13 is the natural number following 12 and preceding 14. Strikingly folkloric aspects of the number 13 have been noted in various cultures around the world: one theory is that this is due to the cultures employing lunar-solar calendars; this can be witnessed, in the "Twelve Days of Christmas" of Western European tradition. The number 13 is: the sixth prime number; the smallest emirp. One of only three known Wilson primes. A Fibonacci number. A happy number; the third centered square number. A lucky number. Equal to the sum of the squares of the digits of its own square in bases 4 and 83; the smallest number whose fourth power can be written as a sum of two consecutive square numbers. Since 52 + 122 = 132, forms a Pythagorean triple. There are 13 Archimedean solids, a standard torus can be sliced into 13 pieces with just 3 plane cuts. There are 13 different ways for the three fastest horses in a horse race to finish, allowing for ties, a fact that can be expressed mathematically by 13 being the third ordered Bell number.
In all Germanic languages, 13 is the first compound number. The Romance languages use different systems: In Italian, 11 is the first compound number, as in Romanian, while in Spanish and Portuguese, the numbers up to and including 15, in French up to and including 16 have their own names; this is the case in most Slavic languages, Hindi-Urdu and other South Asian languages. In Germany, according to an old tradition, 13 -as the first compound number- was the first number written in digits; the Duden now calls this tradition outdated and no longer valid. In Shia Islam, 13 signifies the 13th day of the month of Rajab, the birth of Imam Ali. 13 is a total of 1 Prophet and 12 Imams in the Islamic School of Thought. However, in Sunni Islam, the number 13 bears no symbolic significance; the apparitions of the Virgin of Fátima in 1917 were claimed to occur on the 13th day of six consecutive months. In Catholic devotional practice, the number thirteen is associated with Saint Anthony of Padua, since his feast day falls on June 13.
A traditional devotion called the Thirteen Tuesdays of St. Anthony involves praying to the saint every Tuesday over a period of thirteen weeks. Another devotion, St. Anthony's Chaplet, consists of thirteen decades of three beads each. According to famous Sakhi or story of Guru Nanak Dev Ji, when he was an accountant at a town of Sultanpur Lodhi, he was distributing groceries to people; when he gave groceries to the 13th person, he stopped because in Gurmukhi and Hindi the word 13 is called Terah, which means yours. And Guru Nanak Dev Ji kept saying, "Yours, yours..." remembering God. People reported to the emperor; when treasures were checked, there was more money than before. The Vaisakhi, which commemorates the creation of "Khalsa" or pure Sikh was celebrated on April 13 for many years. In Judaism, 13 signifies the age at which a boy matures and becomes a Bar Mitzvah, i.e. a full member of the Jewish faith. The number of principles of Jewish faith according to Maimonides. According to Rabbinic commentary on the Torah, God has 13 Attributes of Mercy.
The number 13 had been considered sinister and wicked in ancient Iranian civilization and Zoroastrianism. Since beginning of the Nourooz tradition, the 13th day of each new Iranian year is called Sizdah Be-dar, this tradition is still alive among Iranian people both within Iran and abroad. Since Sizdah Be-dar is the 13th day of the year, it is considered a day when evil's power might cause difficulties for people. Therefore, people leave camp in the countryside. In the current post-1979 Revolution era, despite the wishes of the Islamic government, this tradition continues to be practiced by the majority of the population throughout Iran; the Thirteen Classics is considered to be a part of the Chinese classics. The number 13 is considered an unlucky number in some countries; the end of the Mayan calendar's 13th Baktun was superstitiously feared as a harbinger of the apocalyptic 2012 phenomenon. Fear of the number 13 has a recognized phobia, Triskaidekaphobia, a word coined in 1911; the superstitious sufferers of triskaidekaphobia try to avoid bad luck by keeping away from anything numbered or labelled thirteen.
As a result and manufacturers use another way of numbering or labelling to avoid the number, with hotels and tall buildings being conspicuous examples. It is considered unlucky to have thirteen guests at a table. Friday the 13th has been considered an unlucky day. There are a number of theories as to why the number thirteen became associated with bad luck, but none of them have been accepted as likely; the Last Supper: At Jesus Christ's last supper, there were thirteen people around the table, counting Christ and the twelve apostles. Some believe this is unlucky because one of those thirteen, Judas Iscariot, was the betrayer of Jesus Christ. From the 1890s, a number of English language sources relate the "unlucky" thirteen to an idea that at the Last Supper, the disciple who betrayed Jesus, was the 13th to sit at the table. Knights Templar: On Friday 13 October 1307, King Philip IV of
Chinese checkers or Chinese chequers is a strategy board game of German origin which can be played by two, four, or six people, playing individually or with partners. The game is a simplified variation of the game Halma; the objective is to be first to race all of one's pieces across the hexagram-shaped board into "home"—the corner of the star opposite one's starting corner—using single-step moves or moves that jump over other pieces. The remaining players continue the game to establish second-, third-, fourth-, fifth-, last-place finishers; the rules are simple, so young children can play. Despite its name, the game is not a variation of checkers, nor did it originate in China or any part of Asia; the game was invented in Germany in 1892 under the name "Stern-Halma" as a variation of the older American game Halma. The "Stern" refers to the board's star shape; the name "Chinese Checkers" originated in the United States as a marketing scheme by Bill and Jack Pressman in 1928. The Pressman company's game was called "Hop Ching Checkers".
The game was introduced to Chinese-speaking regions by the Japanese. The aim is to race all one's pieces into the star corner on the opposite side of the board before opponents do the same; the destination corner is called home. Each player has 10 pieces, except in games between two players. In "hop across", the most popular variation, each player starts with their colored pieces on one of the six points or corners of the star and attempts to race them all home into the opposite corner. Players take turns moving a single piece, either by moving one step in any direction to an adjacent empty space, or by jumping in one or any number of available consecutive hops over other single pieces. A player may not combine hopping with a single-step move – a move consists of one or the other. There is no capturing in Chinese Checkers, so hopped pieces remain active and in play. Turns proceed clockwise around the board. In the diagram, Green might move the topmost piece one space diagonally forward as shown.
A hop consists of jumping over a single adjacent piece, either one's own or an opponent's, to the empty space directly beyond it in the same line of direction. Red might advance the indicated piece by a chain of three hops in a single move, it is not mandatory to make the most number of hops possible. Can be played "all versus all", or three teams of two; when playing teams, teammates sit at opposite corners of the star, with each team member controlling their own colored set of pieces. The first team to advance both sets to their home destination corners is the winner; the remaining players continue play to determine second- and third-place finishers, etc. The four-player game is the same as the game for six players, except that two opposite corners will be unused. In a three-player game, all players control either two sets of pieces each. If one set is used, pieces race across the board into opposite corners. If two sets are used, each player controls two differently colored sets of pieces at opposite corners of the star.
In a two-player game, each player plays two, or three sets of pieces. If one set is played, the pieces go into the opponent's starting corner, the number of pieces per side is increased to 15. If two sets are played, the pieces can either go into the opponent's starting corners, or one of the players' two sets can go into an opposite empty corner. If three sets are played, the pieces go into the opponent's starting corners. A basic strategy is to create or find the longest hopping path that leads closest to home, or into it. Since either player can make use of any hopping'ladder' or'chain' created, a more advanced strategy involves hindering an opposing player in addition to helping oneself make jumps across the board. Of equal importance are the players' strategies for emptying and filling their starting and home corners. Games between top players are decided by more than a couple of moves. Differing numbers of players result in different starting layouts, in turn imposing different best-game strategies.
For example, if a player's home destination corner starts empty, the player can build a'ladder' or'bridge' with their pieces between the two opposite ends. But if a player's opponent occupies the home corner, the player may need to wait for opponent pieces to clear before filling the home vacancies. While the standard rules allow hopping over only a single adjacent occupied position at a time, this version of the game allows pieces to catapult over multiple adjacent occupied positions in a line when hopping. In the fast-paced or Super Chinese Checkers variant popular in France, a piece may hop over a non-adjacent piece. A hop consists of jumping over a distant piece to a symmetrical position on the opposite side, in the same line of direction; as in the standard rules, a jumping move may consist of any number of a chain of hops. (When making a chain of hops, a piece is allowed to enter an empty corner, as lo
On-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences cited as Sloane's, is an online database of integer sequences. It was maintained by Neil Sloane while a researcher at AT&T Labs. Foreseeing his retirement from AT&T Labs in 2012 and the need for an independent foundation, Sloane agreed to transfer the intellectual property and hosting of the OEIS to the OEIS Foundation in October 2009. Sloane is president of the OEIS Foundation. OEIS records information on integer sequences of interest to both professional mathematicians and amateurs, is cited; as of September 2018 it contains over 300,000 sequences. Each entry contains the leading terms of the sequence, mathematical motivations, literature links, more, including the option to generate a graph or play a musical representation of the sequence; the database is searchable by subsequence. Neil Sloane started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics; the database was at first stored on punched cards.
He published selections from the database in book form twice: A Handbook of Integer Sequences, containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372. The Encyclopedia of Integer Sequences with Simon Plouffe, containing 5,488 sequences and assigned M-numbers from M0000 to M5487; the Encyclopedia includes the references to the corresponding sequences in A Handbook of Integer Sequences as N-numbers from N0001 to N2372 The Encyclopedia includes the A-numbers that are used in the OEIS, whereas the Handbook did not. These books were well received and after the second publication, mathematicians supplied Sloane with a steady flow of new sequences; the collection became unmanageable in book form, when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service, soon after as a web site. As a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998; the database continues to grow at a rate of some 10,000 entries a year.
Sloane has managed'his' sequences for 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database. In 2004, Sloane celebrated the addition of the 100,000th sequence to the database, A100000, which counts the marks on the Ishango bone. In 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS.org was created to simplify the collaboration of the OEIS editors and contributors. The 200,000th sequence, A200000, was added to the database in November 2011. Besides integer sequences, the OEIS catalogs sequences of fractions, the digits of transcendental numbers, complex numbers and so on by transforming them into integer sequences. Sequences of rationals are represented by two sequences: the sequence of numerators and the sequence of denominators. For example, the fifth-order Farey sequence, 1 5, 1 4, 1 3, 2 5, 1 2, 3 5, 2 3, 3 4, 4 5, is catalogued as the numerator sequence 1, 1, 1, 2, 1, 3, 2, 3, 4 and the denominator sequence 5, 4, 3, 5, 2, 5, 3, 4, 5.
Important irrational numbers such as π = 3.1415926535897... are catalogued under representative integer sequences such as decimal expansions, binary expansions, or continued fraction expansions. The OEIS was limited to plain ASCII text until 2011, it still uses a linear form of conventional mathematical notation. Greek letters are represented by their full names, e.g. mu for μ, phi for φ. Every sequence is identified by the letter A followed by six digits always referred to with leading zeros, e.g. A000315 rather than A315. Individual terms of sequences are separated by commas. Digit groups are not separated by periods, or spaces. In comments, etc. A represents the nth term of the sequence. Zero is used to represent non-existent sequence elements. For example, A104157 enumerates the "smallest prime of n² consecutive primes to form an n×n magic square of least magic constant, or 0 if no such magic square exists." The value of a is 2. But there is no such 2×2 magic square, so a is 0; this special usage has a solid mathematical basis in certain counting functions.
For example, the totient valence function. There are 4 solutions for 4, but no solutions for 14, hence a of A014197 is 0—there are no solutions. −1 is used for this purpose instead, as in A094076. The OEIS ma
A hexagram or sexagram is a six-pointed geometric star figure with the Schläfli symbol, 2, or. Since there are no true regular continuous hexagrams, the term is instead used to refer to the compound figure of two equilateral triangles shown to the right; the intersection is a regular hexagon. The hexagram is part of an infinite series of shapes which are compounds of two n-dimensional simplices. In three dimensions, the analogous compound is the stellated octahedron, in four dimensions the compound of two 5-cells is obtained, it has been used in religious and cultural contexts and as decorative motifs. The symbol was used as a decorative motif in medieval Christian churches many centuries before its first known use in a Jewish synagogue, it was first used as a religious symbol by Arabs in the medieval period, known as the Seal of Solomon, depicted as either a hexagram or pentagram, and, adopted by Jewish Kabbalists. In mathematics, the root system for the simple Lie group G2 is in the form of a hexagram, with six long roots and six short roots.
A six-pointed star, like a regular hexagon, can be created using a compass and a straight edge: Make a circle of any size with the compass. Without changing the radius of the compass, set its pivot on the circle's circumference, find one of the two points where a new circle would intersect the first circle. With the pivot on the last point found find a third point on the circumference, repeat until six such points have been marked. With a straight edge, join alternate points on the circumference to form two overlapping equilateral triangles. A regular hexagram can be constructed by orthographically projecting any cube onto a plane through three vertices that are all adjacent to the same vertex; the twelve midpoints to edges of the cube form a hexagram. For example, consider the projection of the unit cube with vertices at the eight possible binary vectors in three dimensions, onto the plane x + y + z = 1; the midpoints are, all points resulting from these by applying a permutation to their entries.
These 12 points project to a hexagram: six vertices around the outer hexagon and six on the inner. It is possible that as a simple geometric shape, like for example the triangle, circle, or square, the hexagram has been created by various peoples with no connection to one another; the hexagram is a mandala symbol called satkona yantra or sadkona yantra found on ancient South Indian Hindu temples. It symbolizes the nara-narayana, or perfect meditative state of balance achieved between Man and God, if maintained, results in "moksha," or "nirvana"; some researchers have theorized that the hexagram represents the astrological chart at the time of David's birth or anointment as king. The hexagram is known as the "King's Star" in astrological circles. In antique papyri, together with stars and other signs, are found on amulets bearing the Jewish names of God, used to guard against fever and other diseases. Curiously the hexagram is not found among these signs. In the Greek Magical Papyri at Paris and London there are 22 signs side by side, a circle with twelve signs, but neither a pentagram nor a hexagram.
Six-pointed stars have been found in cosmological diagrams in Hinduism and Jainism. The reasons behind this symbol's common appearance in Indic religions and the West are unknown. One possibility is; the other possibility is that artists and religious people from several cultures independently created the hexagram shape, a simple geometric design. Within Indic lore, the shape is understood to consist of two triangles—one pointed up and the other down—locked in harmonious embrace; the two components are called "Om" and the "Hrim" in Sanskrit, symbolize man's position between earth and sky. The downward triangle symbolizes Shakti, the sacred embodiment of femininity, the upward triangle symbolizes Shiva, or Agni Tattva, representing the focused aspects of masculinity; the mystical union of the two triangles represents Creation, occurring through the divine union of male and female. The two locked triangles are known as'Shanmukha'—the six-faced, representing the six faces of Shiva & Shakti's progeny Kartikeya.
This symbol is a part of several yantras and has deep significance in Hindu ritual worship and history. In Buddhism, some old versions of