I have grave doubts about the possibility of this problem having a solution as stated. I have (vaguely) heard of Ulam, but have never studied any of his books so I can't say for sure, but certainly within the limits of my knowledge of Physics and my creativity I can't think of a way to construct such a series of non-zero masses.
Until someone can post a workable solution I'm going to have to say that I think it is impossible.
Perhaps you could try posting your question here. If they do manage to come up with a solution I'd be interested in hearing about it.
I suppose there is a possibility. I thought of it more in terms of an electrodynamics problem, rather than gravity.
If we had, say two point masses (m = 1/2 each), one positioned at + infinity and one at - infinity. This would do the trick. We can, in fact, place any single mass at the origin (or any other point in between.)
I must add that this isn't really a constructable problem, but as we are already considering point masses at infinity...
Edit: No, this isn't going to work either, because the masses need to be constrained to [0, 1] on some line, not out at infinity. Ah well.
it is just a question of how much precision you want rather than a question
of it working or not working (at least that is if you can specify the sample
space and distributions).
It is essentially a method of last resort, and as we know these occur almost
all the time in scientific work.