Teaser, answer at weekend

Discussion in 'Math' started by studiot, Feb 10, 2009.

  1. studiot

    Thread Starter AAC Fanatic!

    Nov 9, 2007
    5,005
    513
    Fill in the digits instead of the Xs

    Clue1 There is no remainder

    Clue2 There is only one 7 in the position shown

    XXX)XXXXXX(7XX
    XXX)XXXX
    XXX)X_______
    XXX)XXXXXX
    XXX)XXXXXX
    XXX)XX_________
    XXX)XX0000
     
  2. davebee

    Well-Known Member

    Oct 22, 2008
    539
    46
    Can you explain the notation a little? At first I thought it was a long division problem
    with the first parenthesis separating the divisor and the dividend but the parenthesis
    just before the 7 and lack of a quotient breaks that pattern...
     
  3. Mark44

    Well-Known Member

    Nov 26, 2007
    626
    1
    Your first instinct was right: it is a division problem, but the quotient is written to the right. I'm sure that studiot will confirm this.
     
  4. studiot

    Thread Starter AAC Fanatic!

    Nov 9, 2007
    5,005
    513
    Mark is correct.
    It is an old notation, but I had difficulty laying it out in the post. Sorry.
     
  5. Ratch

    New Member

    Mar 20, 2007
    1,068
    3
    studiot,

    Does that mean that there is no other seven anywhere a "X" is substituted, or only in the quotient?

    Ratch
     
  6. studiot

    Thread Starter AAC Fanatic!

    Nov 9, 2007
    5,005
    513
    Yes it is long division and I have worked it out for you, then replaced the digits with Xs

    If you prefer solve

    \frac{XXXXXX}{XXX} = 7XX

    none of the Xs are the digit 7
     
  7. jpanhalt

    AAC Fanatic!

    Jan 18, 2008
    5,699
    905
    330480/432 = 765

    I still don't understand the problem presented in your first post.

    John
     
  8. studiot

    Thread Starter AAC Fanatic!

    Nov 9, 2007
    5,005
    513
    This is the trouble with trying to simplify to explain. I only posted that to confirm I was looking at a particular long division calculation.


    330480/432 doesn't meet the original spec.

    432)330480(76X
    XXX)3024
    XXX)X_______
    XXX)xx2808
    XXX)xx2592
    XXX)XX_________
    XXX)XX0000

    etc

    has the wrong number of digits in the wrong place for the second line of the calculation.
     
  9. jpanhalt

    AAC Fanatic!

    Jan 18, 2008
    5,699
    905
    Well, I still don't understand the original notation, as I mentioned in my reply.

    Odd thing, most of the X's in the original post (#1) and in your follow up (#8) come across on my viewer (IE7) as ghosts, i.e., white on a light blue background. Initially, I dismissed it as an artifact. However, my e-mail shows:

    Can you state the problem in some wordprocessor or other program, then export as pdf or an image?

    John
     
  10. studiot

    Thread Starter AAC Fanatic!

    Nov 9, 2007
    5,005
    513
    I was trying to avoid wasting storage space on attachments.
     
  11. davebee

    Well-Known Member

    Oct 22, 2008
    539
    46
    142 ) 109056 (768
    994
    -----
    9656
    9656
    -----
    0000
     
  12. Ratch

    New Member

    Mar 20, 2007
    1,068
    3
    studiot,

    There are 164 different solutions. The first and second in dividend-divisor order are 100452/132 = 761 and 100584/132 = 762. Davebee submitted the last or 164th solution in the series. I was going to wait a little longer to submit my entry.

    I have a confession to make. I used my programming skills to brutally extract the solutions with the help of my PC. I know that is not the right thing to do, but I just could not help myself. The PC sure figured out the solutions a lot faster than I could have done. If anyone wants some more solutions or all the solutions, let me know.

    Ratch
     
  13. davebee

    Well-Known Member

    Oct 22, 2008
    539
    46
    Ahh, I did the same thing, except maybe a bit sloppier, as I only ended up with about 150 solutions. I was planning to also show the 100504, 136, 739 result to show that my answer wasn't unique but I'm at work with too much to get done before I go home tonight.
     
  14. studiot

    Thread Starter AAC Fanatic!

    Nov 9, 2007
    5,005
    513
    Congratualtions davebee on being the first to publish.

    Thanks, too Ratch.

    Those who are interested in logic may realise that the fact that the first subtraction cannot be more than 999 (there are three xxxs in this position) means that the first digit on the left cannot be greater than 1 or less than 1 and therefore must be 1.
     
    Last edited: Feb 13, 2009
  15. studiot

    Thread Starter AAC Fanatic!

    Nov 9, 2007
    5,005
    513
    I had better make it clear that davebee's solution fails in the third line, i.e. the second subtraction.

    Oh, and Ratch, your two published solutions also fall at the same hurdle. Sorry.

    But perhaps I've now given too much away?
     
    Last edited: Feb 13, 2009
  16. Ratch

    New Member

    Mar 20, 2007
    1,068
    3
    studiot,

    I don't understand what you are saying. Could you go through one of our faulty solutions point by point so we can understand?

    Ratch
     
  17. studiot

    Thread Starter AAC Fanatic!

    Nov 9, 2007
    5,005
    513
    OK we are dividing a six digit number ABCDEF by a three digit number, GHI to get a quotient JKL

    We are told J=7

    Proceeding throught the long division:

    GHI into ABC won't go so bring down D and move one place right.

    GHI into ABCD goes 7 times.

    We are told the product 7 times GHI is only three digits ie not more than than 999.
    If G is 2 or greater then the product would have four digits.
    So G=1

    This also tells us that A = 1 since the maximum product 1HI times 7HL is 199 times 799 = 159001 and we don't show leading zeros.

    So place 7 times 1HI under the 1BCDEF and do the subtraction.

    We are told this results in a two digit remainder so bring down E to make three digits.

    We are told this is not divisible by 1HI.
    (Davebee's solution would make it divisible 6 times.)

    So we place a zero in the quotient to note this fact

    This means K=0

    Then we bring down the last digit, L to make a four digit number.

    We are told that 1HI divides this exactly with no remainder so this four digit number is equal to L times 1HI.

    This should tell us something about L?????


    You can't bring down two digits at a time, without placing the all important zero in the quotient, by the rules of long division.

    Does this help?
     
    Last edited: Feb 14, 2009
  18. Ratch

    New Member

    Mar 20, 2007
    1,068
    3
    studiot,

    Let's take the last solution of the series 100584/132 = 762 .

    So then we have 1005 - 7*132 = 81 . bringing down the E = 8 gives 818, right?

    Wait a minute. Why or where does it say that particular partial division cannot have zero for a remainder? "6" appears a valid number as far as I can see. I observe no violation of the rules here.

    Anyway, I looked over my computer generated solutions and the only one with a zero for the second digit of the quotient is 100536/142 = 708 . It is the 124th solution of the series.

    Ratch
     
  19. studiot

    Thread Starter AAC Fanatic!

    Nov 9, 2007
    5,005
    513
    Because your calculation would look like my attachment.

    This is not the pattern in the original, which has four digits in the second subtraction and concludes there.

    And yes I've put the quotient on top rather than to the right.
     
  20. davebee

    Well-Known Member

    Oct 22, 2008
    539
    46
    Finally I get it - you need the zero in the quotient in order to allow pulling down both remaining digits in one operation...

    thanks for the puzzle!
     
Loading...