1. The twelve balls. Twelve identical balls. One is a different weight — you do not know whether heavier or lighter. Using a balance scale three times, find it and say which it is.
2. The handshakes. Six people at a table. Everyone shakes hands with everyone else exactly once. How many handshakes?
3. The snail. A snail is at the bottom of a 10-metre well. Each day it climbs 3 metres; each night it slides back 2. How many days to get out?
4. The water tank. A tank fills in 6 hours through pipe A and 4 hours through pipe B. Both open together, how long?
Answers.
1. Weigh 4 against 4. If balanced, the odd ball is in the untouched 4 — weigh 3 of them against 3 known-good balls, which tells you the group and the direction, then one final weighing identifies it. If unbalanced, you know 4 are possibly-heavy and 4 possibly-light. Second weighing: three possibly-heavy plus one possibly-light against the other three possibly-heavy plus a known-good ball. Whichever way it tips narrows you to at most three candidates with a known direction, and the third weighing finds it. Work it through slowly — it is meant to take a while.
2. Fifteen. Each of 6 shakes 5 hands, and every handshake was counted twice: (6 × 5) / 2.
3. Eight days. It gains 1 metre net per day, but on day 8 it starts at 7 metres, climbs 3 to reach 10, and is out before nightfall — so it never slides back.
4. 2 hours 24 minutes. A fills 1/6 per hour, B fills 1/4. Together 5/12 per hour, so 12/5 hours.
