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Checker Board-interesting problem

  1. Jan 26, 2006 #1
    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.


  2. jcsd
  3. Jan 26, 2006 #2


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    Homework Helper

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

    [tex]\sum_{k=0}^{n} 2^k = \frac{2^{n+1}-1}{2-1} =2^{n+1}-1[/tex]

    So you're looking at about 265 cents for the first and 233 for the second, so using 210 [itex]\approx[/itex] 103, this will come out to about 1020 and 1010 cents respectively. Your answers seem close to this.
  4. Jan 26, 2006 #3
    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.
  5. Jan 26, 2006 #4


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    Homework Helper

    Sorry, you're right. I was thinking the first n powers was up to 2^n, but it's actually 2^(n-1).
  6. Jan 26, 2006 #5
    Thank You, I just want to know only if I am in the right direction.

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