While walking around the streets of your local city, you encounter a strange woman sitting atop a gigantic pile of dollar coins. "Would you like to play a game?" she asks. Intrigued by the strange sight you ask, "What kind of game?" She replies:
Well, I've got all these coins that I don't know what to do with. I'd really like to share them with someone. I love games, so let's do it in the form of a game. We'll take turns dividing the pile into two piles and giving the other player the larger half to keep, then continuing play analogously on the smaller pile. If you split the piles equally, then the other player may take either half and play will continue on the other half. It'll be great for our karma - so much giving and whatnot!
Skeptical, you ask her, "Well, there's got to be a catch - you're not just giving away free money, are you? (But I'm in if you are!)". She says,
Well, I'd love to just let you have this pile, but, for tax reasons, I can't just give away so much money. Therefore, you'll have to contribute to my pile before we play. But, if you contribute half as many coins as I have in my pile, I'll let you play (on the whole pile with both of our coins). I'll even let you go first!
That sounds like a fantastic deal, so you agree, and add exactly enough coins to the pile to increase its size by $50\%$. Then, you each play optimally, gaining as many coins as possible. At the end of the day, you count up the coins you earned, and find that you earned back $5 less than you contributed to the pile! At a minimum, how many coins did the woman have at the end of the game?