Jump to content

Magic triangle (mathematics)

From Wikipedia, the free encyclopedia

A magic triangle is a magic arrangement of the integers from 1 to n to triangular figure.

Perimeter magic triangle

[edit]

A magic triangle or 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 + nx.[1]

Examples

[edit]
Order-3 magic triangles

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

Other magic triangles

[edit]

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](sequence A356808 in the OEIS) Its one modification uses triangular numbers instead of square numbers. (sequence A355119 in the 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. (sequence A356643 in the OEIS)

Unsolved problem in mathematics:
How many 2x2 subtriangle-magic triangles for and does it exists for every or finite? if these magic triangles are finite, can we determine the largest n for order-n 2x2 subtriangle-magic triangles are exist?

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. (sequence A375416 in the OEIS)

Magic Triangles have also been discovered, such that when its elements are squared, we obtain another magic triangle.

See also

[edit]

References

[edit]
  1. ^ a b c d e "Perimeter Magic Triangles". www.magic-squares.net. Retrieved 2016-12-27.
  2. ^ "Perimeter Maghic Polygons". www.trottermath.net. Retrieved 2016-12-27.
  3. ^ "Magic Triangle : nrich.maths.org". nrich.maths.org. Retrieved 2016-12-27.
  4. ^ "P4W8: Magic Triangles and Other Figures" (PDF). Retrieved December 27, 2016.
  5. ^ Magic Triangles