Checker Board-interesting problem

[Gamma]

I found this problem on mathworld.com. It is a classic problem so I am hoping some body here must have seen it before.

Consider a checkerboard with 64 squares. If you put 1 penny on the first box, 2 pannies on the second box, 4 pennies on the third one and continue to do this until you fill the whole checker board.

How much total money you need to fill all 64 boxes? I find it to be 1.84 X 10^5 trillion dollars!

How much total money is needed to fill only 32 squares? my answer: 42.9 million


I just want to know if these numbers seems reasonable.


Thanks,

Gamma.

 

[StatusX]

Well, the sum of the first n powers of 2 is:

$$\sum_{k=0}^{n} 2^k = \frac{2^{n+1}-1}{2-1} =2^{n+1}-1$$

So you're looking at about 2⁶⁵ cents for the first and 2³³ for the second, so using 2¹⁰ $\approx$ 10³, this will come out to about 10²⁰ and 10¹⁰ cents respectively. Your answers seem close to this.

 

[Gamma]

Thank you for answering.

I am getting (2^64 -1) and (2^32 - 1)

not (2^65 -1) and (2^33 - 1) (Are sure about this?)

Just a difference of one in the power makes a big difference in the out come.

 

[StatusX]

Sorry, you're right. I was thinking the first n powers was up to 2^n, but it's actually 2^(n-1).

 

[Gamma]

Thank You, I just want to know only if I am in the right direction.

Gamma