Broken Stick Triangle Cut Geometry
Two random cuts form a triangle only one quarter of the time, and the geometry explains every failed stick.
Free theory articleTwo random cuts form a triangle only one quarter of the time, and the geometry explains every failed stick.
Concept & interpretation
If we break a stick at two randomly chosen points, what is the probability that the three resulting pieces can be formed into a triangle? This is a classic probability problem that beautifully illustrates how geometric reasoning simplifies continuous probability.
experiment
Use the interactive model to visualize the ordered-cut domain and see which cut combinations succeed or fail.
- Open the interactive module.
- Change the First cut and Second cut to different values. Watch how the point (x, y) moves in the 2D configuration space.
- Observe the Maximum-piece threshold control. Adjusting it will show you the exact Maximum-piece CDF computed using the alternating sum formula.
- Notice that any point falling inside the inner green triangular region produces a valid triangle (where the longest piece is strictly less than 1/2), while points outside fail the triangle inequality.