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Thread: Least Possible Number of Coins

  1. #1
    Senior Member harpazo's Avatar
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    Least Possible Number of Coins

    Coins are to be put into 7 pockets so that each pocket contains at least one coin. At most 3 of the pockets are to contain the same number of coins, and no two of the remaining pockets are to contain an equal number of coins. What is the least possible number of coins needed for the pockets?

    Seeking for the first-two steps.
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    MHF Contributor MarkFL's Avatar
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    Re: Least Possible Number of Coins

    If we are to minimize the number of coins, then we should take the maximum number allowed to have the same number of coins, which is 3, and these 3 pockets having the same number of coins should have the smallest number allowed, which is 1.

    Since the 4 remaining pockets each has to have a unique number of coins, I would begin with the smallest number allowed, which is 2, and work up from there...what do you find for this minimum number of coins?
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    Senior Member harpazo's Avatar
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    Re: Least Possible Number of Coins

    Quote Originally Posted by MarkFL View Post
    If we are to minimize the number of coins, then we should take the maximum number allowed to have the same number of coins, which is 3, and these 3 pockets having the same number of coins should have the smallest number allowed, which is 1.

    Since the 4 remaining pockets each has to have a unique number of coins, I would begin with the smallest number allowed, which is 2, and work up from there...what do you find for this minimum number of coins?

    After 45 minutes of paper work, I came to the realization that
    since at most 3 of the pockets are to contain the same
    number of coins, then it makes sense to minimize the number of coins in each and by so doing, allowing each to contain just 1 coin.

    Next, I am told that no two of the remaining 4 pockets should contain an equal number of coins. So, they should contain 2, 3, 4, and 5 coins each (also a minimum possibility).

    My Answer for the total: 1+1+1+2+3+4+5=17.

    Correct?
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    MHF Contributor MarkFL's Avatar
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    Re: Least Possible Number of Coins

    Quote Originally Posted by harpazo View Post
    After 45 minutes of paper work, I came to the realization that
    since at most 3 of the pockets are to contain the same
    number of coins, then it makes sense to minimize the number of coins in each and by so doing, allowing each to contain just 1 coin.

    Next, I am told that no two of the remaining 4 pockets should contain an equal number of coins. So, they should contain 2, 3, 4, and 5 coins each (also a minimum possibility).

    My Answer for the total: 1+1+1+2+3+4+5=17.

    Correct?
    17 was my conclusion too.
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    Senior Member harpazo's Avatar
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    Re: Least Possible Number of Coins

    Quote Originally Posted by MarkFL View Post
    17 was my conclusion too.
    It took me 45 minutes to get 17. No such time exist on a timed test. See my point? I would have simply guessed facing this question on a test.
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    Re: Least Possible Number of Coins

    Quote Originally Posted by harpazo View Post
    It took me 45 minutes to get 17. No such time exist on a timed test. See my point? I would have simply guessed facing this question on a test.
    No such problem would appear on a timed test...
    unless the teacher setting up the test was in bad humor or something!
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    Senior Member harpazo's Avatar
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    Re: Least Possible Number of Coins

    Quote Originally Posted by DenisB View Post
    No such problem would appear on a timed test...
    unless the teacher setting up the test was in bad humor or something!
    This question is from a timed test review book called GMAT. I have the test prep book because I like the fact that there are over 200 math practice questions. The GMAT, GRE AND SAT certainly include ridiculous applications like this one that force students to guess their way to a passing grade. This is why I strongly believe that passing a multiple-choice test is no prove of intelligence. Long ago (early 1980s), I got 100 on a test of ten questions by guessing all 10 questions right. What is the probability of that ever happening again. I say about zero.
    Last edited by harpazo; Dec 3rd 2018 at 02:25 PM.
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