1. **State the problem:** A stick of length $l$ units is broken into three pieces. We want to prove that the probability these pieces form a triangle is $\frac{1}{4}$. 2. **Triangle inequality condition:** For three pieces to form a triangle, the length of any piece must be less than half the total length $l$, i.e., each piece $< \frac{l}{2}$. (S1, S2) 3. **Construct an equilateral triangle:** Make an equilateral triangle $ABC$ with side length $l$, where $BC$ represents the stick. (S6) 4. **Identify midpoints:** Let $D, E, F$ be the midpoints of sides $AB, BC,$ and $CA$ respectively. (S4) 5. **Break points at midpoints:** The stick can only break exactly at these midpoints, dividing it into three parts $BX, XY,$ and $YC$. (S5, S7) 6. **Form smaller triangles:** Join $D, E,$ and $F$ to form four smaller equilateral triangles inside $ABC$. (S3) 7. **Location of point Z:** Point $Z$ always lies inside the triangle $DEF$. (S8) 8. **Area relation:** The area of triangle $DEF$ is $\frac{1}{4}$ the area of the larger triangle $ABC$. (S10) 9. **Probability interpretation:** Since the stick is elastic and breaks at midpoints, the probability that the three pieces form a triangle corresponds to the ratio of the area of $DEF$ to $ABC$, which is $\frac{1}{4}$. (S11) **Logical order of statements:** S1, S2, S6, S4, S5, S7, S3, S8, S10, S11