Guess my number 3

What this sequential-guessing and fair-value interview question tests

This is a hard probability and decision-theory problem that combines information theory with game-theoretic valuation. It asks you to reason about the expected cost of an optimal guessing strategy and then back out the prize that makes the game economically fair.

Solving it requires recognizing that the best strategy is to use binary search—each guess should split the remaining possibilities as evenly as possible. From there, you compute the expected number of guesses needed, then set the expected gain (prize minus expected cost) to zero. The problem rewards both algorithmic insight and the ability to work with large numbers and asymptotic reasoning.

• Binary search and information-theoretic lower bounds
• Expected value and fair-game pricing
• Logarithmic growth and order-of-magnitude estimation