If both Bob and Alice are trying to minimize, we let Alice split it into 1/4 and 3/4 lengths, then Bob splits similarly with the 1/4 length and creates 1/16 and 3/16 lengths, and so on with the shortest length getting split in
proportions.
We claim that there are zero triangles at any point in this process. For the purpose of proof by contradiction, we suppose that there is some triangle created. Let the maximum length of this triangle created be
. We see that
is of the form
for some positive integer
because if it was of the form
, the only other possibility, we would have that it was the least of all the lengths, which cannot be for the greatest length in a triangle. Clearly, none of the other lengths of the triangle have the same length, and so the two lesser lengths must both be less than or equal to
where
because otherwise they would be greater than or equal to
, which is clearly impossible. Therefore,
by the Triangle Inequality, but this is impossible because
is positive, and so we see that no triangles exist.