A two-pan scale seems like a simple tool. But when the uncertainty is double—you don't know which ball is different or whether it weighs more or less—a single weighing may not be enough.
The scale and the different ball
Riddle statement
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 only one weighing?
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Solution
Answer: No. One weighing cannot identify the odd ball with certainty.
Label the balls A, B, and C.
For one weighing to distinguish among all three balls, each ball would have to play a different role. If two balls were placed on the same pan—or both left off the scale—swapping them would not change the outcome, so they could not be told apart.
The only arrangement worth considering is therefore to weigh A against B and leave C off the scale.
There are three possible outcomes.
1. The balance remains level.
Then A and B have the same weight. Since one ball is known to be different, C must be the odd ball.
2. A's pan goes down.
There are two possible explanations:
- A is the odd ball and is heavier;
- B is the odd ball and is lighter.
The result does not tell us which of those two balls is different.
3. B's pan goes down.
Symmetrically, B may be heavier or A may be lighter.
Whenever the balance tips, two candidates remain.
Conclusion: one weighing is not enough to identify the odd ball with certainty.