legend of the chessboard rice?

Question

does anyone know of the legend of the chessboard rice. where a sage challenged a king to a chess game and if the sage won he would get rice. the amount of rice was to be figured out by putting rice on every chessboard square and then the double amount of the previous on the next square. how would i figure out the amount of rice the sage was granted?

Answer 1

I remember that legend. As I recall, the king realized that he had been flimflammed by the time he got to the twentieth square and there wasn't enough rice in all the world to pay off the debt. I think the sage was executed at about that point.

Anyway, back to the problem. If you number the squares of a chessboard from 0 to 63, the amount of rice on square n would then be 2^n. The amount of rice on the final square alone would therefore be 2^63 grains, which would require a very big chessboard.

The number of grains on #62 would be half that many, the number on #61 would be half again, and so on. If this series is continued to square #0, the sequence would add up to

N = 2^63 * sum ((1/2)^k, k=0, 63)

I won't prove it here but if |s| < 1, then

sum (s^k, k=0, ∞) = 1 / (1 - s)

It can also be shown that

sum (s^k, k=0, n)
= sum (s^k, k=0, ∞) - sum (s^k, k=n+1, ∞)
= sum (s^k, k=0, ∞) - s^(n+1) * sum (s^k, k=0, ∞)
= (1 - s^(n+1)) * sum (s^k, k=0, ∞)
= (1 - s^(n+1)) / (1 - s)

For our particular problem, s = 1/2 and n = 63. Therefore,

N = 2^63 * (1 - (1/2)^64) / (1 - 1/2)
which simplifies to
N = 2^64 - 1 grains.

If one grain of rice has a mass of 20 mg, this would be a little short of 4 million millions of tons of rice.

Answer 2

the numbers of rice grains in the squares of the chessboard form a GP:1,2,4,8,16,32,..........
here a=1,r=2,n=64
S=a(r^n-1)(r-1)=1[(2^n-1]/1
the king had thought that the sage had asked vry little, but when they actually started putting the rice on the squares, he found that he had lost the entire production of rice for many many years, so he could not fulfill his promise. but a wise man in his kingdom gave him a beautiful suggestion to wriggle out of the situation. the sage was asked to actually count grains of rice one by one and have his bet fulfilled. the poor sage could not count even a small part of the grains due in his whole life. s o the king did not break his promise and yet saved his skin.

Answer 3

There are 64 squares on a chessboard, so you want:

1 + 2 + 4 + 8 + ... + 2^63

The sum of powers of two is one less than the next power of two (2^64 - 1). That number is: 18,446,744,073,709,551,615

Answer 4

t1 = 1
t2 = 2
t3 = 4 and so on. This is a geometric sequence.

To get the geometric series:
Sum = first term*(ratio^n - 1)/(ratio - 1)

t1+t2+t3+...+t64 = 2^64 - 1.

Answer 5

rice, wheat, there are numerous stories

2^64

1.844674407 * 10^19