Explanation
To find the odd ball among N identical-looking balls using a balance scale, we use the principle of tri-sectioning (dividing into three groups). Each weighing on a balance scale has three possible outcomes: the left side is heavier, the right side is heavier, or both sides are equal.
1. Mathematical Formula
The maximum number of balls N for which an odd ball can be identified (where we do not know if it is heavier or lighter) in w weighings is given by the formula:
2. Applying the Formula
We need to find the minimum w such that the capacity N is at least 13.
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If w=2:
This is too small for 13 balls.
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If w=3:
With 3 weighings, we can exactly identify the odd ball out of 13 and determine if it is heavier or lighter.
3. Step-by-Step Logic for 13 Balls
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First Weighing: Divide the balls into three groups: 4,4, and 5. Weigh two groups of 4.
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If they balance, the odd ball is in the remaining 5.
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If they don't balance, the odd ball is among those 8 (and you now have a reference for "standard" weight from the remaining 5).
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Subsequent Weighings: By further dividing the suspected group into three parts in each step, the odd ball is narrowed down. For 13 balls, the mathematical limit confirms 3 weighings are sufficient.
Correct Option:
(a) 3
