Suppose you've rigged the church charity lottery so that you cannot lose (what a swell guy you are). There are 10 drawings, and you start off with 10 dollars exactly.
At each drawing, you must chose an amount to bet, and that amount gets put into a pool (that money then can no longer be used to wager). You then proceed with the next drawing, and you do this until you have no more money left.
The total amount of money won is the product of the amounts wagered each round played that you won (and since it is rigged, everytime you wager you win).
For example, if I wagered all 10 dollars in the first round, I will have won 1 drawing and earned exactly 10 dollars. If I wager 2 in the first round and 8 in the second, I will have won 16 dollars. Wagers must be in whole dollar amounts. Rounds where nothing is wagered are disregarded entirely, depending on how money is distributed amongst the various rounds.
- What is the best strategy to adopt to maximize winnings?
- For $n$ dollars and potentially up to $n$ drawings, what is the generalizable best strategy to maximize winnings?
- Let $\varphi(n)$ represent the maximum amount possible for start amount $n$ and rounds $n$ while observing the rules above. What is the relationship between $\varphi(n)$ and $n$ (can a conclusion be made about its relationship)?