Magic triangle (mathematics)
Template:Short description Template:Other uses A magic triangle is a magic arrangement of the integers from 1 to Template:Mvar to triangular figure.
Perimeter magic triangle
A magic triangle or perimeter magic triangle[1] is an arrangement of the integers from 1 to Template:Mvar 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 Template:Mvar in the triangle with Template:Math.[1]
Examples
Order-3 magic triangles are the simplest (except for trivial magic triangles of order 1).[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 row is same for all k.[5]Template:OEIS Its one modification uses triangular numbers instead of square numbers. Template:OEIS Another magic triangle form is magic triangles with triangular numbers with different summation. In this magic triangle, the sum of the k-th row and the (n-k)-th row is same for all k. Template:OEIS
Another magic triangle form is magic triangles with square numbers with different summation. In this magic triangle, the sum of the 2x2 subtriangles are same for all subtriangles. Template:OEIS
Magic Triangles have also been discovered, such that when its elements are squared, we obtain another magic triangle.