Logical Reasoning Discussion

Q. |
There are 9 coins, out of which one is odd, i.e. its weight is either less or more than that of the other 8 coins. How many iterations of weighing using a pan balance are required to find the odd coin and to find whether it is heavier or lighter? |

✖ A. |
2 |

✔ B. |
3 |

✖ C. |
4 |

✖ D. |
5 |

✖ E. |
It is not possible to find out the odd weight coin |

**Solution:**

Option(**B**) is correct

Divide the 9 coins into 3 groups of 3 coins each.

Now, use the weighing scale to weigh group 1 against group 2.

We can have two cases now:

**Case (i):** Both are equal. Then the odd coin lies in group 3.

**Case (ii):** There is an imbalance. In this case the odd coin lies in either of the two groups.

In both the cases we do not know if the odd coin is heavy or light.

Hence, we need one more turn to determine whether the odd coin is heavy or light.

In the first case, we do so by comparing group 3 with any of groups 1 or 2, because we know that the odd weight coin lies in group 3.

In the second case, we compare either of groups 1 or 2 with group 3, because we know that the odd weight coin is not present in group 3.

Suppose we take group 1 against group 3 and there is an imbalance, then we know that group 1 contains the odd weight coin and from the way the balance tilts, we can know whether it is heavy or light. If both sides are balanced, then we know that group 2 contains the odd weight coin and from the way it tilted when it was weighed against group 1, we can know whether the odd weight coin is heavy or light.

Thus after 2 iterations we know whether the odd weight is coin is heavy or light and a group of 3 coins in which it lies.

Assume that the result obtained is that the odd coin is light.

Now, take 2 coins from the group that has the light coin and weigh them against each other.

The light coin will tilt the balance on the other side. If both coins weigh the same, then the 3rd coin of the group is the light coin.

We thus need 3 iterations to find the odd weight coin and to find if it is heavy or light.

Hence, **option B** is the correct choice.

**Note:** The rule that, if the number of items is $n$ such that $3^a < n \leq 3^{(a+1)}$, then the minimum number of times you need to weigh to find the odd item using a pan balance is $(a + 1)$, is applicable only when it is known whether the odd coin is heavy or light.