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Math Help - Grains of rice on a chess board

  1. #1
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    Grains of rice on a chess board

    Hi, firstly apologies if this is not the correct section.

    At work today I was told that if you took a chess board (with 64 squares) and you were to place 1 grain of rice on the 1st square, 2 on the second square, 4 on the 3rd square, 8 on the 4th and so on doubling the grains of rice on each successive square compared to the previous square - that there would not be enough grains of rice in the world to do this.

    Whilst im skeptical that there would not be enough rice in the world to do this...

    how many grains of rice would be on the 64th square?, and

    what would the formula be for working this out?

    I guess I could work this out just by doubling 1 then 2 then 3 etc, but Id rather learn something new, so if there is a formula could someone explain it aswell.

    As you can probably guess Im not the sharpest when it comes to maths, so thanks for your time and help.

    Chez
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  2. #2
    MHF Contributor arbolis's Avatar
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    Welcome to MHF.
    If you call "n" the nth square, there are 2^{n-1} grains in the nth's square.
    So in the last square there is 2^{63} grains. To see how big or how small this number is, go there: http://www.wolframalpha.com/input/?i=2^63.
    I don't really know the number of rice's grains in the world, though it shouldn't be so hard calculating it. If you can give us the rice world production in tons by year, we will have the order of magnitude of the number of rice grains in the world, assuming that 1 kilo contains a certain number of grains.
    Notice also that on the 63 th square there is half of the grains in the 64th square and that you also have to sum it up.

    This number (of rice on the last square), as huge as it may look like, doesn't scar me. Indeed, there is 100 000 times more atoms of lead in a mole of lead, namely a small fragment of lead that you could have in one hand.
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  3. #3
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    Quote Originally Posted by arbolis View Post
    Welcome to MHF.
    If you call "n" the nth square, there are 2^{n-1} grains in the nth's square.
    So in the last square there is 2^{63} grains. To see how big or how small this number is, go there: http://www.wolframalpha.com/input/?i=2^63.
    I don't really know the number of rice's grains in the world, though it shouldn't be so hard calculating it. If you can give us the rice world production in tons by year, we will have the order of magnitude of the number of rice grains in the world, assuming that 1 kilo contains a certain number of grains.
    Notice also that on the 63 th square there is half of the grains in the 64th square and that you also have to sum it up.

    This number (of rice on the last square), as huge as it may look like, doesn't scar me. Indeed, there is 100 000 times more atoms of lead in a mole of lead, namely a small fragment of lead that you could have in one hand.
    Thanks that answers my question nicely.

    World rice production is around 600 million tons per year (presumably this is a metric ton).
    There are approx 36 950 grains of rice in a kilogram
    so 36 950 000 grains in a ton,
    so grains of rice produced in the world in a year is 36 950 000 * 600 = 22170000000 so something short of whats needed.

    Unless my maths is wrong (likely) so please correct if thats the case.

    Thanks again
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by arbolis View Post
    Welcome to MHF.
    If you call "n" the nth square, there are 2^{n-1} grains in the nth's square.
    So in the last square there is 2^{63} grains. To see how big or how small this number is, go there: http://www.wolframalpha.com/input/?i=2^63.
    I don't really know the number of rice's grains in the world, though it shouldn't be so hard calculating it. If you can give us the rice world production in tons by year, we will have the order of magnitude of the number of rice grains in the world, assuming that 1 kilo contains a certain number of grains.
    Notice also that on the 63 th square there is half of the grains in the 64th square and that you also have to sum it up.

    This number (of rice on the last square), as huge as it may look like, doesn't scar me. Indeed, there is 100 000 times more atoms of lead in a mole of lead, namely a small fragment of lead that you could have in one hand.
    Yes, but the number of grains of rice in total would be \sum_{n=0}^{64}2^n=2^65-1
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  5. #5
    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by Drexel28 View Post
    Yes, but the number of grains of rice in total would be \sum_{n=0}^{64}2^n=2^65-1
    Yes, that's why I wrote "you have to sum it up".
    There's a slight error, if you start at n=0, you must end at n=63 since there are 64 squares but you are right we must sum all the rices on every square.
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    Quote Originally Posted by Drexel28 View Post
    Yes, but the number of grains of rice in total would be \sum_{n=0}^{64}2^n=2^65-1
    The volume ocupied by the grains on the last square alone would fill a sphere of radius the order of 3.6 km (or more if a packing fraction less than 1 is used)

    CB
    Last edited by CaptainBlack; March 10th 2010 at 09:19 PM. Reason: gross error in calculation
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  7. #7
    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by chezza View Post
    Thanks that answers my question nicely.

    World rice production is around 600 million tons per year (presumably this is a metric ton).
    There are approx 36 950 grains of rice in a kilogram
    so 36 950 000 grains in a ton,
    so grains of rice produced in the world in a year is 36 950 000 * 600 = 22170000000 so something short of whats needed.

    Unless my maths is wrong (likely) so please correct if thats the case.

    Thanks again
    I think you made an error in the expression "36 950 000 * 600 = 22170000000". You're multiplying the number of rice grains contained in one ton by 600 instead of 600 millions. Thus the final result should be 10^6 times greater than your result. So yes, it seems that with one year world production, you would have approximately 1/1000th of the grains needed to fill the last square, if I didn't make any arithmetic mental error.
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    Thanks for the replies,
    Its been interesting reading though I have no idea what the formula given by drexel means
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  9. #9
    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by chezza View Post
    Thanks for the replies,
    Its been interesting reading though I have no idea what the formula given by drexel means
    \sum_{n=0}^{63}2^n=2^0+2^1+2^2+2^3+...+2^{62}+2^{6  3}=1+2+4+8+...+9223372036854775808 which is the total number of rice grains. Each term of the sum represent the number of rice grain(s) in each square. Summing them up reach the number of rice grains in the whole board.
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