Puzzle – 31 (XAT-2011)

On 1^st 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 minimum number of pieces of silver bar but did not want to pay any advance rent. How many pieces did he make?

(1)5                 (2)8                 (3)16               (4)20               (5)31

Solution follows here:

Solution:

Beauty of math is illustrated in this problem.
Do you believe it - only five pieces are enough.

If Timon cuts the silver bar of 31 inches in to the pieces with following sizes (in inches), it will result in minimum number of pieces- solution:

1, 2, 4, 8, 16

On day-1 Timon gives 1-inch piece to the landlady

On day-2 he gives 2-inch piece and takes back the 1-inch piece

On day-3 he gives 1-inch piece in addition to the already given 2-inch piece, thus making it 3-inches in total

On day-4 he gives 4-inch piece and takes back the 1-inch and 2-inch pieces

On day-5 he gives 1-inch piece in addition to the already given 4-inch piece, thus making it 5-inches in total

This will continue till day-31.

Hence a minimum of five pieces is enough

Math logic:

Any number up to 2ⁿ-1 can be represented by the combinations of the ‘n’ numbers: 2⁰,2¹,2²,...2^n-1.

For example, if 7 is considered, all numbers from 1 to 7 can be represented as combinations of 1,2, and 4:

1 = 1; 2 = 2; 3 = 1+2; 4 = 4; 5 = 1+4; 6 = 2+4; 7 = 1+2+4  

If 9 is considered, all numbers from 1 to 9 can be represented as combinations of 1,2,4 and 2:

1 = 1; 2 = 2; 3 = 1+2; 4 = 4; 5 = 1+4; 6 = 2+4; 7 = 1+2+4; 8 = 2+2+4; 9 = 1+2+2+4;

If 15 is considered, all numbers from 1 to 15 can be represented as combinations of 1,2,4 and 8:

1 = 1; 2 = 2; 3 = 1+2; 4 = 4; 5 = 1+4; 6 = 2+4; 7 = 1+2+4; 8 = 8; 9 = 1+8; 10 = 2+8;

11 = 1+2+8; 12 = 4+8; 13 = 1+4+8; 14 = 2+4+8; 15 = 1+2+4+8

Answer (1)