PLEASE HELP! MEDAL!!
In a famous story, a wise man in India requests a reward promised by his king. The wise man shows the king a chessboard having 64 squares. "Put 1 grain of rice on the 1st square,2 grains on the 2nd square, 4 on the 3rd square, 8 on the 4th square, and so on. That will be my reward."
The king readily agrees and is ruined because the total number of rice grains turns out to be about 37 trilllion trillion trillion grains of rice- much more than the value of the entire kingdom. How many grains of rice go in the 12th square?
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So starting from 1 grain, keep doubling 11 times (not twelve).
It should be in the thousands.
Square # 1: 1 = 2⁰
Square # 2: 2 = 2¹
Square # 3: 4 = 2²
Square # 4: 8 = 2³
It doubles each time, so square # S will get
N = a•2^S
grains, for some constant value of a. When S=1, N=1, so
1 = a•2^1 = 2a; a = ½, and
N = ½•2^S = 2^(S-1)
So Square # 12 gets
N = 2^11 = 2048 grains.
BTW, the total grains on squares 1 through N is
2^N - 1
so the total on all 64 squares is
2^64 - 1 = 18,446,744,073,709,551,615
which is "only" 18 million trillion. It would still bankrupt the kingdom, because even at only 1 mg per grain, that's a billion grains to the (metric) ton, and that's more than 18 billion tons of rice. And a grain of rice weighs probably more like ten times that.
To get "about 37 trillion trillion trillion grains", would take a mere 125 squares under this rule. Or if the number of grains quadruples instead of doubles from each square to the next, the total on 64 squares will be
(4^64 - 1)/3 ≈ 113.4 trillion trillion trillion grains