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 ''P''-multimagic square (also known as a satanic square) is a

magic square In mathematics, especially History of mathematics, historical and recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diago ...

that remains magic even if all its numbers are replaced by their ''k''th powers for 1 ≤ ''k'' ≤ ''P''. squares are called bimagic, squares are called trimagic, squares tetramagic, and squares pentamagic.

Constants for normal squares

If the squares are normal, the constant for the power-squares can be determined as follows: Bimagic series totals for bimagic squares are also linked to the square-pyramidal number sequence is as follows :-
Squares 0, 1, 4, 9, 16, 25, 36, 49, ....
Sum of Squares 0, 1, 5, 14, 30, 55, 91, 140, 204, 285, ... )number of units in a square-based pyramid)
The bimagic series is the 1st, 4th, 9th in this series (divided by 1, 2, 3, ''n'') etc. so values for the rows and columns in order-1, order-2, order-3 Bimagic squares would be 1, 15, 95, 374, 1105, 2701, 5775, 11180, ... The trimagic series would be related in the same way to the hyper-pyramidal sequence of nested cubes.
Cubes 0, 1, 8, 27, 64, 125, 216, ...
Sum of Cubes 0, 1, 9, 36, 100, ...
Value for Trimagic squares 1, 50, 675, 4624, ... Similarly the tetramagic sequence
4-Power 0, 1, 16, 81, 256, 625, 1296, ...
Sum of 4-Power 0, 1, 17, 98, 354, 979, 2275, ...
Sums for Tetramagic squares 0, 1, 177, ...

Bimagic square

A bimagic square is a magic square that remains magic when all of its numbers are replaced by their

squares In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

. The first known bimagic square has

order Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...

8 and

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 ...

260 and a bimagic constant of 11180. It has been

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

d by Bensen and Jacoby that no nontrivial bimagic squares of order less than 8 exist. This was shown for magic squares containing the elements 1 to ''n''² by Boyer and Trump. However, J. R. Hendricks was able to show in 1998 that no bimagic square of order 3 exists, save for the trivial bimagic square containing the same number nine times. The

proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a co ...

is fairly simple: let the following be our bimagic square. : It is well known that a property of magic squares is that

a+i=2e

. Similarly,

a^2+i^2=2e^2

. Therefore,

(a-i)^2=2(a^2+i^2)-(a+i)^2=4e^2-4e^2=0

. It follows that

a=e=i

. The same holds for all lines going through the center. For 4 × 4 squares, Luke Pebody was able to show by similar methods that the only 4 × 4 bimagic squares (up to symmetry) are of the form or An 8 × 8 bimagic square. Nontrivial bimagic squares are now (2010) known for any order from eight to 64. Li Wen of China created the first known bimagic squares of orders 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62 filling the gaps of the last unknown orders. In 2006 Jaroslaw Wroblewski built a non-normal bimagic square of order 6. Non-normal means that it uses non-consecutive

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s. Also in 2006 Lee Morgenstern built several non-normal bimagic squares of order 7.

Trimagic square

A trimagic square is a magic square that remains magic when all of its numbers are replaced by their

cubes A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

. Trimagic squares of orders 12, 32, 64, 81 and 128 have been discovered so far; the only known trimagic square of order 12, given below, was found in June 2002 by German mathematician Walter Trump.

Higher order

The first 4-magic square was constructed by Charles Devimeux in 1983 and was a 256-order square. A 4-magic square of order 512 was constructed in May 2001 by

André Viricel André — sometimes transliterated as Andre — is the French and Portuguese form of the name Andrew and is now also used in the English-speaking world. It used in France, Quebec, Canada and other French-speaking countries, as well in Portugal, ...

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 ...

. The first 5-magic square, of order 1024 arrived about one month later, in June 2001 again by Viricel and Boyer. They also presented a smaller 4-magic square of order 256 in January 2003. Another 5-magic square, of order 729, was constructed in June 2003 by Li Wen.

References

* * * * *

External links