Arsenal collection: football logic
Problem
These three problems require some logical thinking and a willingness to work systematically.
Can you deduce the missing information in each case below?
Perhaps you might develop techniques in one brainteaser that will help you to solve another.
1. Football Champ
Three teams, Arsenal, Manchester City and Chelsea, have each played two matches.
Three points are given for a win and one point to each team for a draw.
The table below gives the total number of points and goals scored for and against each team.
Fill in the table and find the scores in each match.
| Teams | Games Played | Won | Drawn | Lost | Goals For | Goals Against | Points |
|---|---|---|---|---|---|---|---|
| Arsenal | 2 | 5 | 3 | 3 | |||
| Man City | 2 | 2 | 1 | ||||
| Chelsea | 2 | 3 | 2 | 4 |
2. Friendly Tournament
The youth academy Under-18s competed in a friendly round-robin tournament involving six teams: Arsenal U18, Bolton U18, Chelsea U18, Tottenham U18, Everton U18 and Fulham U18. Each team had to play against every other team. All the matches resulted in either a win or a loss - there were no draws.
- No team lost all their matches, but one team won all their matches.
- Tottenham won their match against Bolton.
- Arsenal and Everton won the same, odd, number of matches, but Arsenal lost to Everton.
- Bolton and Fulham won a total of seven matches
- Chelsea won only one match, against the only other team who also won only one match.
Can you deduce what all of the results were?
3. Qualifying Round
In the qualifying round of the UEFA Under-19 Football Championship, a group of four teams were to play each other once. 2 points were awarded for a win, and 1 point for a draw.
After some of the matches were played, most of the information in the results table was accidentally deleted.
| Team | Played | Won | Drawn | Lost | For | Against | Points |
|---|---|---|---|---|---|---|---|
| A | 4 | 4 | |||||
| B | 5 | 5 | |||||
| C | 0 | 4 | 2 | ||||
| D | 0 | 3 | 0 |
Can you work out the score in each match played?
Getting Started
Work out what each part of the problem tells you, and write it down.
What can you work out from what you know? Write down any new information that you deduce.
Can you think of any clear ways of representing the information?
Student Solutions
1. Football Champ? The Year 9 class at Ounsdale High School showed us their method:
- Josh said that to work out the wins, draws and losses we looked at the points and the only possibilities are given below.
- Man City and Chelsea must have drawn because they each had 1 draw.
- Arsenal must have beaten Man City, and Chelsea must have beaten Arsenal.
- Jordan said: the maximum that Man City could have scored in their draw was 2, so it was either 2 - 2 or 1- 1.
- As Chelsea only let in 2 goals in total, and Chelsea must have scored at least one goal against Arsenal, the same between Man City and Chelsea must have ended up 1-1.
- Total number of goals for and against must be equal, because if Arsenal scored against Man City it would count for Arsenal and against Man City, for example.
And their results were as follows:
| Teams | Games Played | Won | Drawn | Lost | Goals For | Goals Against | Points |
|---|---|---|---|---|---|---|---|
| Arsenal | 2 | 1 | 0 | 1 | 5 | 3 | 3 |
| Man City | 2 | 0 | 1 | 1 | 2 | 5 | 1 |
| Chelsea | 2 | 1 | 1 | 0 | 3 | 2 | 4 |
2. For the Friendly Tournament problem, Szymon from Earl Mortimer College recommended making a table. He explains his method as follows:
Tip 2 says 'Tottenham won their match against Bolton'. Mark this in the table as follows:
| A | B | C | T | E | F | |
| A | ||||||
| B | Image
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| C | ||||||
| T | Image
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| E | ||||||
| F |
- Tip 3: 'Arsenal and Everton won the same, odd, number of matches, but Arsenal lost to Everton'. You know that Everton won against Arsenal, so you can plot these straight away. Now, they won either 1, 3 or 5 games each, but it can't be five, because only one person could have won all their matches.
- Tip 5 says Chelsea won only one match, against the other person who won only one as well. We don't know who this is, but there is only one of them, so it can't be Arsenal and Everton, so they must have won 3 of their matches...
| A | B | C | T | E | F | |
| A | Image
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Image
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Image
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Image
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Image
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| B | Image
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| C | ||||||
| T | ||||||
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| F |
3. Finally, the Qualifying Round problem - this was tough! A bit of careful reading tells you that the results given are only a partial list of results - some games may not have been played yet. Samantha gives her answer as follows:
| Team | Played | Won | Drawn | Lost | For | Against | Points |
| A | 3 | 1 | 1 | 1 | 4 | 4 | 3 |
| B | 3 | 2 | 1 | 0 | 5 | 2 | 5 |
| C | 2 | 0 | 2 | 0 | 4 | 4 | 2 |
| D | 2 | 0 | 0 | 2 | 0 | 3 | 0 |
and the individual results:
A v B: B wins 2-0
A v C: draw 2-2
A v D: A wins 2-0
B v C: draw 2-2
B v D: B wins 1-0
C v D: not played yet.
These solutions were originally submitted for a version of this activity in the context of the Olympics.
Teachers' Resources
Why do this problem?
These problems require some logical thinking and a willingness to work systematically. Routes to a solution are not immediately obvious so working on the problems could help students to develop their resilience, an important quality for mathematical problem-solvers.
Possible approach
Before introducing the task:
"Mathematicians need to be resourceful problem-solvers who don't give up, who talk to each other and share good ideas, who work strategically and systematically. The answers won't be immediately obvious, so these problems will test your ability to work as mathematicians."
The problems could be used in several ways:
Hand out this worksheet and invite students to work on all three problems in pairs.
Cut the worksheet into the three separate problems and invite different groups to work on different problems.
Work on one or two of the questions together as a class and then invite students to use each other's useful insights to solve the remaining problems.
While students are working, circulate and listen for useful insights. Where appropriate, bring the class together so that students can share successful strategies with the rest of the class.
The following key questions or prompts could be offered to students who are stuck:
What do you know? Write it down.
What can you deduce from what you know? Write it down.
Is there a good way of representing what you know, to make it clear?
Possible extension
For other problems that require systematic and logical thinking see Two and Two, Product Sudoku and Cinema Problem.
Possible support
Of the three problems, Football Champ and Qualifying Round use very similar techniques, so one could be solved as a class before students are given the second to apply any useful methods they have devised.