In my previous post, we discussed two common types of games and tournament questions. So, if you are looking for questions on new types of data representation or questions based on seeding in a tennis tournament, do read that post. However, there is another frequently appearing class of questions in the games and tournaments category Football/Hockey tournament questions in which we have to find out Goals scores, winners, ties, etc.
In such tournaments, all competitors play a fixed number of matches. Points are awarded for wins/draws/losses. Then an overall ranking is calculated using the total points or average points per match. Sometimes other factors such as goals scored/goals faced also come into the picture to resolves ties in ranking.
Let us look at a question from CAT 2000 (full set of questions here).
Question: Sixteen teams have been invited to participate in the ABC Gold Cup cricket tournament. The tournament was conducted in two stages. In the first stage, the teams are divided into two groups. Each group consists of eight teams, with each team playing every other team in its group exactly once. At the end of first stage, the top four teams from each group advance to the second stage while the rest are eliminated. The second stage comprised several rounds. A round involves one match for each team. The winner of a match in a round advances to the next round, while the loser is eliminated. The team that remains undefeated in the second stage is declared the winner and claims the Gold Cup.
The tournament rules such that each match results in a winner and a loser with no possibility of a tie. In the first stage, a team earns one point for each win and no points for a loss. At the end of the first stage, teams in each group are ranked on the basis of total points to determine the qualifiers advancing to the next stage. Ties are resolved by a series of complex tie-breaking rules so that exactly four teams form each group advanced to the next stage.
Now questions were asked on: Total number of matches, minimum number of wins required for a team to guarantee advance (or possible advance) to next stage, maximum number of matches that a team can win in the first stage without advancing, etc.
In first stage, teams are divided into two groups of 8 teams each. There they play a match against everyone exactly one ie 8C2 matches in every group. So 2 * 8C2 = 56 matches for the first stage.
In second stage, there are 8 teams in a knockout stage. There will be one winner, so 8 1 = 7.
So, total number of matches is 56 + 7 = 63
For a team to advance to the second stage, it should be among the top 4 in its group. Total points on stake in a group is the same as the total number of matches which is 8C2 = 28. To guarantee advance, it can have 3 teams with the same or more points. There can be 5 teams with 5 wins or 5 points. So, 5 wins is not good enough to ensure a birth in round 2. However, 6 wins will guarantee its advance. This also tells us that a team might have 5 wins but still not advance.
To figure out the minimum wins required to possibly advance, let us look at the method for n teams.
(n/2) 1 teams should win maximum Number of matches.
(n/2) + 1 teams should have exactly the same number of wins.
So in this question, the top 3 teams can have a maximum of 7 + 6 + 5 = 18 points.
All other teams (5) have a combined score of 28 18 = 10 points. Their individual score is 2 points each and one of these five teams will advance to second stage.
So, minimum wins required to advance is 2.
Let us look at another type of question in which we are given a table and we have to fill it. Given below is a random table at the end of hockey tournament. For each win two points were awarded and for a draw one point was given. We also know that the South Africa Spain match was a draw. No two teams have the exact same count for Win/Draw/Loss and Australia has won more matches than Spain. Figure out the result of every match from the table given below,
Team Name Played Won Draw Lost Points India 0 6 Pakistan 6 Australia 3 Spain 3 South Africa
Each team has played 4 matches.
A team can get a score of 6 in two ways,
3 Wins and 1 loss or 2 wins and 2 draws.
India did not lose, so it will have 2 wins and 2 draws whereas on the other hand Pakistan will have 3 wins.
Number of matches played will be 5C2 = 10
The total Number of points at stake is 20. South Africa has the leftover points which is 2.
We also know that the South Africa Spain match was a draw.
So, now our table looks like this,
Team Name Played Won Draw Lost Points India 4 2 2 0 6 Pakistan 4 3 0 1 6 Australia 4 3 Spain 4 Min(1) 3 South Africa 4 Min(1) 2
2 Points can be achieved by 1 Win, 0 Draw, 3 Loss OR 2 Draw, 2 Loss.
We know that both Spain & South Africa have at least 1 Draw. This means that South Africas 2 points are by 2 Draw, 2 Loss.
Team Name Played Won Draw Lost Points India 4 2 2 0 6 Pakistan 4 3 0 1 6 Australia 4 3 Spain 4 Min(1) 3 South Africa 4 0 2 2 2
3 Points can be achieved by 1 Win, 1 Draw, 2 Loss OR 3 Draw, 1 Loss.
As no two teams have the same Win/Draw/Loss count, one of the above applies to Australia whereas the other one applies to Spain. As Australia has won more matches, it will get the 1 Win, 1 Draw, 2 Loss.
Our final table will look like this,
Team Name Played Won Draw Lost Points India 4 2 2 0 6 Pakistan 4 3 0 1 6 Australia 4 1 1 2 3 Spain 4 0 3 1 3 South Africa 4 0 2 2 2
Now, let us try and analyze the match results for the 10 matches (in no special order).
Pakistan has won 3 and lost 1. Pakistan cannot win against India as India did not lose a match. So,
Match 1 India beat Pakistan
Match 2 Pakistan beat Australia
Match 3 Pakistan beat Spain
Match 4 Pakistan beat South Africa
We also know the result of South Africa Vs Spain,
Match 5 South Africa & Spain drew the match.
Spain has lost against Pakistan and it needs to draw all other matches.
Match 6 India & Spain drew the match
Match 7 Australia & Spain drew the match
Australia cannot draw another match as it has only 1 draw. It cannot win against India as India has no losses. So, it must have lost against India and the win must have come against the remaining team ie South Africa.
Match 8 India beat Australia
Match 9 Australia beat South Africa
The only match remaining between India & South Africa must have been a draw as India scored wins against Pakistan and Australia.
Match 10 India & South Africa drew the match
Phew!! I hope that you lasted this long without actually playing the game of Banging head against the wall.
Ravi Handa, an alumnus of IIT Kharagpur, has been teaching for CAT and various other competitive exams for around a decade. He currently runs an online CAT coaching and CAT Preparation course on his website http://www.handakafunda.com