The broken stick triangle

1. Step 1 · A triangle needs every piece under half

   $\displaystyle \text{the longest side must be shorter than the other two together}$

   If one piece is half the stick or more, the other two can’t reach around it.

2. Step 2 · Picture the two breaks as a point in a square

   $\displaystyle \text{across}\colon \text{ first break},\, \text{ up}\colon \text{ second break}$

   Every pair of break points is equally likely, so a chance is a share of the square’s area.

3. Step 3 · Area where every piece is under half

   $\displaystyle 2 \text{ triangles of } \frac{1}{8} \text{ each } = \frac{1}{4}$

   Each orange triangle has legs of 1/2, so area 1/8. Together they cover a quarter of the square.

Answer$\displaystyle \frac{1}{4} = 25\%$