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Magic Triangle (mathematics)
A magic triangle is a magic arrangement of the integers from 1 to to triangular figure. Perimeter magic triangle A magic triangle or perimeter magic triangle is an arrangement of the integers from 1 to on the sides of a triangle with the same number of integers on each side, called the ''order'' of the triangle, so that the sum of integers on each side is a constant, the magic sum of the triangle. Unlike magic squares, there are different magic sums for magic triangles of the same order. Any magic triangle has a complementary triangle obtained by replacing each integer in the triangle with . Examples Order-3 magic triangles are the simplest (except for trivial magic triangles of order 1). Other magic triangles Other magic triangles use Triangular number or square number of vertices to form magic figure. Matthew Wright and his students in St. Olaf College developed magic triangles with square numbers. In their magic triangles, the sum of the k-th row and the (n-k+1)-th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. 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 natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magic Sum
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, a magic square which contains the numbers 1, 2, ..., ''n''2 – the magic constant is M = n \cdot \frac. For normal magic squares of orders ''n'' = 3, 4, 5, 6, 7, and 8, the magic constants are, respectively: 15, 34, 65, 111, 175, and 260 (sequence A006003 in the OEIS). For example, a normal 8 × 8 square will always equate to 260 for each row, column, or diagonal. The normal magic constant of order n is (n^3+n)/2. The largest magic constant of normal magic square which is also a: *triangular number is 15 (solve the Diophantine equation x^2=y^3+16y+16, where y is divisible by 4); *square number is 1 (solve the Diophantine equation x^2=y^3+4y, where y is even); *generalized pentagonal number is 171535 (solve the Diophanti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magic Square
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 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 and the 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). The mathematical study of magic squares typically deals with their construction, classification, and enumeration. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order 3 Magic Triangles
Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of different ways * Hierarchy, an arrangement of items that are represented as being "above", "below", or "at the same level as" one another * an action or inaction that must be obeyed, mandated by someone in authority People * Orders (surname) Arts, entertainment, and media * ''Order'' (album), a 2009 album by Maroon * "Order", a 2016 song from ''Brand New Maid'' by Band-Maid * ''Orders'' (1974 film), a 1974 film by Michel Brault * ''Orders'', a 2010 film by Brian Christopher * ''Orders'', a 2017 film by Eric Marsh and Andrew Stasiulis * ''Jed & Order'', a 2022 film by Jedman Business * Blanket order, purchase order to allow multiple delivery dates over a period of time * Money order or postal order, a financial instrument usually intende ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trivial
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense. Latin Etymology The ancient Romans used the word ''triviae'' to describe where one road split or forked into two roads. Triviae was formed from ''tri'' (three) and ''viae'' (roads) – literally meaning "three roads", and in transferred use "a public place" and hence the meaning "commonplace." The Latin adjective ''triviālis'' in Classical Latin besides its literal meaning could have the meaning "appropriate to the street corner, commonplace, vulgar." In late Latin, it could also simply mean "triple." The pertaining adjective ''trivial'' was adopted in Early Modern English, while the noun ''trivium'' only appears in learned usage from the 19th century, in reference to the ''Artes Liberales'' and the plural ''trivia'' in the sense of "trivialities, trifles" only in the 20th century. Meaning In medieval Latin, the ''trivia'' (singular ''triv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in the triangular arrangement with dots on each side, and is equal to the sum of the natural numbers from 1 to . The sequence of triangular numbers, starting with the 0th triangular number, is (This sequence is included in the On-Line Encyclopedia of Integer Sequences .) Formula The triangular numbers are given by the following explicit formulas: T_n= \sum_^n k = 1+2+3+ \dotsb +n = \frac = , where \textstyle is a binomial coefficient. It represents the number of distinct pairs that can be selected from objects, and it is read aloud as " plus one choose two". The first equation can be illustrated using a visual proof. For every triangular number T_n, imagine a "half-square" arrangement of objects corresponding to the triangular numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usual notation for the square of a number is not the product , but the equivalent exponentiation , usually pronounced as " squared". The name ''square'' number comes from the name of the shape. The unit of area is defined as the area of a unit square (). Hence, a square with side length has area . If a square number is represented by ''n'' points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of ''n''; thus, square numbers are a type of figurate numbers (other examples being Cube (algebra), cube numbers and triangular numbers). Square numbers are non-negative. A non-negative integer is a square number when its square root is again an intege ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magic Hexagon
A magic hexagon of order ''n'' is an arrangement of numbers in a centered hexagonal pattern with ''n'' cells on each edge, in such a way that the numbers in each row, in all three directions, sum to the same magic constant ''M''. A normal magic hexagon contains the consecutive integers from 1 to 3''n''2 − 3''n'' + 1. It turns out that normal magic hexagons exist only for ''n'' = 1 (which is trivial, as it is composed of only 1 cell) and ''n'' = 3. Moreover, the solution of order 3 is essentially unique. Meng also gave a less intricate constructive proof.Meng, F"Research into the Order 3 Magic Hexagon" '' Shing-Tung Yau Awards'', October 2008. Retrieved on 2009-12-16. The order-3 magic hexagon has been published many times as a 'new' discovery. An early reference, and possibly the first discoverer, is Ernst von Haselberg (1887). Proof of normal magic hexagons The numbers in the hexagon are consecutive, and run from 1 to 3n^2-3n+1. He ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Open problems The following questions about antimagic squares have not been solved. * How many antimagic squares of a given order exist? * Do antimagic squares exist for al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magic Polygon
A magic polygon is a polygonal magic graph with integers on its vertices. Perimeter magic polygon A magic polygon, also called a perimeter magic polygon, is a polygon with an integers on its sides that all add up to a magic constant. It is where positive integers (from 1 to ''N'') on a ''k''-sided polygon add up to a constant. Magic polygons are a generalization of other magic shapes such as magic triangles. Magic polygon with a center point Victoria Jakicic and Rachelle Bouchat defined magic polygons as ''n''-sided regular polygons with 2''n''+1 nodes such that the sum of the three nodes are equal. In their definition, a 3 × 3 magic square can be viewed as a magic 4-gon. There are no magic odd-gons with this definition. Magic polygons and degenerated magic polygons Danniel Dias Augusto and Josimar da Silva defined the magic polygon P(''n'',''k'') as a set of vertices of k/2 concentric ''n''-gon and a center point. In this definition, magic polygons of Vict ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
