A **magic triangle** (also called a **perimeter magic triangle**^{ [1] }) is an arrangement of the integers from 1 to n 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.^{ [1] }^{ [2] }^{ [3] }^{ [4] } Unlike magic squares, there are different magic sums for magic triangles of the same order.^{ [1] } Any magic triangle has a complementary triangle obtained by replacing each integer x in the triangle with 1 + *n* − *x*.^{ [1] }

Order-3 magic triangles are the simplest (except for trivial magic triangles of order 1).^{ [1] }

**Area** is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve or the volume of a solid.

In geometry, a **polygon** is a plane figure that is described by a finite number of straight line segments connected to form a closed polygonal chain or *polygonal circuit*. The solid plane region, the bounding circuit, or the two together, may be called a polygon.

A **perimeter** is either a path that encompasses/surrounds/outlines a shape or its length (one-dimensional). The perimeter of a circle or an ellipse is called its circumference.

An *n*-pointed **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 2*n*. No numbers are ever repeated. The magic constant of an *n*-pointed normal magic star is *M* = 4*n* + 2.

A **triangular number** or **triangle number** counts objects arranged in an equilateral triangle. The *n*th triangular number is the number of dots in the triangular arrangement with *n* dots on a side, and is equal to the sum of the *n* natural numbers from 1 to *n*. The sequence of triangular numbers, starting at the 0th triangular number, is

In Euclidean geometry, a **regular polygon** is a polygon that is equiangular and equilateral. Regular polygons may be either **convex** or **star**. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed.

In mathematics, a proof by **infinite descent**, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. It is a method which relies on the well-ordering principle, and is often used to show that a given equation, such as a Diophantine equation, has no solutions.

**68** (**sixty-eight**) is the natural number following 67 and preceding 69. It is an even number.

In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an **interior****angle** if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle per vertex.

In geometry, a **square** is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices *ABCD* would be denoted *ABCD*.

In geometry, a **Heronian triangle** is a triangle that has side lengths and area that are all integers. Heronian triangles are named after Hero of Alexandria. The term is sometimes applied more widely to triangles whose sides and area are all rational numbers, since one can rescale the sides by a common multiple to obtain a triangle that is Heronian in the above sense.

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.

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 and *n* = 3. Moreover, the solution of order 3 is essentially unique. Meng also gave a less intricate constructive proof.

In Euclidean geometry, an **equiangular polygon** is a polygon whose vertex angles are equal. If the lengths of the sides are also equal then it is a regular polygon. Isogonal polygons are equiangular polygons which alternate two edge lengths.

A two-dimensional **equable shape** is one whose area is numerically equal to its perimeter. For example, a right angled triangle with sides 5, 12 and 13 has area and perimeter both have a unitless numerical value of 30.

In the geometry of tessellations, a **rep-tile** or **reptile** is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by recreational mathematician Solomon W. Golomb and popularized by Martin Gardner in his "Mathematical Games" column in the May 1963 issue of *Scientific American*. In 2012 a generalization of rep-tiles called self-tiling tile sets was introduced by Lee Sallows in *Mathematics Magazine*.

In combinatorics, the **Schröder–Hipparchus numbers** form an integer sequence that can be used to count the number of plane trees with a given set of leaves, the number of ways of inserting parentheses into a sequence, and the number of ways of dissecting a convex polygon into smaller polygons by inserting diagonals. These numbers begin

In geometry, the **biggest little polygon** for a number *n* is the *n*-sided polygon that has diameter one and that has the largest area among all diameter-one *n*-gons. One non-unique solution when *n* = 4 is a square, and the solution is a regular polygon when *n* is an odd number, but the solution is irregular otherwise.

**Number Scrabble** is a mathematical game where players take turns to select numbers from 1 to 9 without repeating any numbers previously used, and the first player with a sum of exactly 15 using any three of his number selections wins the game. The game is isomorphic to tic-tac-toe, as can be seen if the game is mapped onto a magic square.

A **magic polygon** is a polygonal magic graph with integers on its vertices.

- 1 2 3 4 5 "Perimeter Magic Triangles".
*www.magic-squares.net*. Retrieved 2016-12-27. - ↑ "Perimeter Maghic Polygons".
*www.trottermath.net*. Retrieved 2016-12-27. - ↑ "Magic Triangle : nrich.maths.org".
*nrich.maths.org*. Retrieved 2016-12-27. - ↑ "P4W8: Magic Triangles and Other Figures" (PDF). Retrieved December 27, 2016.

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