Question

A gentleman decided to treat a few children in the following manner. He gives half of his total stock of toffees and one extra to the first child, and then the half of the remaining stock along with one extra to the second and continues giving away in this fashion. His total stock exhausts after he takes care of 5 children. How many toffees were there in his stock initially?

Answer 1

A person distributes toffees to 5 students. Each student receives one more than half the current number of toffees. After distributing toffees to the fifth student, no toffees remain.

Reverse Calculation Approach

Working backwards from the fifth student:

• At the 5th stage: The person gave away all toffees, resulting in 0 remaining.
• Before giving the extra toffee to the 5th student, there would have been 1. Therefore, the number of toffees before distribution to the 5th student was: \[ \text{Toffees before giving to 5th student} = 2 \text{ (i.e., } 1 \times 2 \text{)} \]
• At the 4th stage: \[ \text{Toffees} = (2 + 1) \times 2 = 6 \]
• At the 3rd stage: \[ \text{Toffees} = (6 + 1) \times 2 = 14 \]
• At the 2nd stage: \[ \text{Toffees} = (14 + 1) \times 2 = 30 \]
• At the 1st stage: \[ \text{Toffees} = (30 + 1) \times 2 = \boxed{62} \]

Final Answer

The person initially possessed \( \boxed{62} \) toffees.