The fake coin, step by step

A balance answers three ways, not two

1. 7 identical-looking coins, one lighter than the rest, and a balance that takes any number of coins in each pan. Nothing has been weighed yet.
2. The obvious plan is to halve: put half in each pan, keep the lighter half, halve again. That really does work, and it gets there in 3 weighings — 7 → 4 → 2 → 1.
3. But look at what a balance actually tells you. It can tip left, tip right, or sit LEVEL — three answers. Halving uses two of them and throws the third away, and the third is the informative one: level means the fake is in neither pan.
4. So split into three roughly equal groups and weigh two of them. If it tips, the fake is in the lighter pan; if it sits level, the fake is in the group you did not weigh at all. Either way one weighing cuts the suspects to a THIRD, not a half.
5. Keep going and the count falls fast: 7 → 3 → 1. That is 2 weighings, 1 fewer than halving — and no fewer is possible, because each weighing has only three outcomes, so 1 of them could only ever separate 3 coins and there are 7.