An antimagic square of order n is an arrangement of the numbers 1 to n^{2} in a square, such that the sums of the n rows, the n columns and the two diagonals form a sequence of 2n + 2 consecutive integers. The smallest antimagic squares have order 4.^{ [1] } Antimagic squares contrast with magic squares, where each row, column, and diagonal sum must have the same value.^{ [2] }


In both of these antimagic squares of order 4, the rows, columns and diagonals sum to ten different numbers in the range 29–38.^{ [2] }


In the antimagic square of order 5 on the left, the rows, columns and diagonals sum up to numbers between 60 and 71.^{ [2] } In the antimagic square on the right, the rows, columns and diagonals add up to numbers between 5970.^{ [1] }
The following questions about antimagic squares have not been solved.^{[ citation needed ]}
A sparse antimagic square (SAM) is a square matrix of size n by n of nonnegative integers whose nonzero entries are the consecutive integers for some , and whose rowsums and columnsums constitute a set of consecutive integers.^{ [3] } If the diagonals are included in the set of consecutive integers, the array is known as a sparse totally antimagic square (STAM). Note that a STAM is not necessarily a SAM, and vice versa.
A filling of the n × n square with the numbers 1 to n^{2} in a square, such that the rows, columns, and diagonals all sum to different values has been called a heterosquare.^{ [4] } (Thus, they are the relaxation in which no particular values are required for the row, column, and diagonal sums.) There are no heterosquares of order 2, but heterosquares exist for any order n ≥ 3: if n is odd, filling the square in a spiral pattern will produce a heterosquare.^{ [4] } And if n is even, a heterosquare results from writing the numbers 1 to n^{2} in order, then exchanging 1 and 2. It is suspected that there are exactly 3120 essentially different heterosquares of order 3.^{ [5] }
Amicable numbers are two different numbers related in such a way that the sum of the proper divisors of each is equal to the other number.
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, and Italy.
In 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 diagonals are the same. The order of the magic square is the number of integers along one side (n), and the constant sum is called the magic constant. If the array includes just the positive integers , the magic square is said to be normal. Some authors take magic square to mean normal magic square.
In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan. In an example of Stigler's law of eponymy, they are named after Eric Temple Bell, who wrote about them in the 1930s.
An npointed magic star is a star polygon with Schläfli symbol {n/2} in which numbers are placed at each of the n vertices and n intersections, such that the four numbers on each line sum to the same magic constant. A normal magic star contains the consecutive integers 1 to 2n. No numbers are ever repeated. The magic constant of an npointed normal magic star is M = 4n + 2.
In mathematics, a magic cube is the 3dimensional equivalent of a magic square, that is, a number of integers arranged in a n × n × n pattern such that the sums of the numbers on each row, on each column, on each pillar and on each of the four main space diagonals are equal to the same number, the socalled magic constant of the cube, denoted M_{3}(n). It can be shown that if a magic cube consists of the numbers 1, 2, ..., n^{3}, then it has magic constant
In mathematics, a perfect magic cube is a magic cube in which not only the columns, rows, pillars, and main space diagonals, but also the cross section diagonals sum up to the cube's magic constant.
In mathematics, a Pmultimagic square is a magic square that remains magic even if all its numbers are replaced by their kth power for 1 ≤ k ≤ P. Thus, a magic square is bimagic if it is 2multimagic, and trimagic if it is 3multimagic; tetramagic for 4multimagic; and pentamagic for a 5multimagic square.
The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–91), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences.
A pandiagonal magic square or panmagic 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 square, also add up to the 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. In general where is the side length of the square.
In geometry, a space 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 as each other.
In mathematics, the boustrophedon transform is a procedure which maps one sequence to another. The transformed sequence is computed by an "addition" operation, implemented as if filling a triangular array in a boustrophedon manner—as opposed to a "Raster Scan" sawtoothlike manner.
In number theory, friendly numbers are two or more natural numbers with a common abundancy index, the ratio between the sum of divisors of a number and the number itself. Two numbers with the same "abundancy" form a friendly pair; n numbers with the same "abundancy" form a friendly ntuple.
In mathematics, a perfect power is a positive integer that can be resolved into equal factors, and whose root can be exactly extracted, i.e., a positive integer that can be expressed as an integer power of another positive integer. More formally, n is a perfect power if there exist natural numbers m > 1, and k > 1 such that m^{k} = n. In this case, n may be called a perfect kth power. If k = 2 or k = 3, then n is called a perfect square or perfect cube, respectively. Sometimes 0 and 1 are also considered perfect powers.
A Pythagorean quadruple is a tuple of integers a, b, c, and d, such that a^{2} + b^{2} + c^{2} = d^{2}. They are solutions of a Diophantine equation and often only positive integer values are considered. However, to provide a more complete geometric interpretation, the integer values can be allowed to be negative and zero with the only condition being that d > 0. In this setting, a Pythagorean quadruple (a, b, c, d) defines a cuboid with integer side lengths a, b, and c, whose space diagonal has integer length d; with this interpretation, Pythagorean quadruples are thus also called Pythagorean boxes. In this article we will assume, unless otherwise stated, that the values of a Pythagorean quadruple are all positive integers.
In mathematics, a Delannoy number describes the number of paths from the southwest corner of a rectangular grid to the northeast corner, using only single steps north, northeast, or east. The Delannoy numbers are named after French army officer and amateur mathematician Henri Delannoy.
The Siamese method, or De la Loubère method, is a simple method to construct any size of nodd magic squares. The method was brought to France in 1688 by the French mathematician and diplomat Simon de la Loubère, as he was returning from his 1687 embassy to the kingdom of Siam. The Siamese method makes the creation of magic squares straightforward.
In mathematics, the Wythoff array is an infinite matrix of integers derived from the Fibonacci sequence and named after Dutch mathematician Willem Abraham Wythoff. Every positive integer occurs exactly once in the array, and every integer sequence defined by the Fibonacci recurrence can be derived by shifting a row of the array.