Tackling Games & Tournament Problems in LR - Vikas Saini



  • Q. There are 200 teams participating in a knockout tournament where they form groups & play against each other.So in round 1 there will be 100 teams,resulting in 100 winners.Round 2 will then have 50 each in a group and so on.If there are an odd numbers in any rounds,which is selected at random and granted a bye.The tournament continues till a winner is decided.No match ends in a draw.(old cat question)

    (i)How many matches must be played to decide the winner ?

    (ii)How many rounds took place ?

    (iii)How many byes were granted ?

    Alternate approach :-

    It's bit lengthy & ridiculous.

    200 teams

    round 1 :- 100 match

    round 2 :- 50 match

    round 3 :- 25 match

    round 4 :- 12 match(1 bye)

    round 5 :- 6 match(1 bye)

    round 6 :- 3 match(1 bye)

    round 7 :- 2 match

    round 8 :- final

    Total = 100+50+25+12+6+3+2+1

    = 199 matches.

    Short approach :-

    Direct formula to find no of matches in knockout tournament =

    " if 'n' teams are participating then number of matches will be played 'n-1'.

    The concept behind this logic is - To decide winner rest of teams have to lose thier matches.

    In our question there are 200 teams,to decide winner 199 teams must lose their matches.

    So no of matches = 199.

    (ii) no of round are 8 as above data.

    short approach to find round

    2^k < N.

    rounds = k+1.

    In our question N = 200.

    nearest 2^k = 2^7.

    k = 7.

    rounds = 7 + 1 = 8.

    (iii) No of byes

    200 100 50 25 13 7 4 2 1

    three odd numbers here 25,13 & 7.

    So no of byes = 3.

    Q. There are 8 teams in IPL.Each team plays with another team twice.

    Winning team gets 2 points,losing team gets 0 points each match.No match ends with drawn or tie.Top 4 team advanced to semifinal.Then winning team of semifinal play against each other in final.

    (i) How many minimum point a team get,still  qualify for semifinal.

    (ii) How many maximum points required for a team but didn't reach into semifinal.

    (iii) How many minimum match winning required to win IPL trophy.

    (iv) How many maximum match a team can win but didn't win IPL trophy.

    In a match winning team 2 points,losing team 0 points.

    So 1 match = 2 points.

    Each team play against every other team twice.

    So total no will be played before semifinal = 2 x 8c2 = 56.

    56 matches = 112 points.

    Each team match = 2 x 7 = 14 match.

    (i)

    so it's clear 4 team will getting into semi.So top 3 had no problems at all for reaching there..

    One team will be there who will win all 14 matches.

    Hence team no 1 points = 2 x 14 = 28.

    Second team will be the one who will lose only by top 1 team..

    So team no 2 points = 2 x 12 = 24.

    Third team will be the one who will lose by top 2 teams.

    So team no 3 points = 2 x 10 = 20.

    Top 3 teams total points = 28+24+20 = 72 points.

    Points left = 112 - 72 = 40 points.

    Teams left = 5.

    so per team points = 40/5 = 8 points..

    So a team can reach into semifinal by just 8 points ( 4 wins only).

    (ii) Now the question is how many maximum points but still didn't reach into semifinal.

    So 3 teams will be there who can  never qualify for semifinal..

    team no 8 will lose all matches.

    team no 8 earned = 0.

    team no 7 will be the one just win by team no 8.

    team no 7 earned = 2 x 2 = 4 points.

    team no 6 will be the one who just win by team no 8 & team no 7.

    team no 6 earned = 2 x 4 = 8 points.

    Total points by last three teams = 0+4+8 = 12.

    points left = 112-12 = 100.

    teams left = 5.

    per team point = 100/5 = 20.

    So even 20 points not enough to  predict to be in semi final.

    (iii) minimum win required to win IPL trophy..

    To win IPL trophy any team must win semifinal & final.To get into semifinal only 4 win enough.

    So minimum win = 4 + 2 = 6.

    (iv) Maximum win but didn't win IPL trophy

    The team had to lose in final only whether team has won all the matches before..

    So 14+1 = 15 win.

    Q. In an Indian institute of management league,4 football teams are going to play against each other once.After some matches have been played a table was being made to display the result of those matches.However much of information of in the table was accidentally erased,as shown below.

    It is known that teams are awarded 2 points for a win,0 points for loss,1 points for drawn.

    Team

    played

    won

    drawn

    lost

    Goals for

    Goals conceded

    points

    IIM A

     

     

     

     

    4

    4

     

    IIM B

     

     

     

     

    5

    2

    5

    IIM C

     

    0

     

     

     

    4

    2

    IIM k

     

     

     

     

    0

    3

    0

     

    Q. 1 . What was the score of match between IIM A and IIM B ?

    Q.2.   What was the score of match between IIM B and IIM C ?

    Q.3.   What was the score of match between IIM A  and IIM K ?

    Basic concept

    1.No of goals for = No of goals conceded

    2.  goals for – goals conceded > 0

        Team win.

    3. goals for – goals conceded < 0

        Team lose.

    4. goals for – goals conceded = 0.

       Match drawn.

    Firstly we need to find no of goals for IIM C.

    Total no of goals =  total no of goals conceded

    4+5+0+IIM C goals  = 4+2+4+3

    9 + IIM C goals = 13

    IIM C goals = 4.

    Here given IIM B have 5 points. It means it won 2 matches & 1 match was ended with drawn.

    Any team can play maximum 3 matches here cause number of teams are 4.Every team have to play exactly match against each other.

    In table it is shown that IIM C had won 0 matches,but it have 2 points.

    It means IIM C have played 2 drawn.

    It is cleared that match IIM C and IIM B was drawn,and own second drawn match IIM C had played with IIM A.

    If IIM C would have been played against IIM K then obviously either IIM K or IIM C could get points,but IIM K had 0 points.

    It is clear IIM B had played against all 3 teams where it won against IIM A & IIM K,and played drawn with IIM C.

    It is clear IIM K had played 2 matches against IIM A & IIM B and in both matches it lost.

    For IIM A it is clear that that won lost against IIM B and played a drwn against IIM C.

    In such case it no of goals were more than no of goals for,but clearly given data of IIM A is 4-4.

    It means IIM A defeated IIM K.

    Now table becomes as per above information –

    Team

    played

    win

    drawn

    lose

    Goals for

    Goals conceded

    points

    IIM A

    3

    1

    1

    1

    4

    4

    3

    IIM B

    3

    2

    1

    0

    5

    2

    5

    IIM C

    2

    0

    2

    0

    4

    4

    2

    IIM K

    2

    0

    0

    2

    0

    3

    0

    Match results observations 1 :-

    1. IIM K (0) – IIM A (1)
    2. IIM K (0) – IIM B (2)
    3. IIM C(2) -IIM B(2)
    4. IIM C(2) - IIM A(2)
    5. IIM B(1) – IIM A(1)

    Above observation we get  match between IIM B and IIM A was drawn,which is not true.

    We need to  exchange data of 1 & 2.

    Match results observations 2 :-

    1. IIM K (0) – IIM A (2)
    2. IIM K (0) – IIM B (1)
    3. IIM C (2) – IIM B (2)
    4. IIM C (2) – IIM A(2)
    5. IIM B (2) – IIM A (0)

    Answers

    1. Score of IIM A – IIM B (0 – 2)
    2. Score of IIM B – IIM C (2 – 2)
    3. Score of IIM A – IIM K (2 – 0)

    Q. In a sport event, 6 teams A,B,C,D,E,F are competing each other.Matches are scheduled in two stage. Each team play three match in stage 1 & 2 matches in stage 2.No team play against the same team more than once in the event. No ties are permitted in any of the match.

    Stage 1

    One team won all the 3 matches.

    Two teams lost all the matches.

    D lost to A but won against C & F.

    E lost to B but won against C & F.

    B lost atleast one match.

    F did not play against top teams of stage 1.

    Stage 2

    The leader of stage 1 lost the next two matches.

    Of the two teams of the bottom team of stage 1,one team won both matches,while the other lost both matches.

    One more team lost both matches in stage 2.

    1. The teams with the most wins in the event is/are

    1. A             2.            A & C                     3.    F     4. E              5.  B & E

    2. The two teams that defeated the leader of stage 1 are

    1. F & D     2.            E & F        3. B & D        4.     E & D     5.      F & D

    3. The only team(s) that won both matches in stage 2

    1.    B.            2.            E & F                      3.A,E & F     4.    B,E & F     5.  B & F.

    4.The teams that won exactly two matches in the events are

    1. A,D & c     2. D & E    3. D & B     4. D,E & F     5. D & F

    Solutions :-

    Stage 1

    Let’s try to fill the table as given lucid information -

     

    A

    B

    C

    D

    E

    F

    A

    -

     

     

    won

    Din’t play

     

    B

     

    -

     

    Din’t play

    Won

     

    C

     

     

    -

    Lost

    Lost

     

    D

    Lost

    Din’t play

    won

    -

    Din’t play

    won

    E

    Din’t play

    Lost

    Won

    Din’t play

    -

    Won

    F

     

     

     

    Lost

    Lost

    -

           

      Now we have given information that two teams lost all 3 matches.These two teams are certainly F & C,because every other teams have won atleast 1 match.In addition,given B lost atleast onematch.So A is the only team who won all 3 matches.F did not play against A.

    Now results can be further matches :-

    1. A v/s B ( A won)

    2. A v/s C  ( A won)

    3. B v/s F ( B won )

    So our table becomes after stage 1

     

    A

    B

    C

    D

    E

    F

    A

    -

    won

    won

    won

    Din’t play

    Din’t play

    B

    Lost

    -

    Din’t play

    Din’t play

    won

    Won

    C

    Lost

    Din’t play

    -

    Lost

    Lost

    Din’t play

    D

    Lost

    Din’t Play

    won

    -

    Din’t play

    won

    E

    Din’t play

    Lost

    Won

    Din’t play

    -

    Won

    F

    Din’t play

    Lost

    Din’t play

    Lost

    Lost

    -

     

    As per given information of stage 2

    A will lose both matches.( against E & F)

    F will win both matches.(against A & C)

    C will lose both matches.(against B & F)

    D will lose both matches.(against B & E)

    B will win both two matches.(against C & D)

    Stage 2 table

     

    A

    B

    C

    D

    E

    F

    A

     

     

     

     

    Lost

    Lost

    B

     

     

    won

    won

     

     

    C

     

    Lost

     

     

     

    Lost

    D

     

    Lost

     

     

    Lost

     

    E

    Won

     

     

    won

     

     

    F

    won

     

    won

     

     

     

    So total win by teams are  :-

    A – 4,     B – 4,     C – 0,     D-2,        E-4,        F-2

    Answers :-

    1. B & E

    2. E & F

    3. B,E & F

    4. D & F                

    Q. 7 players A,B,C,D,E,F,G participate in chess tournament in which each player plays exactly once against each player plays exactly once against each of the six players.The tournament starts on monday,finishes on Wednesday and an equal number of matches is played on all the three days.Some of the observations made on each day are given below :

    Monday

    * F loses to C,D,E and G.

    * Only one player,who is not C,wins more than one match on Monday.

    Tuesday

    * B wins against C,D and E.

    * E loses to A & C.

    * G wins exactly two matches two matches on Tuesday.

    Wednesday

    * G loses to B but wins against A & C.

    * D wins against C & E.

    * F loses to A & B.

    Note :- None of the matches end in a draw.

    1. Who wins the highest number of matches in the tournament ?

    1. C     2. A   3. D   4. B

    2. Who wins more than one match on Monday ?

    1. D  2. B  3. C  4.CBD

    3. Popat,an avid chess buff,watched all the matches of the tournament except the ones that took place on Monday.How many  times did he see either B or D winning a match ?

    1. 10  2.9  3.8  4.7

    4. On how many days during the tournament does at least one player win more than two matches ?

    1. 0  2. 1  3. 2  4. CBD

    Solution :-

    Total match in tournament = 7c2 = 21.

    per day match = 21/3 = 7.

    Every team has won distinct matches.

    So no of wins lies from 0 - 6.

    Observation :-

    F lost his all matches.(0)

    B has won at least 5 matches against C,D,E,G,F.

    G lost his matches against B,but won against A,C,G & 2 more matches.

    It means B beats A.

    So number of wins

    A(4),B(6),F(0),G(5).

    D won against C,E,F.

    E lost by A,C,D,B,G.

    Hence C won against E & F.

    No of win in tournament.

    A (4),B(6),C(2),D(3),E(1),F(0),G(5)

    Monday matches observation

    (i) F - C (C)

    (ii) F - D (D)

    (iii) F - E (E)

    (iv) F - G (G)

    (v) B - A (B )

    (vi) A - C (A)

    (vii) A - D (A)

    Tuesday observation

    (i) B - C ( B )

    (ii) B - D (B )

    (iii) B - E (B )

    (iv) E - A (A)

    (v) E - C (C)

    (vi) G - D (G)

    (vii) G - E ( G)

    Wednesday observation

    (i)G - B (B )

    (ii) G - A (G)

    (iii) G - C (G)

    (iv) D - C (D)

    (v) D - E (D)

    (vi) F - A (A)

    (vii) F - B ( B )

    Answers

    1. B (6 won)

    2. A ( 2 won,1 lost)

    3. 7 times

    4. 1 ( only Tuesday)

    Q. Given below data is indicating the individual scores of 10 players of two different basketball teams in single match between them.Each had 5 players and all the scoring shots carried either two or three points.The team scoring the maximum points won.Players scored for their teams only.

    Player - points

    A  - 15

    B  - 17

    C  - 23

    D  - 18

    E   - 13

    F   - 16

    G   - 19

    H  - 24

    I   -  15

    J   - 20

    The difference between the lowest and the highest individual scores in both the teams in the same.The highest individual scores from the teams had the same number of scoring shots.

    A,D,F & G had an even number of scoring shots comprising both 2 & 3 points.

    B,E & I had an odd number of scoring shots comprising both 2 & 3 pointers.

    C was another player in his team to score 8 points in 4 shots.

    1. What is difference b/w the number of 3 pointer shots by both the teams.

    2. What is difference between the total 2 pointers and 3 pointers shots scored by both the teams together ?

    3. Who  had the highest ratio of points scored to number of scoring shots ?

    solution :-

    As per given data individual scores

    in descending order

    24,23,20,19,18,17,16,15,15,13.

    suppose two teams y & z

    first case

    y - 24,17

    z - 23,16

    but in first case if 13 point scorer has to be team y,but it impossible.

    second case

    y - H(24),B(17)

    z - J(20),E(13)

    in team y number of scoring shots must be 24.

    player-shot-point

    A  - 6(3+3) - 15

    D -  8(6+2) - 18

    F -  6(2+4) -  16

    G - 8 (5+3) - 19

    C - 9(4+5) - 23

    B - 7(4+3) - 17

    E - 5 (2+3) - 13

    I - 7 ( 6+1) - 15

    So it is clear that y is winning team.

    & three pointers score between 24-17 are C,G & D.

    and y team three pointer score between 20-13 are A,F,I.

    Players & score of both teams

    team y

    H - 24 (0+8,3+6,6+4)

    C - 23 (4+5)

    G - 19(5+3)

    D - 18(2+6)

    B - 17(4+3)

    team z

    J - 20(4+4,7+2,10+0)

    F - 16(2+4)

    A - 15(3+3)

    I - 15(1+6)

    E - 13(2+3)

    answers :-

    1. no of 3 pointers of team y

    (i) 8+5+3+2+3 = 21

    (ii) 6+5+3+2+3 = 19

    (iii) 4+5+3+2+3 = 17.

    no of 3 pointers of team z

    (i)4+4+3+1+3 = 15

    (ii) 2+4+3+1+3 = 13

    (iii) 0+4+3+1+3 = 11.

    Difference of 3 pointers of both teams in every case is 6.

    2. 10.

    3. Fardeen.


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