Question

Question 170

Long back in ancient times, a farmer saved the life of a king's daughter. The king decided to reward the farmer with whatever he wished. The farmer, who was a chess champion, made an unusal request
"I would like you to place 1 rupee on the first square of my chessboard . 2 rupees on the second square, 4 on the third square, 8 on the fourth square and so on, until you have covered all 64 squares. Each square should have twice as many rupees as the previous square ' The king thought this to be too less and asked the farmer to think of some better reward, but the farmer didn't agree.

How much money has the farmer earned?

[Hint The following table may help you. What is the first square on which the king will place at least 10 lakh?]

Answer 1

Given, $a8\times 8$ grid.
Now, find the sum of each row,
e.g. 1st row $=2^0+2^1+2^2+2^3+2^4+2^5+2^6+2^7=2552ndrow=2^8+2^9+2^{10}+2^{11}+2^{12}+2^{13}+2^{14}+2^{15}=2^8(2^0+2^1+2^2+2^3+2^4+2^5+2^6+2^7)=2^8\times 255=255\times 256=652803rdrow=2^{16}\times 255=16711680$ [$\because 2^8=256and{2}^{16}=2^8\times 2^8=256\times 256andsoon$]