Question

On 1st March, Timon arrived in a new city and was looking for a place to stay. He met a landlady who offered to rent her apartment at a reasonable price but wanted him to pay the rent on a daily basis. Timon had a silver bar of 31 inches, and an inch of the silver bar was exactly equal to a day’s rent. He agreed to pay an inch of the silver bar towards the daily rent. Timon wanted to make the minimum number of pieces of the silver bar but did not want to pay any advance rent. How many pieces did he make?

• A. 5
• B. 8
• C. 16
• D. 20
• E. 31

Hint:

Problems involving “minimum number of parts to represent daily payments” usually map to binary decomposition of the total. Always think in powers of 2.

Answer 1

The Correct Option is A

Step 1: Understand the requirement.
- Timon must pay

exactly 1 inch per day, from Day 1 to Day 31.
- He cannot give advance rent, so he must be able to pay each day’s rent \emph{exactly on that day}.
- Objective: Break the 31-inch bar into the \emph{minimum number of pieces}.

Step 2: Apply binary decomposition logic.
This is a classic case of representing daily rent using different piece sizes, similar to the binary system: \[ 31 = 1 + 2 + 4 + 8 + 16 \] Each day’s rent can be constructed by combining these pieces. For example: - Day 1: give 1-inch piece.
- Day 2: return 1-inch, give 2-inch.
- Day 3: give 1-inch + 2-inch.
- Day 4: return both, give 4-inch.
- Continue similarly until Day 31 = 16+8+4+2+1.

Step 3: Minimum number of pieces.
Thus, Timon only needs 5 pieces: \[ 1, \; 2, \; 4, \; 8, \; 16 \]

Step 4: Final answer.
\[ \boxed{5} \]