Addendum
I hope you enjoyed that. The entire story, in readable chronological order, may be found here. Now I'm going to sit back, close my eyes and let my brain dribble out of my ears for a while.
I am so not a mathematical person. Summoning the effort to inject required amounts of cleverness into that story has nearly caused my brain to implode, quietly, like an overripe banana under an iron slab.
All worship my sister Kristel, for at least half the genius was hers.