# Homework Help: Checker Board-interesting problem

1. Jan 26, 2006

### 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.

2. Jan 26, 2006

### 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.

3. Jan 26, 2006

### Gamma

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

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

4. Jan 26, 2006

### StatusX

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

5. Jan 26, 2006

### Gamma

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

Gamma