Problem of the Week 1202

Magic Triangles

Find a 3×3 array of distinct positive integers so that:

• the three numbers in each row, column, and two main diagonals form the edge lengths of a triangle; and
• the perimeters of the eight triangles that arise are all the same.

The idea is to minimize the sum of the nine entries. Here a "triangle" is a "nondegenerate triangle": a < b + c, b < a + c, and c < a + b.

Unsolved problem: Same thing, but with "perimeter" replaced by "area." For this, consider only the rows and columns (not the diagonals; such is called a "semi-magic square"). Further, allow the entries to be any real numbers, but still aim for distinct entries (or possibly the weaker condition, that the triangles be distinct).

Source: Lee Sallows, Nijmegen, The Netherlands.

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