A unit stick is broken at two independent uniform points into three pieces. Probability they form a triangle?

How to answer

1/4. The pieces form a triangle iff every piece is < 1/2 (longest < sum of others). The success region {all three < 1/2} in the unit square has area 1/4; the failure region {some piece > 1/2} is three disjoint corner triangles of total area 3/4.

Key idea: Forgetting the inequality must hold for ALL three pieces (longest < 1/2), or miscomputing the geometry — the success region is 1/4, not 3/8.