mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a perfect magic cube is a

magic cube In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a collection of integers arranged in an ''n'' × ''n'' × ''n'' pattern such that the sums of the numbers on each row, on each c ...

in which not only the columns, rows, pillars, and main

space diagonal In geometry, a space diagonal (also interior diagonal or body diagonal) of a polyhedron is a line connecting two vertices that are not on the same face. Space diagonals contrast with '' face diagonals'', which connect vertices on the same face (b ...

s, but also the

cross section Cross section may refer to: * Cross section (geometry) ** Cross-sectional views in architecture and engineering 3D *Cross section (geology) * Cross section (electronics) * Radar cross section, measure of detectability * Cross section (physics) **A ...

diagonals sum up to the cube's

magic constant The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. For a normal magic square of order ''n'' – that is ...

. Perfect magic cubes of order one are trivial; cubes of orders two to four can be

proven Proven is a rural village in the Belgian province of West Flanders, and a "deelgemeente" of the municipality Poperinge. The village has about 1400 inhabitants. The church and parish A parish is a territorial entity in many Christianity, Chr ...

not to exist, and cubes of orders five and six were first discovered by

Walter Trump Walter Trump (born 1953) is a German mathematician and retired high school teacher. He is known for his work in recreational mathematics. He has made contributions working on both the square packing problem and the magic tile problem. In 1979 he ...

and

Christian Boyer A Christian () is a person who follows or adheres to Christianity, a monotheistic Abrahamic religion based on the life and teachings of Jesus Christ. Christians form the largest religious community in the world. The words ''Christ'' and ''Chr ...

on November 13 and September 1, 2003, respectively. A perfect magic cube of order seven was given by A. H. Frost in 1866, and on March 11, 1875, an article was published in the

Cincinnati Commercial The ''Cincinnati Commercial Tribune'' was a major daily newspaper in Cincinnati, Ohio, formed in 1896, and folded in 1930.(3 December 1930)OLDEST NEWSPAPER IN CINCINNATI QUITS; Commercial Tribune Stopped by McLean Interests After Political Shift ...

newspaper on the discovery of a perfect magic cube of order 8 by Gustavus Frankenstein. Perfect magic cubes of orders nine and eleven have also been constructed. The first perfect cube of order 10 was constructed in 1988 (Li Wen, China).

An alternative definition

In recent years, an alternative definition for the perfect magic cube was proposed by

John R. Hendricks John Robert Hendricks (September 4, 1929 – July 7, 2007) was a Canadian amateur mathematician notable for his work in magic squares and hypercubes. He published many articles in the Journal of Recreational Mathematics as well as other mathemati ...

. By this definition, a perfect magic cube is one in which all possible lines through each cell sum to the magic constant. The name

Nasik magic hypercube In mathematics, a magic hypercube is the ''k''-dimensional generalization of magic squares and magic cubes, that is, an ''n'' × ''n'' × ''n'' × ... × ''n'' array of integers such that the sums of the numbers on each pillar (along any axis) as ...

is another, unambiguous, name for such a cube. This definition is based on the fact that a

pandiagonal magic square A pandiagonal magic square or panmagic square (also diabolic square, diabolical square or diabolical magic square) is a magic square with the additional property that the broken diagonals, i.e. the diagonals that wrap round at the edges of the squ ...

has traditionally been called 'perfect', because all possible lines sum correctly. This same reasoning may be applied to

hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...

s of any dimension. Simply stated; in an order ''m'' magic hypercube, if all possible lines of ''m'' cells sum to the magic constant, the hypercube is perfect. All lower dimension hypercubes contained in this hypercube will then also be perfect. This is not the case with the original definition, which does not require that the planar and diagonal squares be a

pandiagonal magic cube In recreational mathematics, a pandiagonal magic cube is a magic cube with the additional property that all broken diagonals (parallel to exactly two of the three coordinate axes) have the same sum as each other. Pandiagonal magic cubes are exten ...

. For example, a magic cube of order 8 has 244 correct lines by the ''old'' definition of "perfect", but 832 correct lines by this ''new'' definition. The smallest perfect magic cube has order 8, and none can exist for double odd orders. Gabriel Arnoux constructed an order 17 perfect magic cube in 1887. F.A.P.Barnard published order 8 and order 11 perfect cubes in 1888. By the modern (given by J.R. Hendricks) definition, there are actually six classes of magic cube;

simple magic cube A simple magic cube is the lowest of six basic classes of magic cubes. These classes are based on extra features required. The simple magic cube requires only the basic features a cube requires to be magic. Namely, all lines parallel to the face ...

pantriagonal magic cube {{no footnotes, date=December 2014 A pantriagonal magic cube is a magic cube where all 4''m''2 pantriagonals sum correctly. There are 4 one-segment pantriagonals, 12(''m'' − 1) two-segment pantriagonals, and 4(''m'' − 2)( ...

diagonal magic cube The class of diagonal magic cubes is the second of the six magic cube classes (when ranked by the number of lines summing correctly), coming after the simple magic cubes. In a diagonal magic cube of order ''m'',Traditionally, ''n'' has been used ...

s, pantriagonal diagonal magic cubes,

s, and perfect magic cubes.

Examples

1. Order 4 cube by Thomas Krijgsman, 1982; magic constant 130. 2. Order 5 cube by

and Christian Boyer, 2003-11-13; magic constant 315.

References

* * Planck, C., The Theory of Paths Nasik, Printed for private circulation, A.J. Lawrence, Printer, Rugby,(England), 1905 * H.D, Heinz & J.R. Hendricks, ''Magic Square Lexicon: Illustrated'', hdh, 2000, 0-9687985-0-0

External links

* * {{cite web, first1=Walter, last1=Trump, url=http://www.trump.de/magic-squares/magic-cubes/cubes-1.html, title=Perfect magic cube of order 6 found
Christian Boyer: Perfect magic cubes

MathWorld news: Perfect magic cube of order 5 discovered

Aale de Winkel: The Magic Encyclopedia

* Most-perfect cube https://oeis.org/A270205 Magic squares

An alternative definition

Examples

See also

References

External links