Question

Ruben has two congruent wooden dowels. He cuts one dowel in two in order to have three pieces to make a triangle. Explain why, despite having three sides, Ruben will not be able to make a triangle with his three pieces.

Answer 1

Let each original dowel have length L. After cutting one dowel into lengths a and b we have a + b = L, and the three pieces are a, b, and L.

A triangle is possible only when every side is strictly less than the sum of the other two. But here
L = a + b,
so the longest piece L is equal to the sum of the other two, not less. That gives a degenerate triangle (the three pieces would lie flat in a straight line), not a genuine triangle. Hence Ruben cannot form a triangle.