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Math Help - Good problem

  1. #1
    Newbie
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    Mar 2009
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    1

    Good problem

    Each of three cards has an integer written on it. The three integers
    p, q, r satisfy
    the condition 0
    p < q < r. Three players A, B, C mix the cards and pick one each.
    The number on the card they select is added to their scores.
    This process is repeated at most ten times, after which
    A has 20 points, B has 10
    points, and
    C has 9 points. Also we know that B got the r card in the last round.

    Who received the
    q card in the first round?
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  2. #2
    Super Member
    Joined
    Jan 2009
    Posts
    591
    With these assumptions also:
    There are only 3 cards, and their initial score each is zero (0).


    The total value at the end of all the rounds is 39 (20+10+9).

    Total value is (p+q+r)*rounds = 39.
    Number of rounds: 1, or 3, or 39 (the factors of 39)

    Since the number of rounds is "at most ten", can't have 39 rounds.

    If there were only 1 round, then B could not have received the
    highest valued card, since player A has a higher score.

    There were three rounds.

    Since all three cards total to 13, player B could not have received
    all three cards, nor could he have received the same card 3 times to
    get a total of 10.
    Thus B received 1 of the cards twice.

    No cards are equal. Player B scores a 10 count this possible way:
    10 = 1+1+8
    10 = 2+2+6
    10 = 3+3+4
    10 = 4+4+2
    The remaining card is:
    13 = 8+1+ 4
    13 = 6+2+ 5
    13 = 4+3+ 6
    13 = 4+2+ 7

    Since player B received the highest card in the last round, then
    remaining card cannot be a 6 or 7.
    Thus the cards are numbered:
    :: p=1, q=4, r=8
    or
    :: p=2, q=5, r=6 <-- no combination of these gives a score of 20.
    The max score possible in three rounds is 18 (3*6).

    Therefore
    player A: 8+8+4 = 20
    Player B: 8+1+1 = 10
    Player C: 4+4+1 = 9

    Player B received the 8 in the last round, Player A the 4, and Player C the 1.

    The order of rounds must be:
    A..B..C
    8..1..4 first round
    8..1..4 second round
    4..8..1 third round

    Player C received card q in the first round.
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