To be honest, I got the problem you (t.b) answered as a bonus problem from my prof... I didn't fully understand the question, and asked it here in hopes a hint toward a solution might help me understand the question better - after all stackexchange seems to hate questions promoting discussions. Sadly, that wasn't the case - to be frank, I didn't really understand your solution, but based on the upvotes, I chose your (sole) solution as an answer. I will be taking time in the future to reread it.