In front of you, there are 9 coins. They all look absolutely identical, but…
2023
In front of you, there are 9 coins. They all look absolutely identical, but one of the coins is fake. However, you know that the fake coin is lighter than the rest, and in front of you is a balance scale.
What is the least number of weighings you can use to find the counterfeit coin?
- A.
0
- B.
1
- C.
2
- D.
3
Show answer & explanation
Correct answer: C
Concept:
A balance scale weighing has exactly three possible outcomes — left side goes down, right side goes down, or the two sides balance. So k weighings can distinguish among at most 3k different situations. To guarantee finding one lighter counterfeit coin among n identical-looking coins, you need the smallest whole number k such that 3k is greater than or equal to n.
Applying this to the puzzle:
Here n = 9, and 31 = 3 is too small (3 < 9), while 32 = 9 exactly matches 9, so the minimum number of weighings needed is k = 2.
First weighing: split the 9 coins into three groups of 3 and place two of the groups on the two pans, leaving the third group aside. If one pan rises (its group is lighter), the lighter coin is in that group. If the pans balance, the lighter coin is in the group set aside.
Second weighing: take the 3 coins in the identified group, place one coin on each pan and set the third aside. If one pan rises, that coin is the lighter fake. If the pans balance, the coin set aside is the fake.
Cross-check:
This matches the general rule exactly: two weighings offer 3 × 3 = 9 distinguishable outcomes, covering all 9 coins with none to spare, so 2 is both sufficient and the true minimum — a single weighing offers only 3 outcomes, which is not enough to pin down one coin among 9.