Sometimes it's good the keep the problem in the back of your mind while you do other stuff that appears irrelevant. Here is Stanislaw Ulam's account of the invention of the Monte Carle Method--from [Los Alamos Science Special Issue 1987](http://www-star.st-and.ac.uk/~kw25/teaching/mcrt/MC_history_3.pdf). Anything may suggest a way to tackle your problem.

> The first thoughts and attempts I made ... were suggested by a question which occurred to me in 1946 as I was convalescing from an illness and playing solitaires. The question was what are the chances that a Canfield solitaire laid out with 52 cards will come out successfully? After spending a lot of time trying to estimate them by pure
combinatorial calculations, I wondered whether a more practical method than
“abstract thinking” might not be to lay it out say one hundred times and
simply observe and count the number of successful plays. This was already possible to envisage with the beginning of the new era of fast computers, and I immediately thought of problems of neutron diffusion and other questions of mathematical physics, and more generally how to change processes described by certain differential equations into an equivalent form interpretable
as a succession of random operations. Later...  described the idea to John von Neumann and we began to plan actual calculations.”