Today we look at another scenario in the games and tournament category where in some teams play each other exactly once and wins-loss data is presented to us.
Let's take the following problem as our example and take it forward from there.
Q Eight players – A through H qualified for the final round of the world Snooker Championship which is a round robin competition, i.e., each player plays with every other player exactly once. At the start of the tournament, these eight players were seeded from 1 to 8, with seed 1 being considered as the top seed. No match ended as a tie and after each match, the winner was awarded one point and the loser was not awarded any point. At the end of the tournament, it was found that each player scored exactly one point less than his seeding. The following table gives the results of some of the matches played.
For example, in the match between A and D, A won over D.
It was also known that C was seeded 3 and E was not among the top five seeds.
For such round robin competition happening in a single round, we take the following template which in this case was actually provided with partial data filled.
Key notes for such a template :
1) The matches along the the diagonal marked as X are not possible as a team/player cannot play himself.
2) An entry of W on one side of diagonal for a row say A(entry in column D) becomes an entry of L for the column A(in the row D). Since If A beats D it implicitly means D loses to A.
3) After all standard details have been filled in the template, you make use of the conditions given in the questions.
Condition 1 : Points earned is less than the seed number assigned
Seed 1 : 0 points ; No games won
Seed 2 : 1 points ; Exactly 1 game won..and so on.
Also, note here in this context, all rows except F has 1 win indicated. Hence F is Seed 1 with 0 points.
Fill all empty entries in row F as L and make corresponding entries on other side of diagonal as the case may be to be W.
Condition 2 : C is seed 3 => Won 2 games only. Initial table shows victory against H and also he wins against F from condition 1.
Fill the table to indicate the necessary changes.You should have the following intermediate scenario now :
Condition 3 : Now except the decided seeds C,F and H, all have more than 1 wins.
Hence H is the seed 2 with exactly 1 win.So fill the remaining fields in row H as L and make necessary entries in column H to complete the scenario.
You would reach this point.
Also, E is not amongst first 5 seeds. Now observing from table let's write down the possible seed numbers.
A : 5/6/7/8
B : 4/5/6/7
D : 5/6/7
E : 6/7
G : 5/6/7/8
If D : 5 => 1 win more each for E and G
A : 6/7/8 , B : 4 , D : 5 , E : 6/7 , G : 7/8
If D : 6 => 1 win more for E or G and 1 loss for the other.
If E wins over D, D will win over G and we have,
A : 8 , B : 4 , D : 6 , E : 7 , G : 5. But If A = 8 implies A also beats E apart from G. Rejected.
If G wins over D, D will win over E and we have,
A : 5 , B : 4 , D : 6 , E : 7 , G : 8. E already loses from G and now from D. So can't be 7. Rejected
If D : 7 => 1 loss more each for E and G
A : 8 , B : 4 , D : 7 , E : 6 , G : 5. If A is 8 then A beats E. D also beats E and E already beaten against G.So can't be 6 seeded. Rejected.
1. Which player was seeded 1 at the start of the tournament ?
(1) A (2) F (3) H (4) Cannot be determined
2. How many matches did B win?
(1) 1 (2) 2 (3) 3 (4) 4
3. How many points did A score?
(1) 5 (2) 6 (3) 7 (4) Cannot be determined
4. At the end of the tournament, the eight players were ranked based on the number of points scored such that the person with the highest number of points is ranked 1, the one with the second highest number of points is ranked 2 and so on. For which player was his rank the same as the number of points he scored?
(1) D (2) A (3) E (4) Cannot be determined
This brings us to close of this article of Games and Tournaments involving round robin win-loss scenarios. Next time we' ll see another type. All questions hold the key in choosing the right template to organize the data given.If the articles are helping you learn better, do like and share the posts and blog.
Cheers!
AS
Let's take the following problem as our example and take it forward from there.
Q Eight players – A through H qualified for the final round of the world Snooker Championship which is a round robin competition, i.e., each player plays with every other player exactly once. At the start of the tournament, these eight players were seeded from 1 to 8, with seed 1 being considered as the top seed. No match ended as a tie and after each match, the winner was awarded one point and the loser was not awarded any point. At the end of the tournament, it was found that each player scored exactly one point less than his seeding. The following table gives the results of some of the matches played.
For example, in the match between A and D, A won over D.
It was also known that C was seeded 3 and E was not among the top five seeds.
For such round robin competition happening in a single round, we take the following template which in this case was actually provided with partial data filled.
Key notes for such a template :
1) The matches along the the diagonal marked as X are not possible as a team/player cannot play himself.
2) An entry of W on one side of diagonal for a row say A(entry in column D) becomes an entry of L for the column A(in the row D). Since If A beats D it implicitly means D loses to A.
3) After all standard details have been filled in the template, you make use of the conditions given in the questions.
Condition 1 : Points earned is less than the seed number assigned
Seed 1 : 0 points ; No games won
Seed 2 : 1 points ; Exactly 1 game won..and so on.
Also, note here in this context, all rows except F has 1 win indicated. Hence F is Seed 1 with 0 points.
Fill all empty entries in row F as L and make corresponding entries on other side of diagonal as the case may be to be W.
Condition 2 : C is seed 3 => Won 2 games only. Initial table shows victory against H and also he wins against F from condition 1.
Fill the table to indicate the necessary changes.You should have the following intermediate scenario now :
Condition 3 : Now except the decided seeds C,F and H, all have more than 1 wins.
Hence H is the seed 2 with exactly 1 win.So fill the remaining fields in row H as L and make necessary entries in column H to complete the scenario.
You would reach this point.
Also, E is not amongst first 5 seeds. Now observing from table let's write down the possible seed numbers.
A : 5/6/7/8
B : 4/5/6/7
D : 5/6/7
E : 6/7
G : 5/6/7/8
If D : 5 => 1 win more each for E and G
A : 6/7/8 , B : 4 , D : 5 , E : 6/7 , G : 7/8
If D : 6 => 1 win more for E or G and 1 loss for the other.
If E wins over D, D will win over G and we have,
A : 8 , B : 4 , D : 6 , E : 7 , G : 5. But If A = 8 implies A also beats E apart from G. Rejected.
If G wins over D, D will win over E and we have,
A : 5 , B : 4 , D : 6 , E : 7 , G : 8. E already loses from G and now from D. So can't be 7. Rejected
If D : 7 => 1 loss more each for E and G
A : 8 , B : 4 , D : 7 , E : 6 , G : 5. If A is 8 then A beats E. D also beats E and E already beaten against G.So can't be 6 seeded. Rejected.
1. Which player was seeded 1 at the start of the tournament ?
(1) A (2) F (3) H (4) Cannot be determined
2. How many matches did B win?
(1) 1 (2) 2 (3) 3 (4) 4
3. How many points did A score?
(1) 5 (2) 6 (3) 7 (4) Cannot be determined
4. At the end of the tournament, the eight players were ranked based on the number of points scored such that the person with the highest number of points is ranked 1, the one with the second highest number of points is ranked 2 and so on. For which player was his rank the same as the number of points he scored?
(1) D (2) A (3) E (4) Cannot be determined
This brings us to close of this article of Games and Tournaments involving round robin win-loss scenarios. Next time we' ll see another type. All questions hold the key in choosing the right template to organize the data given.If the articles are helping you learn better, do like and share the posts and blog.
Cheers!
AS
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