I18-01

Question

The initial standing of the 16 teams coming into the football tournament is as follows: 
     Standing     Team Name       1   Team 1      2   Team 2      3   Team 3      4   Team 4      5   Team 5      6   Team 6      7   Team 7      8   Team 8      9   Team 9      10   Team 10      11   Team 11      12   Team 12      13   Team 13      14   Team 14      15   Team 15      16   Team 16      
These teams enter a football tournament where they are divided into four groups of four teams each. The format of the tournament consists of Group matches followed by Quarter Finals and Semi Finals. The winners of the Semi Finals then play for the trophy.In the Group stage, each team in a group plays against every other team in that group. A win earns the winning team 3 points, while a draw gives 1 point to each team. The losing team gets no points. The top two teams from each group advance to the Quarter Finals. In each group, all teams finished with different point totals, so there were no ties in the final group standings.
 
The position of the teams at the end of the Group stage is as given below:
      Position   Group 1   Group 2   Group 3   Group 4       1   Team 14   Team 13   Team 7   Team 9       2   Team 5   Team 12   Team 16   Team 11       3   Team 10   Team 2   Team 1   Team 3       4   Team 6   Team 8   Team 15   Team 4Format of the matches in the knockout stage is as given below:
 
 Quarter-finals: 
 
Match A: Winner of Group 1 vs Runner-up of Group 2
 
Match B: Runner-up of Group 1 vs Winner of Group 2
 
Match C: Winner of Group 3 vs Runner-up of Group 4
 
Match D: Runner-up of Group 3 vs Winner of Group 4
 
 Semi-finals: 
 
SF1: Winner of Match A vs Winner of Match D
 
SF2: Winner of Match B vs Winner of Match C
 
 Final:  Winner of SF1 vs Winner of SF2
 
In the knockout stage, there were no upsets — that is, a team with a better initial standing (as shown in the Introduction tab) always won against a team with a lower standing.

Bunuel · Answer

Official Solution:

Question 1: How many of the top 8 teams in the initial standings advanced to the Quarter Finals? 
 
We know that the teams in the Quarter Finals are the top two teams from each group.
 
These teams are: Team 14, Team 13, Team 7, Team 9, Team 5, Team 12, Team 16, Team 11.
 
The top 8 teams based on the initial standings are: Team 1, Team 2, Team 3, Team 4, Team 5, Team 6, Team 7, Team 8.
 
The teams common to both lists are: Team 5 and Team 7. So, only 2 of the top 8 teams reached the Quarter Finals.
 
Thus, this question can be answered.
 
 Question 2: Which team won the tournament? 
 
Since there were no upsets in the knockout stage, the team with the better initial standing always won.
 
Following the bracket and applying this rule:
 
 Quarter-finals: 
 
– Match A: Team 14 (Winner of Group 1) vs Team 12 (Runner-up of Group 2) → Team 12 wins (Team 12 has better initial standing)
 
– Match B: Team 5 (Runner-up of Group 1) vs Team 13 (Winner of Group 2) → Team 5 wins
 
– Match C: Team 7 (Winner of Group 3) vs Team 11 (Runner-up of Group 4) → Team 7 wins
 
– Match D: Team 16 (Runner-up of Group 3) vs Team 9 (Winner of Group 4) → Team 9 wins
 
 Semi-finals: 
 
– SF1: Team 12 vs Team 9 → Team 9 wins
 
– SF2: Team 5 vs Team 7 → Team 5 wins
 
 Final: 
 
– Final: Team 5 vs Team 9 → Team 5 wins (as Team 5 had a better initial standing)
 
Therefore, Team 5 won the tournament.
 
Thus, this question can be answered.
 
 Question 3: Did Team 5 remain unbeaten throughout the tournament? 
 
We know that Team 5 won the tournament and that there were no upsets in the knockout stage, meaning Team 5 won all of its knockout matches.
 
However, we do not have the results of individual group-stage matches. It's possible that Team 5 could have drawn or even lost a match in the group stage and still advanced as one of the top two teams in its group.
 
Therefore, we cannot determine whether Team 5 remained unbeaten throughout the tournament.
 
Thus, this question cannot be answered.

Correct answer:

How many of the top 8 teams in the initial standings advanced to the Quarter Finals?  "Yes"

Which team won the tournament?  "Yes"

Did Team 5 remain unbeaten throughout the tournament?  "No"

Bunuel · Answer

This MSR question and its solutions have been completely revised and improved.

akshatv45 · Answer

For Q3 part 3 - I don't think the definition of unbeaten is very clear in this context. Team 5 definitely lost or tied matches in any scenario. It means that it was definitely not unbeaten. And hence we can answer this part correctly.
So answer should be yes.

Bunuel  - could you please let me know if I am missing something?

Bunuel · Answer

The term unbeaten is clear, it means no losses, while ties are allowed.

Team 5 finished 2nd in its group, so it definitely did not win all matches.

However, we don’t know if it tied or lost in the group stage, both are possible.

Since the question asks whether Team 5 remained unbeaten, and we don’t know for sure, the answer is No, the question cannot be answered with certainty.

Goldenfuture · Answer

Bunuel  - Since there are no upsets:

Match B: Team 5 (Runner-up of Group 1) vs Team 13 (Winner of Group 2) → Team 5 wins
Won't Team 13 (1st position) win here as it had a higher position than Team 5 (2nd position). How does being in a group 1 makes you have a better "standing". Standing is driven by position - not grouping

Bunuel · Answer

“No upsets” refers to the initial standings (Introduction), not group finish.

In the knockout stage, there were no upsets — that is, a team with a better  initial  standing ( as shown in the Introduction tab ) always won against a team with a lower standing.

Luca46 · Answer

I like the solution - it’s helpful.

zitipiti · Answer

But just in the previous question where we were asked about the score of Team 5, you mentioned this info: "  Team 5 finished second in the group, so it must have had more than 3 points, and Team 14, which finished first, had more than Team 5. Since both Team 14 and Team 5 drew against Team 10, Team 14 must have had 7 points (1 draw with Team 10 and 2 wins), and Team 5 must have had 4 points (1 draw with Team 10, 1 loss to Team 14, and 1 win against Team 6).      Therefore, Team 5 got 4 points in the Group stage. "
  It is obvious from this that team 5 must have lost a game. Please let me know if I'm thinking incorrectly.

Bunuel · Answer

The condition  “If Team 10 drew all three of its group-stage matches”  applies only to that question. It is not given in the question about whether Team 5 remained unbeaten. Therefore, you cannot use the conclusion that Team 5 lost to Team 14 there.

zitipiti · Answer

Oh, I see. Thank you, I was thinking of tagging along with that information.

I18-01

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Standing	Team Name
1	Team 1
2	Team 2
3	Team 3
4	Team 4
5	Team 5
6	Team 6
7	Team 7
8	Team 8
9	Team 9
10	Team 10
11	Team 11
12	Team 12
13	Team 13
14	Team 14
15	Team 15
16	Team 16

Yes	No
		How many of the top 8 teams in the initial standings advanced to the Quarter Finals?
		Which team won the tournament?
		Did Team 5 remain unbeaten throughout the tournament?