1. The two guards. Two doors: one leads out, one does not. One guard always lies, one always tells the truth, and you do not know which is which. You may ask one guard one question. What do you ask?
2. The burning ropes. Two ropes, each takes exactly one hour to burn, but neither burns evenly. Measure exactly 45 minutes.
3. The eight coins. Eight coins, one of them lighter. You have a balance scale. Find the light one in two weighings.
4. The hard one. Three people are each given a hat, red or blue, chosen by a coin flip. Each sees the other two hats but not their own. Simultaneously, each must either guess their own colour or pass. They win if at least one guesses correctly and nobody guesses wrongly. They may agree a strategy beforehand. Random guessing wins 50% of the time. Can they do better?
Answers.
1. Ask either guard: “If I asked the other guard which door leads out, what would he say?” Both answers point to the wrong door. Take the other one.
2. Light rope A at both ends and rope B at one end simultaneously. When A is gone, 30 minutes have passed. At that instant light B’s other end — its remaining 30 minutes now burns in 15. Total: 45.
3. Weigh 3 against 3. If balanced, the light coin is among the remaining 2 — weigh those. If not, take the lighter group of 3, weigh 1 against 1; balanced means it is the third.
4. Yes — 75%. The rule: if you see two hats of the same colour, guess the opposite; otherwise pass. This fails only when all three hats match, which happens 2 times in 8. The other 6 times, exactly one person sees a matching pair and is right.
