Q1: You have 12 identical-looking coins, but one is counterfeit. It is either heavier or lighter than the others, though you do not know which. Using a balance scale and only 3 weighings, how can you identify the counterfeit coin and determine whether it is heavier or lighter?
Try dividing the 12 coins into 3 equal groups of 4.
Note: For a visual explanation, check out this video: https://www.youtube.com/watch?v=tE2dZLDJSjA&pp=0gcJCdgAo7VqN5tD 1. Label the marbles from 1 to 12. 2. First Weighing: - Left pan: marbles 1, 2, 3, 4 - Right pan: marbles 5, 6, 7, 8 3. If they balance: - The different marble is among 9, 10, 11, 12. 4. Second Weighing (for the balanced case): - Weigh 9, 10, 11 against 1, 2, 3 (known to be normal). - (1) If they balance: - The different marble is 12. - Weigh 12 against any known normal marble to determine if it is heavy or light. - (2) If 9, 10, 11 is heavy: - The different marble is 9, 10, or 11 and it is heavy. Weigh 9 against 10: - If they balance, 11 is the heavy marble. - If not, the heavier one is the different marble. - (3) If 9, 10, 11 is light: - The different marble is 9, 10, or 11 and it is light. Weigh 9 against 10: - If they balance, 11 is the light marble. - If not, the lighter one is the different marble. 5. If the first weighing does not balance: - Assume 5, 6, 7, 8 is the heavy side. 6. Second Weighing (for the unbalanced case): - Weigh 1, 5, 6 against 2, 7, 8. - If they balance: - The different marble is either 3 or 4. Weigh 4 against 9 (a known normal marble): - If they balance, the different marble is 3. - If not, the direction of the tilt will reveal whether 4 is heavier or lighter. - If 2, 7, 8 is the heavy side: - The different marble is either: - 7 or 8 and heavy, or - 1 and light. Weigh 7 against 8: - If one side is heavier, that is the different marble. - If they balance, then 1 is the different marble (light). - If 1, 5, 6 is the heavy side: - The different marble is either: - 5 or 6 and heavy, or - 2 and light. Weigh 5 against 6: - If one side is heavier, that is the different marble. - If they balance, then 2 is the different marble (light).
Q2: You stand before two doors. One leads to freedom, the other to doom. Two guards stand by the doors — one always lies, the other always tells the truth. You can ask only one yes/no question to one guard. What do you ask to find the safe door?
Ask what the other guard would say about the door to freedom.
If I asked the other guard whether this door leads to freedom, would he say yes?
Q3: You are given two ropes. Each rope takes exactly 1 hour to burn completely, but they do not burn at a consistent rate. For example, half the rope might take 59 minutes and the rest just 1 minute. Using only these two ropes and a lighter, how can you measure exactly 45 minutes?
Try lighting both ends of one rope at the same time.
1. Light Rope A at both ends and Rope B at one end — simultaneously. 2. Rope A will burn in 30 minutes, because lighting both ends makes it burn twice as fast (unevenness doesn't matter). 3. When Rope A is completely burnt (after 30 minutes), immediately light the other end of Rope B as well. 4. At this point, Rope B has been burning from one end for 30 minutes — so it has 30 minutes' worth of burn left, but unevenly distributed. 5. Lighting the other end causes the remaining rope to burn in 15 minutes (again, it burns from both ends). 6. When Rope B finishes burning, a total of 45 minutes has passed.