You have a balance with two pans and three visually identical balls. You know that one of them weighs differently than the other two, but you don't know if it is heavier or lighter. Can the different ball be identified with certainty using a single weighing?
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The scale and the bola distinta
The beautiful thing here is the double uncertainty: you don't know which ball is different or if it weighs more or less. With just one weighing, every possibility has to be placed in its place.
Hints
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- A single weighing on a two-pan scale only has three possible results.
- But here the cases that you must distinguish are more than three.
- Count how many different hypotheses there are: for each ball, it could be heavier or lighter.
Solution
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Answer: No, it is impossible in a single weighing. Model the possible states: - rare ball = 1, 2 or 3 (3 options),
- and also rare = heavier or lighter (2 options). Total:
$$ 3\times2=6\ \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\text{estados}. $$ A single weighing of a dish scale only has 3 results: 1. left low, 2. balance, 3. low right. Like $3<6$, there is not enough information capacity to distinguish all cases. **Conclusion:** in a weighing you cannot simultaneously determine identity and direction (plus/minus). $$
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