Friday, February 15, 2008

This Number is a Two-Timer...

Two times a number...Here's a math puzzle for you to ponder. Feel free to post your answer in the comments.
Q: A positive integer has a unknown number of digits but it ends in a two. If the two is moved to the front of the number the new number will be exactly double the old. What is the number?

8 comments:

  1. I played around with it but I keep ending up with equations that look something like this:

    2000A+200B+20C+2=2000+100A+10B+C
    1900A + 190B + 19C = 1998
    19(100A + 10B + C) = 1998

    1998 not being evenly divisible by 19, there doesn't seem to be any integer combination, even if you use a zero, that will get the job done. The same holds true for four variables, etc.

    Guess I'm on the wrong track?

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  2. Ben,

    It has an unknown number of digits which is why I included an ellipsis ...

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  3. You tried 3 or 4 variables but have you gone higher?

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  4. Well, the first digit is a 1. The next-to-last digit is a 4. That fulfills the basic requirements given by the problem statement.

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  5. You're on the right track when you start with "the second to last digit must be 4". You know that because you know the last digit is 2, and doubling a number that ends in 2 gives a number that ends in 4.

    Now continue from there. If you know it ends in 42, then the next digit on the left of 4 must be 8. This is because doubling any number that ends in 42 gives a number that ends in 84. Hence the last 3 digits are 842

    Keep going with this reasoning. The trick is not to give up!

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  6. I did it in 18 places. Any takers??

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  7. Don and Pantheist are exactly correct. The last digit is 2 which will get moved to the front. Therefore the next digit has to be double this:
    ...42
    Then double the 4:
    ...842
    Then double the 8 (remember the carry):
    ...(1)6842
    Then double the 6 and add the carry (13), that gets you another carry:
    ...(1)36842
    Double the 3 and add the carry (7)
    ...736842
    Double the 7 (remember the carry)
    ...(1)4736842
    Double the 4 and add the carry (9)
    ...94736842
    Double the 9 and remember the carry
    ...(1)894736842
    Double the 8 and add the carry (17), remember the new carry:
    ...(1)7894736842
    etc.
    ...(1)57894736842
    ...(1)157894736842
    ...3157894736842
    ...63157894736842
    ..(1)263157894736842
    .5263157894736842
    (1)05263157894736842
    Now when you double you get back to the digit 2 so you are done.

    105,263,157,894,736,842

    P.S. This is the shortest number. You could actually repeat this sequence of digits as much as you like.

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