Question

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 Rs 10 lakh?]

Position of Square on chess board	Amount (in Rs)
1st square	1
2nd square	2
3rd square	4

Answer 1

Given, a 8 × 8 grid.

Now, find the sum of each row,

E.g. 1^st row = 2⁰ + 2¹ + 2² + 2³ + 2⁴ + 2⁵ + 2⁶ + 2⁷

= 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255

2^nd row = 2⁸ + 2⁹ + 2¹⁰ + 2¹¹ + 2¹² + 2¹³ + 2¹⁴ + 2¹⁵

= 2⁸ (2⁰ + 2¹ + 2² + 2³ + 2⁴ + 2⁵ + 2⁶ + 2⁷)

= 2⁸ × 255 = 255 × 256 = 65280

3^rd row = 2¹⁶ × 255 = 16711680   ...[∵ 2⁸ = 256 and 2¹⁶ = 2⁸ × 2⁸ = 256 × 256 and so on]