> If I break a stick into 6 pieces, what is the > probability that they > can form a tetrahedron? > > In general, if I break a stick into (n*n+n)/2 pieces, > what is the > probability that they can form an n-simplex? > > I know that the longest piece must be no more than > 0.5 of the lengh of > the original unbroken stick. > > Thanks in advance.
let the stick be of unit length. We generate five random numbers uniformly distributed between 0 and 1, and after we sort them in ascending order we have some r1, r2, r3, r4, r5. Let a=r1, b=r2-r1, c=r3-r2, d=r4-r3, e=r5-r4, f=1-r5 be the edges of a possible tetrahedron. The edges should satisfy the triangle inequalities for the four faces of tetrahedron. A program based on this reasoning for 1000000 trials yields an approximate probability p = 0.0185 .