Magic triangle (mathematics)

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

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Perimeter magic triangle

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 + n − x. [1]

Examples

Order-3 magic triangles Order 3 Magic Triangles.gif
Order-3 magic triangles

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

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

See also

Related Research Articles

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<span class="mw-page-title-main">Triangular number</span> Figurate number

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<span class="mw-page-title-main">1,000,000</span> Natural number

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<span class="mw-page-title-main">Bernoulli's triangle</span> Array of partial sums of the binomial coefficients

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<span class="mw-page-title-main">Doubly triangular number</span> Type of triangular number

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References

  1. 1 2 3 4 5 "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
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