Brilli the ant has a block of cheese that is in the shape of an equilateral triangle. She is hosting a party, and has invited her two-legged house mates. She insists that all of her guests get a piece of cheese that is in the shape of an isosceles triangle. What is the maximal number N such that she cannot split the equilateral triangle block of cheese into N non-degenerate isosceles triangles?

Answer 1

She cannot subdivide the equilateral triangle into 2 isosceles triangles. This is the smallest N which is impossible.

She can subdivide it into 3 congruent isosceles triangles by joining the centroid with each of the vertices. Each triangle is then 30-30-120 as angles.

Each of the 30-30-120 triangles can in turn be subdivided into 3 isosceles triangles with angles 60-60-60, and 30-30-120, thus adding two more triangles.

So the next possible N are then 5, 7, 9.

Since each of the smaller 30-30-120 triangles can again be subdivided, indefinitely, we conclude that the original equilateral triangle can be subdivided into odd N above 1. Even N are therefore not possible. Thus there is no maximum number N, since N->∞, as long as N=2k, where k∈integers.

Answer 2

MathMate, I would disagree with you because 4 triangles are possible. We can break up the equilateral triangle to 4 different equilateral triangles.