A Puzzle by Midnight Math

midnightmathImagine a game that involves betting on the color of a single card in a standard 52-card deck. Each card is turned over one by one, and before each card is flipped, you may do one of two things:

1) Bet: If the next card is red, you earn $1. If the next card is black, you lose $1.

2) Pass: The next card is turned over and shown to you, and play continues.

Once you bet on a card, the deck is reshuffled and may play again. If you reach the end of the deck, you are forced to bet on the last card.

In a naive strategy, you could bet on the first card of each deck, winning 50% of the time and earning, on average, $0. Can you produce a better strategy, or a proof that one does not exist?

Send in your solutions (with proofs) to midnight.math@gmail.com or talk to Kevin O’Toole or Ian Hoover. If you are correct, you will be given the highest of accolades: your name mentioned here, next issue.