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Math Help - more puzzle

  1. #1
    Junior Member samsum's Avatar
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    more puzzle

    First, i want to thanks CAPTAINBLACK. last time the monkey puzzle gave us a hint to solve that puzzle, here i want to thank a lot.

    Here another puzzle. I don't know why that guys want us play with puzzles all the time. We love puzzle, i mean it.

    1. Arrange the digits 0, 1, 2, , 9 in an order to form a ten digit number that is divisible by every number from 1 through 18 (inclusive). Every digit must be used exactly once.

    2. One-hundred marbles are separated into five bags such that the first and second bags contain a total of 51 marbles, the second and third bags contain a total of 42 marbles, the third and fourth bags contain a total of 35 marbles, and the fourth and fifth bags contain a total of 31 marbles. How many marbles are in each bag?

    3. Find a ten digit number of the form ABCDEFGHIJ so that A is the number of 0s in the number, B is the number of 1s in the number, and so on.
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    Quote Originally Posted by samsum View Post
    2. One-hundred marbles are separated into five bags such that the first and second bags contain a total of 51 marbles, the second and third bags contain a total of 42 marbles, the third and fourth bags contain a total of 35 marbles, and the fourth and fifth bags contain a total of 31 marbles. How many marbles are in each bag?
    Let bi be the number of marbles in bag i.

    Then write out the equations:

    b1+b2+b3+b4+b5=100
    b1+b2...............=51
    .....b2+b3..........=42
    ..........b3+b4.....=35
    ...............b4+b5=31

    Now subtract the second of these from the first to get:

    b3+b4+b5=100-51=49

    Now subtract the foruth from this to get:

    b5=49-35=14

    Now back substitution of this into the fifth to second equations to will
    give the values for the remaining bags.

    RonL
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by samsum View Post
    1. Arrange the digits 0, 1, 2, , 9 in an order to form a ten digit number that is divisible by every number from 1 through 18 (inclusive). Every digit must be used exactly once.
    The smallest number divisibke by all the number from 1 to 18 is:

    N=2^4 3^2 5 7 11 13 17=12252240

    As the number is to have 10 digits it is k*N, where 82<=k<=816.

    Now I leave it to you to complete this (I do know the k and the 10
    digit number, but you might like to find them your self).

    RonL
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    Hello, samsum!

    3. Find a ten-digit number of the form ABCDEFGHIJ so that A is the number of 0s
    in the number, B is the number of 1s in the number, and so on.

    I found one solution: .
    6210001000

    Are there any others?

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  5. #5
    Junior Member samsum's Avatar
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    First and second solution, i understand it. However the last one i am confused. A is 0's, i dont' even understand that mean. Captainblack might know that and help me out...Thanks
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  6. #6
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    Quote Originally Posted by Soroban View Post
    Hello, samsum!


    I found one solution: . 6210001000

    Are there any others?
    Quote Originally Posted by samsum View Post
    First and second solution, i understand it. However the last one i am confused. A is 0's, i dont' even understand that mean. Captainblack might know that and help me out...Thanks
    The most significant digit is A, and it is the number of zeros in the number.
    Count them, there are 6, and A is 6, so that checks out.

    The second digit is B, and it is the number of "1"s in the number. Count them
    there are 2, and B is 2, so that checks out.

    The third digit is C, and it is the number of "2"s in the number. Count them
    there is 1, and C is 1, so that checks out.

    and so on...

    RonL
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  7. #7
    Junior Member samsum's Avatar
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    if the fouth is D. It's number of 4 in the number. , how can u determine D ?
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    Quote Originally Posted by samsum View Post
    if the fouth is D. It's number of 4 in the number. , how can u determine D ?
    In Sorobans number there are no 3's so D is 0, same for 4's and 5's so E
    and F are 0, there is one 6, so G is 1, and there are no 7's, 8's or 9's so
    H, I and J are all zero.

    Now the interesting question is how do you find this number if you don't
    already know it, and also are there any others?

    RonL
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  9. #9
    Grand Panjandrum
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    Quote Originally Posted by CaptainBlack View Post
    Now the interesting question is how do you find this number if you don't
    already know it, and also are there any others?
    I was looking at this at work today, and it looks like the solution is unique,
    and that you can get to it by a process of eliminating all other possibilities.
    Unfortunatly the demonstration is too fiddly for me to want to be bothered
    with typing it!

    However it is obvious that the sum of the digits of the number must be 10
    which can be used to elliminate a large number of candidate solutions.

    RonL
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  10. #10
    Junior Member samsum's Avatar
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    the numbers are really complex, what did you based on that you can tell the number from the letter ? Is the solution long and lazy to type right ? It's ok. Let everyone else solve it, you solved a lot puzzle already. Don't u want to solve some more ?
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  11. #11
    Junior Member samsum's Avatar
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    i got it now. thanks
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  12. #12
    Junior Member samsum's Avatar
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    1. Arrange the digits 0, 1, 2, , 9 in an order to form a ten digit number that is divisible by every number from 1 through 18 (inclusive). Every digit must be used exactly once.


    i tried to do as Captainblack has gone ahead. However, it's get very complicated how to get that number, and i tried to use Microsoft Excel. and messy again...hehehe...!
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    Quote Originally Posted by samsum View Post
    1. Arrange the digits 0, 1, 2, , 9 in an order to form a ten digit number that is divisible by every number from 1 through 18 (inclusive). Every digit must be used exactly once.


    i tried to do as Captainblack has gone ahead. However, it's get very complicated how to get that number, and i tried to use Microsoft Excel. and messy again...hehehe...!
    Trial and error shows that

    12252240*199 = 2438195760

    RonL
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  14. #14
    Junior Member samsum's Avatar
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    is that a program. One of my groupmate solve that by using the program Trial and error.
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  15. #15
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    Quote Originally Posted by samsum View Post
    is that a program. One of my groupmate solve that by using the program Trial and error.
    No, I looked at batches of 10 at a time on something a bit like Excel.

    I do have tools that would allow me to do this automaticaly but I did
    not use them for this problem.

    I'm sure there are tricks that would allow the search space to be refined
    but I didn't need them as the search space was small enough for semi-automatic
    methods..

    RonL
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