In one of our Brain Potions from last year (found here), we explored a betting strategy to supposedly win "Infinite Money" playing a version of Roulette.
In this week's Brain Potion we present a simple card game and explore how much money we can win!
Start with a deck of $2N$ cards, with $N$ black cards and $N$ red cards and start with $\$1.00$. The dealer of the game flips the cards over, one by one, until all $2N$ cards are dealt. Before he flips over each card, you have a chance to bet/predict the color of the next card. You can choose to risk some/all/none of your money, and if you are correct you win double what you bet.
Your goal is to come up with a strategy to guarantee you win as much money as possible, leaving nothing to chance.
Try to come up with a strategy for $N = 2$. That means you have to have a strategy that deals will all possible ways the cards could be dealt: $$BBRR, BRBR, BRRB, RBBR, RBRB, RRBB$$ so that you always end up with the same amount of money. Hint: You can guarantee you win more than $\$2$!
As a challenge, once you've solved the problem for $N=2$, try $N=3$, etc. There is a nice pattern for how much money you can win, depending on $N$. Try to find (and prove) it!
Have your own request, idea, or feedback for the Brain Potion series? Share with us in our Request and Idea Thread available here.