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* At a fantasy auction, a dragon egg has a 60% chance to hatch a gold-producing dragon (yielding 100kg gold) and a 40% chance to explode (destroying all gold). The auction uses a "Vickrey" rule: you pay the average of your bid and the runner-up’s bid. How much should you bid if gold is worth $10,000/kg and you’re risk-neutral?
* A spaceship travelling at near relativistic speeds experiences time dilation. If the ship’s stock volatility scales with Lorentz factor γ, derive the adjusted Black-Scholes PDE for a call option expiring in "ship time."
* How many digits are in 99 to the 99th power?
* A line of 100 passengers is waiting to board a plane. They each hold a ticket to one of the 100 seats on that flight. (For convenience, let's say that the nth passenger in line has a ticket for the seat number n.) Unfortunately, the first person in line is crazy, and will ignore the seat number on their ticket, picking a random seat to occupy. All of the other passengers are quite normal, and will go to their proper seat unless it is already occupied. If it is occupied, they will then find a free seat to sit in, at random. What is the probability that the last (100th) person to board the plane will sit in their proper seat (#100)?
* What is the sum of the numbers one to 100?
* Two entangled quantum prisoners choose Cooperate or Defect in superposition. If measured in a Bell basis, payoffs are superposed. Derive the Nash equilibrium when payoffs are \( \sqrt{2} \$ \) for mutual cooperation.
* You have a 3 gallon jug and 5 gallon jug. How do you measure out exactly 4 gallons? Is this possible?