tilly,
Thanks! After having read your code and your proof I see why it wasn't immediately obvious to me. I was stuck looking at it a different way. I stubbornly had to convince myself that my way was in fact correct. After I did, I realized we were saying the same thing. This is the frustrating part - I sometimes can't get out of my own way.

Unfortunately, I now lack all ambition to implement my solution.

I only got interested in the problem because glancing at the commentary of existing posts - a number thought brute forcing was necessary. I had a hunch that this wasn't true.