How many cricket matches have to be played?

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There are N teams in a cricket match. How many matches have to be played to know the winner and the runner-up?

Generalized solution for N teams.

 

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  • 6 Answer(s)

    Answer is “n-1”.
    Assumption: question is to find the minimum matches to be played
    How:

    for each match played one team is eliminated. thus in first round, n/2 matches are played.
    in second round n/4 are played
    .
    .
    process is continued still one team is left.
    so total=n/2+n/4+n/8+….+1
    =1+2+4+…+n/2
    this is in geometric progression with a=1, r=2,  no. of terms =??(say k)

           for each match played one team is eliminated. thus after first round, n/2 teams are left.
           after 2nd round , n/4 teams are left
           .
           .
           after ‘k’th round n/(2^k) teams are left
           Now for n/(2^k)=1 team
           =>n=2^k
           =>k=log(n)
    a=1,r=2,k=log n (base 2)

    total matches = a(r^n – 1)/(r-1)
    =2^(log n) -1
    =n-1

    bsm Curious Answered on 5th September 2015.

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    on 8th September 2015.
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    n-1.
    when each match is played a team gets eliminate so after 1st round n/2 teams remains
    in next  n/4

    and so on
    till one team remains.
    therefore
    =n/2+n/4+…………+1
    =1+2+4+………….._n/4+n/2
    this is a GP sequence
    a=1 r=2
    and no. of terms = x(assumed)

    after each match, one team is eliminated. thus after first round, n/2 teams are left.
    after 2nd round , n/4 teams are left

    and so on

    after k th round

    n/(2^k) teams are left

    Now for n/(2^k)=1 team
    n=2^k
    k=log(n)
    a=1
    r=2
    k=log n (base 2)

    total matches = a(r^n – 1)/(r-1)
    =2^(log n) -1
    =n-1

    hence answer is n-1

    akky8055 Starter Answered on 5th September 2015.
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    n = number of teams,
    x = a variable

    When n is in order of 2^x,

    for example, n = 2^4 = 16, then number of matches will be n-1 = 16-1 = 15 (simple GP)

    But when n is not in order of 2^x,

    for example, n = 21,

    then number of matches will be n-1 = 21-1 = 20???

    Lets check this, n = 21 = 16 + 5 = 16 + 4 + 1 (all in order of 2^x like, 2^4+2^2+2^0)

    when n is in order of 2^x, then i can use n-1 for finding the matches.

    for 16 teams = 15 matches
    for 4 teams = 3 matches
    for 1 team = 0 match

    winner of these 2 groups needs to play again, along with the remaining 1 team. (total 3 teams remaining, say N)….

    hence, N = 3 = 2+1 (again in order of 2^x)

    for 3 teams = 2 matches are required for deciding the winner, who is none but the winner of all 21 teams.

    Total matches to decide a winner =15+3+2 = 20 matches. that is (n-1).

    Genarilized solution will be n-1.

    Kumaran Ilangovan Curious Answered on 6th September 2015.
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    Its just similar to 25 Horse and 5 track puzzle (First solve it)

    Lets say N=10
    Here we have,
    12 Teams(Horses) and 2 (Teams can play a match)Track and find Top 2

    MATCH #1 – 5 Matches
    A1 – B1
    A2 – B2
    A3 – B3
    A4 – B4
    A5  – B5
    A6  – B6

    Lets say, all A’s are winner
    So,
    MATCH #2 – Next tournament(3 Matches)-
    A1-A2
    A3-A4
    A5-A6
    Now we have winner- A1,A3 and A5

    MATCH #3 – Next Tournament – (3 Matches)
    A1-A3
    A3-A5
    A1-A5
    Now we’ll get 2 winners lets say – A1,A3(A1 is winner and A3 runnerup)

    In MATCH #4 – (1 Matches)
    We’ll take winners opponent in MATCH #2 – A1’s opponent is A4
    A4 – A3
    Lets say A4 win

    In MATCH #5-(1 Matches)
    We’ll take winners opponent in MATCH #1- A1’s Opponent B1
    B1 – A4

    Lets B1 wins So, B1 is 1st Runnerup.

    A total of 14 Matches required to find out Winner and runner up in 12 Teams.

    It is not a generalized solution but only a way

    SherlockHolmes Expert Answered on 3rd September 2015.
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    n-1 is the correct answer
    After each match is played a team will get eliminated so after 1st round n/2 teams remain
    in next  n/4 will remain in tournament and so on…………..till one team remains.

    therefore
    =n/2+n/4+…………+1
    =1+2+4+…………..n/4+n/2
    this is a Geometric Progression sequence with a=1 & r=2

    after each match, one team will get eliminated. thus after first round, n/2 teams are left.
    after 2nd round , n/4 teams are left

    This implies after k th round
    n/(2^k) teams are left

    Now for n/(2^k)=1 team
    n=2^k
    k=log(n)
    a=1
    r=2
    k=log n (base 2)

    Total matches = a(r^n – 1)/(r-1)
    =2^(log n) -1
    =n-1

    Prateek Jain Curious Answered on 7th September 2015.

    Complete Copy of akky8055 answer below.

    on 7th September 2015.
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    n-1 is final answer

    anujarora Default Answered on 7th September 2015.
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