Antimagic Square
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 2''n'' + 2 consecutive integers. The smallest antimagic squares have order 4. Antimagic squares contrast with magic squares, where each row, column, and diagonal sum must have the same value. Examples Order 4 antimagic squares In both of these antimagic squares of order 4, the rows, columns and diagonals sum to ten different numbers in the range 29–38. Order 5 antimagic squares In the antimagic square of order 5 on the left, the rows, columns and diagonals sum up to numbers between 60 and 71. In the antimagic square on the right, the rows, columns and diagonals add up to numbers in the range 59–70. Generalizations A sparse antimagic square (SAM) is a square matrix of size ''n'' by ''n'' of nonnegative integers whose nonzero entries are the consecutive integers 1,\ldots ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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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 integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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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 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 1,2,...,n^2, the magic square is said to be "normal". Some authors take "magic square" to mean "normal magic square". Magic squares that include repeated entries do not fall under this definition and are referred to as "trivial". Some well-known examples, including the #Sagrada Família magic square, Sagrada Família magic square and the #Parker square, Parker square, are trivial in this sense. When all the rows and columns but not both diagonals sum to the magic constant, this gives a semimagic square (sometimes called orthomagic square). ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Parity (mathematics)
In mathematics, parity is the Property (mathematics), property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers with decimals or fractions like 1/2 or 4.6978. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Spiral
In mathematics, a spiral is a curve which emanates from a point, moving further away as it revolves around the point. It is a subtype of whorled patterns, a broad group that also includes concentric objects. Two-dimensional A two-dimensional, or plane, spiral may be easily described using polar coordinates, where the radius r is a monotonic continuous function of angle \varphi: * r=r(\varphi)\; . The circle would be regarded as a degenerate case (the function not being strictly monotonic, but rather constant). In ''x-y-coordinates'' the curve has the parametric representation: * x=r(\varphi)\cos\varphi \ ,\qquad y=r(\varphi)\sin\varphi\; . Examples Some of the most important sorts of two-dimensional spirals include: * The Archimedean spiral: r=a \varphi * The hyperbolic spiral: r = a/ \varphi * Fermat's spiral: r= a\varphi^ * The lituus: r = a\varphi^ * The logarithmic spiral: r=ae^ * The Cornu spiral or ''clothoid'' * The Fibonacci spiral and golden spiral * ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Frénicle Standard Form
A magic square is in the Frénicle standard form, named for Bernard Frénicle de Bessy, if the following two conditions hold: # the element at position ,1(top left corner) is the smallest of the four corner elements; and # the element at position ,2(top edge, second from left) is smaller than the element in ,1 In 1693, Frénicle described all the 880 essentially different order-4 magic squares. Properties This standard form was devised since a magic square remains "essentially similar" if it is rotated or transposed, or flipped so that the order of rows is reversed. There exist 8 different magic squares sharing one standard form. For example, the following magic squares are all essentially similar, with only the final square being in the Frénicle standard form: 8 1 6 8 3 4 4 9 2 4 3 8 6 7 2 6 1 8 2 9 4 2 7 6 3 5 7 1 5 9 3 5 7 9 5 1 1 5 9 7 5 3 7 5 3 9 5 1 4 9 2 6 7 2 8 1 6 2 7 6 8 3 4 2 9 4 6 1 8 4 3 8 Generali ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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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 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 1,2,...,n^2, the magic square is said to be "normal". Some authors take "magic square" to mean "normal magic square". Magic squares that include repeated entries do not fall under this definition and are referred to as "trivial". Some well-known examples, including the #Sagrada Família magic square, Sagrada Família magic square and the #Parker square, Parker square, are trivial in this sense. When all the rows and columns but not both diagonals sum to the magic constant, this gives a semimagic square (sometimes called orthomagic square). ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |