What is the probability that 3 pieces make a triangle?

The area of the triangle formed by the axes and the line x+y= l corresponds to all the possibilities of breaking the stick into three parts i.e. x,y>0 & x+y<l.

Now, for the three pieces to form the sides of a triangle, the triangle equality should be true.

x+y>l-x-y ⇒ x+y> l/2.
x+l-x-y>y ⇒ y<l/2
y+l-x-y>x⇒ x<l/2

The shaded area in the figure above corresponds to all the three conditions above.

probability = favorable outcome / all possible outcomes
                     = shaded area/ total area
                     = (1/2 *l/2*l/2)/1/2 *l*l
                     = 1/4

Wow. that is a really cool question.