A magic series is a set of distinct positive integers which add up to the magic constant of a magic square and a magic cube , thus potentially making up lines in magic tesseracts .
31-572: So, in an n × n magic square using the numbers from 1 to n , a magic series is a set of n distinct numbers adding up to n ( n + 1)/2. For n = 2, there are just two magic series, 1+4 and 2+3. The eight magic series when n = 3 all appear in the rows, columns and diagonals of a 3 × 3 magic square. Maurice Kraitchik gave the number of magic series up to n = 7 in Mathematical Recreations in 1942 (sequence A052456 in
62-478: A denominator of n 3 − 3 5 n 2 + ( 2 7 + 1 2100 ) n + ⋯ . {\displaystyle n^{3}-{\tfrac {3}{5}}n^{2}+\left({\tfrac {2}{7}}+{\tfrac {1}{2100}}\right)\!n+\cdots .} Richard Schroeppel in 1973 published the complete enumeration of the order 5 magic squares at 275,305,224. This recent magic series work gives hope that
93-734: A periodical devoted to recreational mathematics . During World War II, he emigrated to the United States , where he taught a course at the New School for Social Research in New York City on the general topic of "mathematical recreations." Kraïtchik was agrégé of the Free University of Brussels , engineer at the Société Financière de Transports et d'Entreprises Industrielles (Sofina) , and director of
124-409: A finite even number, and odd otherwise. Let R be a commutative ring and let I be an ideal of R whose index is 2. Elements of the coset 0 + I {\displaystyle 0+I} may be called even , while elements of the coset 1 + I {\displaystyle 1+I} may be called odd . As an example, let R = Z (2) be the localization of Z at
155-410: A number expressed in the binary numeral system is odd if its last digit is 1; and it is even if its last digit is 0. In an odd base, the number is even according to the sum of its digits—it is even if and only if the sum of its digits is even. An even number is an integer of the form x = 2 k {\displaystyle x=2k} where k is an integer; an odd number is an integer of
186-468: A permutation (as defined in abstract algebra ) is the parity of the number of transpositions into which the permutation can be decomposed. For example (ABC) to (BCA) is even because it can be done by swapping A and B then C and A (two transpositions). It can be shown that no permutation can be decomposed both in an even and in an odd number of transpositions. Hence the above is a suitable definition. In Rubik's Cube , Megaminx , and other twisting puzzles,
217-533: Is divisible by 2, and odd if it is not. For example, −4, 0, and 82 are even numbers, while −3, 5, 7, and 21 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. 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,
248-614: Is a prime ideal of Z {\displaystyle \mathbb {Z} } and the quotient ring Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } is the field with two elements . Parity can then be defined as the unique ring homomorphism from Z {\displaystyle \mathbb {Z} } to Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } where odd numbers are 1 and even numbers are 0. The consequences of this homomorphism are covered below. The following laws can be verified using
279-410: Is a number that has an odd number of 1's in its binary representation; these numbers play an important role in the strategy for the game Kayles . The parity function maps a number to the number of 1's in its binary representation, modulo 2 , so its value is zero for evil numbers and one for odious numbers. The Thue–Morse sequence , an infinite sequence of 0's and 1's, has a 0 in position i when i
310-427: Is an integer, it will be even if and only if the dividend has more factors of two than the divisor. The ancient Greeks considered 1, the monad , to be neither fully odd nor fully even. Some of this sentiment survived into the 19th century: Friedrich Wilhelm August Fröbel 's 1826 The Education of Man instructs the teacher to drill students with the claim that 1 is neither even nor odd, to which Fröbel attaches
341-529: Is congruent to 0 modulo 2, and odd if it is congruent to 1 modulo 2. All prime numbers are odd, with one exception: the prime number 2. All known perfect numbers are even; it is unknown whether any odd perfect numbers exist. Goldbach's conjecture states that every even integer greater than 2 can be represented as a sum of two prime numbers. Modern computer calculations have shown this conjecture to be true for integers up to at least 4 × 10 , but still no general proof has been found. The parity of
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#1732776163010372-413: Is evil, and a 1 in that position when i is odious. In information theory , a parity bit appended to a binary number provides the simplest form of error detecting code . If a single bit in the resulting value is changed, then it will no longer have the correct parity: changing a bit in the original number gives it a different parity than the recorded one, and changing the parity bit while not changing
403-400: Is far from obvious. The parity of a function describes how its values change when its arguments are exchanged with their negations. An even function, such as an even power of a variable, gives the same result for any argument as for its negation. An odd function, such as an odd power of a variable, gives for any argument the negation of its result when given the negation of that argument. It
434-399: Is not true for normal integer arithmetic. By construction in the previous section, the structure ({even, odd}, +, ×) is in fact the field with two elements . The division of two whole numbers does not necessarily result in a whole number. For example, 1 divided by 4 equals 1/4, which is neither even nor odd, since the concepts of even and odd apply only to integers. But when the quotient
465-511: Is possible for a function to be neither odd nor even, and for the case f ( x ) = 0, to be both odd and even. The Taylor series of an even function contains only terms whose exponent is an even number, and the Taylor series of an odd function contains only terms whose exponent is an odd number. In combinatorial game theory , an evil number is a number that has an even number of 1's in its binary representation , and an odious number
496-558: The Institut des Hautes Etudes de Belgique . He died in Brussels . Kraïtchik is famous for having inspired the two envelopes problem in 1953, with the following puzzle in La mathématique des jeux : Among his publications were the following: Parity (mathematics) In mathematics , parity is the property of an integer of whether it is even or odd . An integer is even if it
527-485: The OEIS ). In 2002, Henry Bottomley extended this up to n = 36 and independently Walter Trump up to n = 32. In 2005, Trump extended this to n = 54 (over 2 × 10) while Bottomley gave an experimental approximation for the numbers of magic series: In July 2006, Robert Gerbicz extended this sequence up to n = 150. In 2013 Dirk Kinnaes was able to exploit his insight that
558-422: The face-centered cubic lattice and its higher-dimensional generalizations (the D n lattices ) consist of all of the integer points whose coordinates have an even sum. This feature also manifests itself in chess , where the parity of a square is indicated by its color: bishops are constrained to moving between squares of the same parity, whereas knights alternate parity between moves. This form of parity
589-420: 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 the last digit of any even number is 0, 2, 4, 6, or 8. The same idea will work using any even base. In particular,
620-400: The prime ideal (2). Then an element of R is even or odd if and only if its numerator is so in Z . The even numbers form an ideal in the ring of integers, but the odd numbers do not—this is clear from the fact that the identity element for addition, zero, is an element of the even numbers only. An integer is even if it is congruent to 0 modulo this ideal, in other words if it
651-702: The form x = 2 k + 1. {\displaystyle x=2k+1.} An equivalent definition is that an even number is divisible by 2: 2 | x {\displaystyle 2\ |\ x} and an odd number is not: 2 ⧸ | x {\displaystyle 2\not |\ x} The sets of even and odd numbers can be defined as following: { 2 k : k ∈ Z } {\displaystyle \{2k:k\in \mathbb {Z} \}} { 2 k + 1 : k ∈ Z } {\displaystyle \{2k+1:k\in \mathbb {Z} \}} The set of even numbers
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#1732776163010682-536: The magic series could be related to the volume of a polytope . Trump used this new approach to extend the sequence up to n = 1000. Mike Quist showed that the exact second-order count has a multiplicative factor of 1 n 3 ( 1 + 3 5 n + 31 420 n 2 + ⋯ ) {\displaystyle {\tfrac {1}{n^{3}}}\!\left(1+{\tfrac {3}{5n}}+{\tfrac {31}{420n^{2}}}+\cdots \right)} equivalent to
713-506: The mouthpiece, the harmonics produced are odd multiples of the fundamental frequency . (With cylindrical pipes open at both ends, used for example in some organ stops such as the open diapason , the harmonics are even multiples of the same frequency for the given bore length, but this has the effect of the fundamental frequency being doubled and all multiples of this fundamental frequency being produced.) See harmonic series (music) . In some countries, house numberings are chosen so that
744-432: The moves of the puzzle allow only even permutations of the puzzle pieces, so parity is important in understanding the configuration space of these puzzles. The Feit–Thompson theorem states that a finite group is always solvable if its order is an odd number. This is an example of odd numbers playing a role in an advanced mathematical theorem where the method of application of the simple hypothesis of "odd order"
775-400: The number it was derived from again produces an incorrect result. In this way, all single-bit transmission errors may be reliably detected. Some more sophisticated error detecting codes are also based on the use of multiple parity bits for subsets of the bits of the original encoded value. In wind instruments with a cylindrical bore and in effect closed at one end, such as the clarinet at
806-401: The philosophical afterthought, It is well to direct the pupil's attention here at once to a great far-reaching law of nature and of thought. It is this, that between two relatively different things or ideas there stands always a third, in a sort of balance, seeming to unite the two. Thus, there is here between odd and even numbers one number (one) which is neither of the two. Similarly, in form,
837-473: The properties of divisibility . They are a special case of rules in modular arithmetic , and are commonly used to check if an equality is likely to be correct by testing the parity of each side. As with ordinary arithmetic, multiplication and addition are commutative and associative in modulo 2 arithmetic, and multiplication is distributive over addition. However, subtraction in modulo 2 is identical to addition, so subtraction also possesses these properties, which
868-484: The relationship between the magic series and the magic square may provide an exact count for order 6 or order 7 magic squares. Consider an intermediate structure that lies in complexity between the magic series and the magic square. It might be described as an amalgamation of 4 magic series that have only one unique common integer. This structure forms the two major diagonals and the central row and column for an odd order magic square. Building blocks such as these could be
899-485: The right angle stands between the acute and obtuse angles; and in language, the semi-vowels or aspirants between the mutes and vowels. A thoughtful teacher and a pupil taught to think for himself can scarcely help noticing this and other important laws. Integer coordinates of points in Euclidean spaces of two or more dimensions also have a parity, usually defined as the parity of the sum of the coordinates. For instance,
930-461: The way forward. Maurice Kraitchik Maurice Borisovich Kraitchik (21 April 1882 – 19 August 1957) was a Belgian mathematician and populariser . His main interests were the theory of numbers and recreational mathematics . He was born to a Jewish family in Minsk . He wrote several books on number theory during 1922–1930 and after the war, and from 1931 to 1939 edited Sphinx ,
961-418: Was famously used to solve the mutilated chessboard problem : if two opposite corner squares are removed from a chessboard, then the remaining board cannot be covered by dominoes, because each domino covers one square of each parity and there are two more squares of one parity than of the other. The parity of an ordinal number may be defined to be even if the number is a limit ordinal, or a limit ordinal plus