Survivor

Question: Players A, B, C, and D are playing a game which starts with A saying the number $1.$ From that point on, each player says a number between $1$ and $4$ greater than the number spoken by the preceding player. The player who says $20$ wins the round, the player after them is removed from the game, the player after them starts the next round. Any other players survive to the next round. Each player’s top aim is to win the whole game but, if they realize that’s impossible, they’ll prioritize surviving to the next round. Who will survive the $4$ players game? [Solution]

Cornhole Connoisseur

Question: it’s the end of a close game of cornhole, and your team is $3$ points away from the win. By house rules, you have to hit the $3$ points exactly, or you lose. On your team are three athletes. The first is The Aggressor who has a $40%$ chance to get the cornhole, a $30%$ chance to hit the board and a $30%$ chance to miss entirely. The second is The Conservative who has a $10%$ chance to get the cornhole, a whopping $80%$ chance to hit the board, and a $10%$ chance to miss entirely. Finally, there’s The Waste who’s blind drunk at every game and will always miss the board. Given your roster and the free choice to use any of them in any situation, and you play optimally, what is the chance that you win the game? [Solution]

Election Comeback

Question: it’s election night and your candidate is behind in the count. However, a significant fraction of the vote is still out in uncounted mail-in ballots. What are the chances that your candidates come back for the stunning victory if a whole bunch of people vote? What happens when the polls are tilted in one direction? [Solution]

Crescent Observatory

Question: you’re watching the Moon from your room, alone, like every night. Seeking a higher purpose, you decide to track the projected area of the illuminated portion of the Moon throughout the month. If your data is accurate, how much faster will the illuminated Moon’s area be growing at half Moon as compared to a crescent Moon of $1/6^\text{th}$ area? [Solution]

Beating the Sphinx

Question: an oracular Sphinx gives you a starting capital of a dollar and the opportunity to wager any portion of your stake on a series of $4$ coin flips. The coin flips are completely random, except for the fact that the Sphinx guarantees there aren’t any runs of $3$ like outcomes in a row (no $\mathbf{H}\rightarrow\mathbf{H}\rightarrow\mathbf{H}$ or $\mathbf{T}\rightarrow\mathbf{T}\rightarrow\mathbf{T}$). If you set your wagers right, what is your maximum guaranteed profit for the worst-case outcome? If the Sphinx now offers you $N$ wagers, and guarantees that no $Q$ flips in a row will have the same outcome, what is your maximum guaranteed profit? [Solution]