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:

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

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

__Solution:__

Beauty of math is illustrated in
this problem.

Do you believe it - only five pieces are enough.

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 2

^{n}-1 can be represented by the combinations of the ‘n’ numbers: 2^{0},2^{1},2^{2},...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)

Very nicely illustrated.. :)

ReplyDeleteDear nayan, thanks for the complements

ReplyDeleteA very impressive solution... Thanks a lot Mr. Vemuri :-)

ReplyDeleteawesome puzzle....

ReplyDeletechutiya waala logic hai

ReplyDelete