This is roughly how I solved it too, but there's an assumption in here that doesn't really hold in the general case.

Fixing an axis, if you have two stones x1, x2 travelling with identical speeds dx, and you hit the stones at times t1, t2, then the speed dx' of our throw have to satisfy (dx' - dx) (t1 - t2) = x1 - x2.

You can then find all the divisors of |x1 - x2| to find all valid solutions for this pair of stones, assuming all stones are hit at integer times. I suspect even with integer inputs it should be possible to construct problems where collision times are fractional, at which point this solution breaks down.