Cricket World Cup puzzle

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There are 9 team participating in ICC Cricket World Cup, every team will play two matches with every other team. Top 4 teams will be qualified for playoffs.

How many minimum wins required for a team to ensure a place in playoffs?

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    Solution:

    Let us first calculate the total number of matches that will be played:
    9C2 * 2
    = 9! / (2!7!) * 2
    = 8 * 9
    = 72
    Therefore, every team will play 16 matches.

    Let us now glance over the cases that are possible ==>

    Case 1:
    Let us say that Team A, Team B and Team C wins 14 matches each
    (they lose 1 match each from the other two teams),
    then the Team D can win only (16 – 3 * 2) = 10 matches.

    Case 2:
    Suppose that, the top 4 teams win 13 matches each.
    Team A will win 13 matches and will lose 3 matches from Team B, Team C and Team D.
    Team B will win 13 matches and will lose 3 matches from Team A, Team C and Team D.
    Team C will win 13 matches and will lose 3 matches from Team A, Team B and Team D.
    Team D will win 13 matches and will lose 3 matches from Team A, Team B and Team C.

    If this case actually happens, then the Team E will be able to win 8 matches at maximum
    as Team A, Team B, Team C and Team D will win all the matches played against Team E.

    Case 3:
    Suppose that, the top 5 teams win 12 matches each.
    Team A will win 12 matches and will lose 3 matches from Team B, Team C, Team D and Team E.
    Team B will win 12 matches and will lose 3 matches from Team A, Team C, Team D and Team E.
    Team C will win 12 matches and will lose 3 matches from Team A, Team B, Team D and Team E.
    Team D will win 12 matches and will lose 3 matches from Team A, Team B, Team C and Team E.
    Team E will win 12 matches and will lose 3 matches from Team A, Team B, Team C and Team D.

    If this case actually happens, Team E will win 12 matches but it still won’t be able to get into the
    playoffs assuming that it has the lowest run rate among all.

    This implies that from case 2, it is evident that a team must win at least 13 matches to confirm
    its position in the playoffs.

    Thank You.

    NitinGurbani Genius Answered on 20th December 2015.
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