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A magic triangle is a magic arrangement of the
Order-3 magic triangles are the simplest (except for
Magic Triangles
/ref> Its one modification uses triangular numbers instead of square nubers. Another magic tringle form is magic triangles with triangluar numbers with different summation. In this magic triangles, the sum of the k-th row and the (n-k)-th row is same for all k.
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 ...
s from 1 to to triangular figure.
Perimeter magic triangle
A magic triangle or perimeter magic triangle is an arrangement of theinteger
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 ...
s 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 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 ...
s, 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
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 ...
magic triangles of order 1).
Other magic triangles
Other magic triangles useTriangular 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 i ...
or square number
In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as .
The u ...
of vertices to form magic figure. Matthew Wright and his students in St. Olaf College
St. Olaf College is a private liberal arts college in Northfield, Minnesota. It was founded in 1874 by a group of Norwegian-American pastors and farmers led by Pastor Bernt Julius Muus. The college is named after the King and the Patron Saint Ola ...
developed magic triangles with square numbers. In their magic triangles, the sum of the k-th row and the (n-k+1)-th row is same for all k./ref> Its one modification uses triangular numbers instead of square nubers. Another magic tringle form is magic triangles with triangluar numbers with different summation. In this magic triangles, the sum of the k-th row and the (n-k)-th row is same for all k.
See also
*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 he ...
*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 a ...
* Magic polygon
References