QR Technical Round — Brainteasers & Estimation

A traditional QR technical round mixing probability/statistics brainteasers with Fermi estimation: invariant-style stone games, expected waiting-time questions, and market-sizing with confidence intervals. Requirements A verbal technical round (one of ~6 QR rounds) testing probability, statistics, and quantitative reasoning through brainteasers and estimation. Examples Reported questions: Two piles of stones, one with 1 and one with 24. Each turn, take from the larger pile an amount equal to the smaller pile (doubling the smaller): (1, 24) → (2, 23) → … Can repeated moves ever empty one pile? What about starting (1, 25) or (1, 23)? Prove under which initial conditions a pile can be emptied. A bus comes exactly every 10 minutes; you arrive at a random time. Expected wait for the next bus? And if the bus is late with 50% probability, how does the expected wait change? Estimate how many houses the total annual U.S. spend on Netflix could buy; give a 90% confidence interval. If told there are 81M U.S. subscribers, how does your interval change? Notes The stone-game question is an invariant/number-theory proof (think in terms of parity / the binary representation of the pile sizes), not a calculation. The estimation questions want a structured Fermi breakdown plus an honest confidence interval that tightens when you're given a concrete data point. Preparation Drill invariant and parity arguments for two-pile / stone-moving games and practice stating the exact initial-condition characterization with a proof. Practice expected-waiting-time reasoning (including the inspection-paradox flavor) and structured Fermi estimation with explicit confidence intervals.