semimagic squares of order 3 are illustrated above.

An magic square for which every pair of numbers symmetrically opposite the center sum to . The Lo Shu is associative but not panmagic. The numbers of associative magic squares of order 1, 2, ... are 1, 0, 1, 48, ... (OEIS A081262).Order four squares can be panmagic or associative, but not both. Order five squares are the smallest which can be both associative and panmagic, and 16 distinct associative panmagic squares exist, one of which is illustrated above (Gardner 1988). The numbers of associative panmagic squares of order 1, 2, ... are therefore 1, 0, 0, 0, 16, ... (OEIS A081263).

If all the diagonals--including those obtained by "wrapping around" the edges--of a magic square sum to the same magic constant, the square is said to be a panmagic square (Kraitchik 1942, pp. 143 and 189-191). (Only the rows, columns, and main diagonals must sum to the same constant for the usual type of magic square.) The terms diabolic square (Gardner 1961, pp. 135-137; Hunter and Madachy 1975, p. 24; Madachy 1979, p. 87), pandiagonal square (Hunter and Madachy 1975, p. 24), and Nasik square (Madachy 1979, p. 87) are sometimes also used.No panmagic squares exist of order 3 or any order for an integer. The Siamese method for generating magic squares produces panmagic squares for orders with ordinary vector (2, 1) and break vector (1, ).The Lo Shu is not panmagic, but it is an associative magic square. Order four squares can be panmagic or associative, but not both. Order five squares are the smallest..

A magic square for which the number of letters in the word for each number generates another magic square. This definition depends, of course, on the language being used. In English, for example,where the magic square on the right corresponds tothe number of letters in

A magic square in which the elements in each corner have the same sum. Dürer's magic square, illustrated above, is an example of a gnomon magic square since the sums in any of the four quadrants (as well as the sum of the middle four numbers) are all 34 (Hunter and Madachy 1975, p. 24).

A set distinct numbers taken from the interval form a magic series if their sum is the th magic constant(Kraitchik 1942, p. 143). If the sum of the th powers of these numbers is the magic constant of degree for all , then they are said to form a th order multimagic series. Here, the magic constant of degree is defined as times the sum of the first th powers,where is a generalized harmonic number of order .For example is bimagic since and . It is also trimagic since . Similarly, is trimagic.The numbers of magic series of various lengths are gives in the following table for small orders (Kraitchik 1942, p. 76; Boyer), where the , 3, and 4 values were corrected and extended by Boyer and Trump in 2002.SloaneA052456A052457A052458A090037111112200038000486220513948206321349800795733218440083515434038039121091537408202949738126010781325415282464323600114528684996756947689757311870122950111860062822226896106..

A set of distinct numbers taken from the interval form a magic series if their sum is the th magic constant(Kraitchik 1942, p. 143). The numbers of magic series of orders , 2, ..., are 1, 2, 8, 86, 1394, ... (OEIS A052456). The following table gives the first few magic series of small order.magic series12, 3, , , , , , , If the sum of the th powers of these number is the magic constant of degree for all , then they are said to form a th order multimagic series. Here, the magic constant of degree is defined as times the sum of the first th powers,where is a harmonic number of order .

A tetramagic cube is a magic cube that remains magicwhen all its numbers are squared, cubed, and taken to the fourth power.Only two tetramagic cubes are known, and both were found by C. Boyer in 2003. The smaller of these is a tetramagic cube of order 1024; this cube, its square, and its cube are perfect, while its fourth power is only semiperfect. The larger is a perfect tetramagic cube of order 8192; this cube, its square, its cube, and its fourth power are perfect.

A semiperfect magic cube, sometimes also called an "Andrews cube" (Gardner 1976; Gardner 1988, p. 219) is a magic cube for which the cross section diagonals do not sum to the magic constant. Some care is needed with terminology, as some authors drop the "semiperfect" and refer to such cubes simple as "magic cubes" (e.g., Benson and Jacoby 1981, p. 4).A semiperfect magic cube of order 3 has magic constant 42. It must be associative with opposite elements summing to (Andrews 1960, p. 65) and have as its center (Gardner 1976; Benson and Jacoby 1981, p. 4; Andrews 1960, p. 65). Hendricks (1972) proved that there are four distinct semiperfect magic cubes excluding rotations and reflections (Gardner 1976; Benson and Jacoby 1981, pp. 4 and 11-13), illustrated above. These cubes were described by Andrews (1960, pp. 66-70), although he seems not to have noted that they represent..

A pandiagonal semiperfect magic cube is a semiperfect magic cube that remains semiperfect when any single orthogonal section is "restacked" cyclically so that the ordering of any set of plane sections becomes , , , ..., or (Benson and Jacoby 1981, p. 4).There is no pandiagonal semiperfect magic cube of order 3 (Benson and Jacoby 1988, p. 14).The order 4 and 5 magic cubes shown above are pandiagonal semiperfect magic (Benson and Jacoby 1988, pp. 15-23 and 30).

A pandiagonal perfect magic cube is a perfect magic cube that remains perfect when any single orthogonal section is "restacked" cyclically so that the ordering of any set of plane sections becomes , , , ..., or (Benson and Jacoby 1981, p. 4).Pandiagonal perfect magic cubes are possible for orders 8 and 9, but no smaller orders. They are also not possible for orders 12 or 14, but are possible for all orders that are multiples of 8 and odd orders greater than or equal to 9 (Benson and Jacoby 1981, p. 5). Benson and Jacoby (1981, pp. 76-78) explicitly construct a pandiagonal perfect magic cube of order 9.Planck (1950; cited in Gardner 1988) constructed a perfect pandiagonal magic cube.

An -fold multimagic cube is a magic cube that remains magic when each element is squared, cubed, etc., up to th power. (Of course, when the elements of a cube are taken to a power, the numbers are no longer consecutive, so the resulting magic cube is no longer simple.) A 2-fold multimagic cube is called a bimagic cube, a 3-fold multimagic cube is called a trimagic cube, a 4-fold multimagic cube is called a tetramagic cube, and so on.

A magic cube is an version of a magic square in which the rows, columns, pillars, and four space diagonals each sum to a single number known as the cube's magic constant. Magic cubes are most commonly assumed to be "normal," i.e., to have elements that are the consecutive integers 1, 2, ..., . However, this requirement is dropped (as it must be) in the consideration of so-called multimagic cubes.If it exists, a normal magic cube has magic constantFor , 2, ..., the magic constants are given by 1, 9, 42, 130, 315, 651, ... (OEIS A027441).If only rows, columns, pillars, and space diagonals sum to , a magic cube is called a semiperfect magic cube, or sometimes an Andrews cube (Gardner 1988, p. 219). If, in addition, the diagonals of each orthogonal slice sum to , then the magic cube is called a perfect magic cube. If a perfect or semiperfect magic cube is magic not only along the main space diagonals, but also on the broken space diagonals, it is..

A bimagic cube is a (normal) magic cube that remains magic when all its elements are squared. Of course, even a normal magic cubic becomes nonnormal (i.e., contains nonconsecutive elements) upon squaring.Cazalas (1934) attempted but failed to construct a bimagic cube (Boyer). David M. Collison apparently constructed a bimagic cube of order 25 in an unpublished paper (Hendricks 1992), but it was not until the year 2000 that John Hendricks published an order 25 perfect magic cube whose square is a semiperfect magic cube.On January 20, 2003, Christian Boyer discovered an order 16 bimagic cube (where the cube itself is perfect magic, but its square is only semiperfect magic). This was rapidly followed by another order 16 bimagic cube (where the base cube is perfect and its square semiperfect) on January 23, an order 32 bimagic cube (where both the base cube and its square are perfect) on January 27, and an order 27 bimagic cube (where the base..

The number(1)(2)to which the numbers in any horizontal, vertical, or main diagonal line must sum in a magic square. The first few values are 1, 5, 15, 34, 65, 111, 175, 260, ... (OEIS A006003). The magic constant for an th order magic square starting with an integer and with entries in an increasing arithmetic series with difference between terms is(3)(Hunter and Madachy 1975, Madachy 1979). In a panmagic square, in addition to the main diagonals, the broken diagonals also sum to .For a magic cube, magic tesseract, etc., the magic -D constant is(4)(5)The first few magic constants are summarized in the following table.SloaneA006003A027441A021003111125917315421234341305145653151565There is a corresponding multiplicative magic constant for multiplicationmagic squares.A similar magic constant of degree is defined for magic series and multimagic series as times the sum of the first th powers,(6)(7)where is a harmonic number of order..

If replacing each number by its square in a magic square produces another magic square, the square is said to be a bimagic square. Bimagic squares are also called doubly magic squares, and are 2-multimagic squares.Lucas (1891) and later Hendricks (1998) showed that a bimagic square of order 3 is impossible for any set of numbers except the trivial case of using the same number 9 times.The first known bimagic square, constructed by Pfeffermann (1891a; left figure), had order 8 with magic constant 260 for the base square and after squaring. Another order 8 bimagic square is shown at right.Benson and Jacoby (1976) stated their belief that no bimagic squares of order less than 8 exist, and this was subsequently proved by Boyer and Trump in 2002 (Boyer).Pfeffermann (1891b) also published the first 9th-order bimagic square. Only a part of the first Pfeffermann's bimagic squares of both order 8 and of order 9 were published, with their completion left as..

A prime magic square is a magic square consisting only of prime numbers (although the number 1 is sometimes allowed in such squares). The left square is the prime magic square (containing a 1) having the smallest possible magic constant, and was discovered by Dudeney in 1917 (Dudeney 1970; Gardner 1984, p. 86). The second square is the magic square consisting of primes only having the smallest possible magic constant (Madachy 1979, p. 95; attributed to R. Ondrejka). The third square is the prime magic square consisting of primes in arithmetic progression () having the smallest possible magic constant of 3117 (Madachy 1979, p. 95; attributed to R. Ondrejka). The prime magic square on the right was found by A. W. Johnson, Jr. (Dewdney 1988).According to a 1913 proof of J. N. Muncey (cited in Gardner 1984, pp. 86-87), the smallest magic square composed of consecutive odd primes including..

An edge-magic graph is a labeled graph with graph edges labeled with distinct elements so that the sum of the graph edge labels at each graph vertex is the same.A vertex-magic graph labeled graph vertices which give the same sum along every straight line segment. No magic pentagrams can be formed with the number 1, 2, ..., 10 (Trigg 1960; Langman 1962, pp. 80-83; Dongre 1971; Richards 1975; Buckley and Rubin 1977-1978; Trigg 1998), but 168 almost magic pentagrams (in which the sums are the same for four of the five lines) can. The figure above show a magic pentagram with sums 24 built using the labels 1, 2, 3, 4, 5, 6, 8, 9, 10, and 12 (Madachy 1979).

A magic hexagon of order is an arrangement of close-packed hexagons containing the numbers 1, 2, ..., , where is the th hex number such that the numbers along each straight line add up to the same sum. (Here, the hex numbers are i.e., 1, 7, 19, 37, 61, 91, 127, ...; OEIS A003215). In the above magic hexagon of order , each line (those of lengths 3, 4, and 5) adds up to 38.It was discovered independently by Ernst von Haselberg in 1887 (Bauch 1990, Hemme 1990), W. Radcliffe in 1895 (Tapson 1987, Hemme 1990, Heinz), H. Lulli (Hendricks, Heinz), Martin Kühl in 1940 (Gardner 1963, 1984; Honsberger 1973), Clifford W. Adams, who worked on the problem from 1910 to 1957 (Gardner 1963, 1984; Honsberger 1973), and Vickers (1958; Trigg 1964).This problem and the solution have a long history. Adams came across the problem in 1910. He worked on the problem by trial and error and after many years arrived at the solution which he transmitted to M. Gardner,..

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