Arsenal Collection: Football Logic

8824 /www/nrich/html/content/id/8824/ 1 2012-09-24T17:39:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><p>These three problems require some logical thinking and a willingness to work systematically.</p> <p>Can you deduce the missing information in each case below?<br></br> Perhaps you might develop techniques in one brainteaser that will help you to solve another.</p> <p> </p> <h3>1. Football Champ</h3> <mdo:image alt="" src="Arteta%20goal%20crop.jpg" style="width: 250px; height: 250px; margin: 5px 10px; float: right;"></mdo:image> <p>Three teams, Arsenal, Manchester City and Chelsea, have each played two matches.</p> <p>Three points are given for a win and one point to each team for a draw.</p> <p>The table below gives the total number of points and goals scored for and against each team.</p> <p>Fill in the table and find the scores in each match.<br></br> </p> <p> </p> <p> </p> <br></br> <table style="width: 680px;border-spacing:1px;" border="1"> <thead> <tr> <th scope="col" style="text-align: center; vertical-align: middle;"> Teams</th> <th scope="col" style="text-align: center; vertical-align: middle;">Games Played</th> <th scope="col" style="text-align: center; vertical-align: middle;"> Won</th> <th scope="col" style="text-align: center; vertical-align: middle;"> Drawn</th> <th scope="col" style="text-align: center; vertical-align: middle;"> Lost</th> <th scope="col" style="text-align: center; vertical-align: middle;">Goals For</th> <th scope="col" style="text-align: center; vertical-align: middle;">Goals Against</th> <th scope="col" style="text-align: center; vertical-align: middle;"> Points</th> </tr> </thead> <tbody> <tr> <th style="text-align: center; vertical-align: middle;">Arsenal</th> <th style="text-align: center; vertical-align: middle;">2</th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;">5</th> <th style="text-align: center; vertical-align: middle;">3</th> <th style="text-align: center; vertical-align: middle;">3</th> </tr> <tr> <th style="text-align: center; vertical-align: middle;">Man City</th> <th style="text-align: center; vertical-align: middle;">2</th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;">2</th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;">1</th> </tr> <tr> <th style="text-align: center; vertical-align: middle;">Chelsea</th> <th style="text-align: center; vertical-align: middle;">2</th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;">3</th> <th style="text-align: center; vertical-align: middle;">2</th> <th style="text-align: center; vertical-align: middle;">4</th> </tr> </tbody> </table> <p> </p> <p> </p> <h3> 2. Friendly Tournament</h3> <p>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. </p> <ul> <li>No team lost all their matches, but one team won all their matches.</li> <li>Tottenham won their match against Bolton.</li> <li>Arsenal and Everton won the same, odd, number of matches, but Arsenal lost to Everton.</li> <li>Bolton and Fulham won a total of seven matches</li> <li>Chelsea won only one match, against the only other team who also won only one match.</li> </ul> <p>Can you deduce what all of the results were?</p> <p> </p> <h3>3. Qualifying Round</h3> <p><br></br> 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.</p> <p>After some of the matches were played, most of the information in the results table was accidentally deleted.</p> <p> </p> <p> </p> <table style="border-spacing: 1px; width: 600px;" border="1"> <thead> <tr> <th scope="col" style="text-align: center; vertical-align: middle;">Team</th> <th scope="col" style="text-align: center; vertical-align: middle;">Played</th> <th scope="col" style="text-align: center; vertical-align: middle;">Won</th> <th scope="col" style="text-align: center; vertical-align: middle;">Drawn</th> <th scope="col" style="text-align: center; vertical-align: middle;">Lost</th> <th scope="col" style="text-align: center; vertical-align: middle;">For</th> <th scope="col" style="text-align: center; vertical-align: middle;">Against</th> <th scope="col" style="text-align: center; vertical-align: middle;">Points</th> </tr> </thead> <tbody> <tr> <th style="text-align: center; vertical-align: middle;">A</th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;">4</th> <th style="text-align: center; vertical-align: middle;">4</th> <th style="text-align: center; vertical-align: middle;"> </th> </tr> <tr> <th style="text-align: center; vertical-align: middle;">B</th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;">5</th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;">5</th> </tr> <tr> <th style="text-align: center; vertical-align: middle;">C</th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;">0</th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;">4</th> <th style="text-align: center; vertical-align: middle;">2</th> </tr> <tr> <th style="text-align: center; vertical-align: middle;">D</th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;"> </th> <th style="text-align: center; vertical-align: middle;">0</th> <th style="text-align: center; vertical-align: middle;">3</th> <th style="text-align: center; vertical-align: middle;">0</th> </tr> </tbody> </table> <p><br></br> Can you work out the score in each match played?<br></br> <br></br> <a href=""><mdo:image alt="" src="NEW%20AITC%20LOGO%20JPEG-1.jpg" style="width: 250px; height: 86px; margin: 5px; float: right;"></mdo:image></a></p></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <span class="editorial">1. Football Champ? The Year 9 class at Ounsdale High School showed us their method:</span><br></br> <ul> <li>Josh said that to work out the wins, draws and losses we looked at the points and the only possibilities are given below.</li> <li>Man City and Chelsea must have drawn because they each had 1 draw.</li> <li>Arsenal must have beaten Man City, and Chelsea must have beaten Arsenal.</li> <li>Jordan said: the maximum that Man City could have scored in their draw was 2, so it was either 2 - 2 or 1- 1.</li> <li>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.</li> <li>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.</li> </ul> <br></br> <span class="editorial">And their results were as follows:</span><br></br> <br></br> <br></br> <table style="width: 680px;border-spacing:1px;" border="1"> <thead> <tr> <th scope="col" style="text-align: center; vertical-align: middle;"> Teams</th> <th scope="col" style="text-align: center; vertical-align: middle;">Games Played</th> <th scope="col" style="text-align: center; vertical-align: middle;"> Won</th> <th scope="col" style="text-align: center; vertical-align: middle;"> Drawn</th> <th scope="col" style="text-align: center; vertical-align: middle;"> Lost</th> <th scope="col" style="text-align: center; vertical-align: middle;">Goals For</th> <th scope="col" style="text-align: center; vertical-align: middle;">Goals Against</th> <th scope="col" style="text-align: center; vertical-align: middle;"> Points</th> </tr> </thead> <tbody> <tr> <th style="text-align: center; vertical-align: middle;">Arsenal</th> <th style="text-align: center; vertical-align: middle;">2</th> <th style="text-align: center; vertical-align: middle;">1</th> <th style="text-align: center; vertical-align: middle;">0</th> <th style="text-align: center; vertical-align: middle;">1</th> <th style="text-align: center; vertical-align: middle;">5</th> <th style="text-align: center; vertical-align: middle;">3</th> <th style="text-align: center; vertical-align: middle;">3</th> </tr> <tr> <th style="text-align: center; vertical-align: middle;">Man City</th> <th style="text-align: center; vertical-align: middle;">2</th> <th style="text-align: center; vertical-align: middle;">0</th> <th style="text-align: center; vertical-align: middle;">1</th> <th style="text-align: center; vertical-align: middle;">1</th> <th style="text-align: center; vertical-align: middle;">2</th> <th style="text-align: center; vertical-align: middle;">5</th> <th style="text-align: center; vertical-align: middle;">1</th> </tr> <tr> <th style="text-align: center; vertical-align: middle;">Chelsea</th> <th style="text-align: center; vertical-align: middle;">2</th> <th style="text-align: center; vertical-align: middle;">1</th> <th style="text-align: center; vertical-align: middle;">1</th> <th style="text-align: center; vertical-align: middle;">0</th> <th style="text-align: center; vertical-align: middle;">3</th> <th style="text-align: center; vertical-align: middle;">2</th> <th style="text-align: center; vertical-align: middle;">4</th> </tr> </tbody> </table> <br></br> <br></br> <span class="editorial">2. For the Friendly Tournament problem, Szymon from Earl Mortimer College recommended making a table. He explains his method as follows:</span><br></br> <br></br> Tip 2 says 'Tottenham won their match against Bolton'. Mark this in the table as follows:<br></br> <br></br> <table style="width: 300px;border-spacing:1px;" border="1"> <tbody> <tr> <td> </td> <td>A</td> <td>B</td> <td>C</td> <td>T</td> <td>E</td> <td>F</td> </tr> <tr> <td>A</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td>B</td> <td> </td> <td> </td> <td> </td> <td><mdo:image alt="" src="cross.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td> </td> <td> </td> </tr> <tr> <td>C</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td>T</td> <td> </td> <td><mdo:image alt="" src="tick.jpg" style="font-size: 16px; width: 31px; height: 29px;"></mdo:image></td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td>E</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td>F</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> </tbody> </table> <br></br> <ul> <li>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.</li> <li>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...</li> </ul> <span class="editorial">If you keep using the tips in this logical way, the eventual table of answers looks as follows:</span><br></br> <br></br> <p> </p> <table style="width: 300px;border-spacing:1px;" border="1"> <tbody> <tr> <td> </td> <td>A</td> <td>B</td> <td>C</td> <td>T</td> <td>E</td> <td>F</td> </tr> <tr> <td>A</td> <td> </td> <td><mdo:image alt="" src="tick.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td><mdo:image alt="" src="tick.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td><mdo:image alt="" src="tick.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td><mdo:image alt="" src="cross.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td><mdo:image alt="" src="cross.jpg" style="width: 31px; height: 29px;"></mdo:image></td> </tr> <tr> <td>B</td> <td><mdo:image alt="" src="cross.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td> </td> <td><mdo:image alt="" src="tick.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td><mdo:image alt="" src="cross.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td><mdo:image alt="" src="tick.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td><mdo:image alt="" src="cross.jpg" style="width: 31px; height: 29px;"></mdo:image> </td> </tr> <tr> <td>C</td> <td><mdo:image alt="" src="cross.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td><mdo:image alt="" src="cross.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td> </td> <td><mdo:image alt="" src="tick.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td><mdo:image alt="" src="cross.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td><mdo:image alt="" src="cross.jpg" style="width: 31px; height: 29px;"></mdo:image></td> </tr> <tr> <td>T</td> <td><mdo:image alt="" src="cross.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td><font size="3"><mdo:image alt="" src="tick.jpg" style="width: 31px; height: 29px;"></mdo:image></font></td> <td><mdo:image alt="" src="cross.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td> </td> <td><mdo:image alt="" src="cross.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td><mdo:image alt="" src="cross.jpg" style="width: 31px; height: 29px;"></mdo:image></td> </tr> <tr> <td>E</td> <td><mdo:image alt="" src="tick.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td><mdo:image alt="" src="cross.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td><mdo:image alt="" src="tick.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td><mdo:image alt="" src="tick.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td> </td> <td><mdo:image alt="" src="cross.jpg" style="width: 31px; height: 29px;"></mdo:image></td> </tr> <tr> <td>F</td> <td><mdo:image alt="" src="tick.jpg" style="font-size: 16px; width: 31px; height: 29px;"></mdo:image></td> <td><mdo:image alt="" src="tick.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td><mdo:image alt="" src="tick.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td><mdo:image alt="" src="tick.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td><mdo:image alt="" src="tick.jpg" style="width: 31px; height: 29px;"></mdo:image></td> <td> </td> </tr> </tbody> </table> <br></br> <br></br> <span class="editorial">3. Finally, the Qualifying Round problem - this was tough! A bit of careful reading tells you that the results given are only a <strong>partial</strong> list of results - <strong>some games may not have been played yet</strong>. Samantha gives her answer as follows:</span><br></br> <br></br> <br></br> <table border="1"> <tbody> <tr align="center"> <td align="center" valign="left" width="50">Team</td> <td align="center" valign="left" width="50">Played</td> <td align="center" valign="left" width="50">Won</td> <td align="center" valign="left" width="50">Drawn</td> <td align="center" valign="left" width="50">Lost</td> <td align="center" valign="left" width="50">For</td> <td align="center" valign="left" width="50">Against</td> <td align="center" valign="left" width="50">Points</td> </tr> <tr> <td align="center" valign="left" width="50">A</td> <td align="center" valign="left" width="50">3</td> <td align="center" valign="left" width="50">1</td> <td align="center" valign="left" width="50">1</td> <td align="center" valign="left" width="50">1</td> <td align="center" valign="left" width="50">4</td> <td align="center" valign="left" width="50">4</td> <td align="center" valign="left" width="50">3</td> </tr> <tr> <td align="center" valign="left" width="50">B</td> <td align="center" valign="left" width="50">3</td> <td align="center" valign="left" width="50">2</td> <td align="center" valign="left" width="50">1</td> <td align="center" valign="left" width="50">0</td> <td align="center" valign="left" width="50">5</td> <td align="center" valign="left" width="50">2</td> <td align="center" valign="left" width="50">5</td> </tr> <tr> <td align="center" valign="left" width="50">C</td> <td align="center" valign="left" width="50">2</td> <td align="center" valign="left" width="50">0</td> <td align="center" valign="left" width="50">2</td> <td align="center" valign="left" width="50">0</td> <td align="center" valign="left" width="50">4</td> <td align="center" valign="left" width="50">4</td> <td align="center" valign="left" width="50">2</td> </tr> <tr> <td align="center" valign="left" width="50">D</td> <td align="center" valign="left" width="50">2</td> <td align="center" valign="left" width="50">0</td> <td align="center" valign="left" width="50">0</td> <td align="center" valign="left" width="50">2</td> <td align="center" valign="left" width="50">0</td> <td align="center" valign="left" width="50">3</td> <td align="center" valign="left" width="50">0</td> </tr> </tbody> </table> <br></br> <span class="editorial">and the individual results:</span><br></br> <br></br> A v B: B wins 2-0<br></br> A v C: draw 2-2<br></br> A v D: A wins 2-0<br></br> B v C: draw 2-2<br></br> B v D: B wins 1-0<br></br> C v D: not played yet.<br></br> <br></br> <em>These solutions were originally submitted for a version of this activity in the context of the Olympics.</em><br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><h3>Why do this problem?</h3> <p>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.</p> <p> </p> <h3>Possible approach</h3> <p>Before introducing the task:</p> <p>"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."</p> <p>The problems could be used in several ways:</p> <p>Hand out <a href="/content/id/8824/Arsenal_worksheet_Football-Logic.pdf">this worksheet</a> and invite students to work on all three problems in pairs.</p> <p>Cut the worksheet into the three separate problems and invite different groups to work on different problems.</p> <p>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.</p> <p>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.</p> <p> </p> <p>The following <strong>key questions</strong> or prompts could be offered to students who are stuck:</p> <p>What do you know? Write it down.<br></br> What can you deduce from what you know? Write it down.<br></br> Is there a good way of representing what you know, to make it clear?</p> <p> </p> <h3>Possible extension</h3> <p>For other problems that require systematic and logical thinking see <a href="/781">Two and Two</a>, <a href="/4903">Product Sudoku</a> and <a href="/1864">Cinema Problem</a>.</p> <p> </p> <h3>Possible support</h3> <p>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.</p></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><p>Work out what each part of the problem tells you, and write it down.</p> <p>What can you work out from what you know? Write down any new information that you deduce.</p> <p>Can you think of any clear ways of representing the information?</p></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><h3>Fencing Tournament</h3> <p>F wins all 5.</p> <p>A wins against B, C and D</p> <p>E wins against A, C and D</p> <p>B wins against C and E only</p> <p>D wins agains B only</p> <p>C wins against D only</p> <h3> </h3> <h3>Football Champ</h3> <p> </p> <table style="width: 600px;border-spacing:1px;" border="1"> <thead> <tr> <th scope="col">Teams</th> <th scope="col">Games Played</th> <th scope="col">Won</th> <th scope="col">Drawn</th> <th scope="col">Lost</th> <th scope="col">Goals For</th> <th scope="col">Goals Against</th> <th scope="col">Points</th> </tr> </thead> <tbody> <tr> <td>A</td> <td>2</td> <td>1</td> <td>0</td> <td>1</td> <td>5</td> <td>3</td> <td>3</td> </tr> <tr> <td>B</td> <td>2</td> <td>0</td> <td>1</td> <td>1</td> <td>2</td> <td>5</td> <td>1</td> </tr> <tr> <td>C</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>3</td> <td>2</td> <td>4</td> </tr> </tbody> </table> <p>The scores are A versus B = 4-1, B versus C = 1-1 and C versus A= 2-1.</p> <p>Solution to original version by Kenneth Macleod, Forres Academy</p> <p>Team A have obviously won 1 game and lost 1 game because they have three points which means they have won 1 game and because all the teams have played 2 games, they have lost a game too. If team A had drawn a game they would have 4 points. So team A beat team B(4-1) and lost to team C(1-2).</p> <p> </p> <p> </p> <h3>Medals Count</h3> <p> </p> <table style="width: 500px;border-spacing:1px;" border="1"> <thead> <tr> <th scope="col">Country</th> <th scope="col">Gold</th> <th scope="col">Silver</th> <th scope="col">Bronze</th> <th scope="col">Total</th> </tr> </thead> <tbody> <tr> <td>Japan</td> <td>9</td> <td>6</td> <td>10</td> <td>25</td> </tr> <tr> <td>Italy</td> <td>8</td> <td>9</td> <td>10</td> <td>27</td> </tr> <tr> <td>France</td> <td>7</td> <td>16</td> <td>18</td> <td>41</td> </tr> </tbody> </table> <p> </p> <h3>Hockey</h3> <p> </p> <table style="width: 600px;border-spacing:1px;" border="1"> <thead> <tr> <th scope="col">Team</th> <th scope="col">Played</th> <th scope="col">Won</th> <th scope="col">Drawn</th> <th scope="col">Lost</th> <th scope="col">For</th> <th scope="col">Against</th> <th scope="col">Points</th> </tr> </thead> <tbody> <tr> <td>A</td> <td>3</td> <td>1</td> <td>1</td> <td>1</td> <td>4</td> <td>4</td> <td>3</td> </tr> <tr> <td>B</td> <td>3</td> <td>2</td> <td>1</td> <td>0</td> <td>5</td> <td>2</td> <td>5</td> </tr> <tr> <td>C</td> <td>2</td> <td>0</td> <td>2</td> <td>0</td> <td>4</td> <td>4</td> <td>2</td> </tr> <tr> <td>D</td> <td>2</td> <td>0</td> <td>0</td> <td>2</td> <td>0</td> <td>3</td> <td>0</td> </tr> </tbody> </table> <p>Original solution when this first appeared on the site:</p> <p> </p> <p><span class="editorial">This solution was submitted by</span> <strong class="editorial">Wymondham High School</strong> <span class="editorial">who reasoned their solution in the following way:</span></p> <p>To start off we tried to work out the logical answers. For example where it says 'D had no points', it was obvious they had neither a win or a draw.</p> <p>Then we found out that 'For' and 'Against' columns must add up to the same number. In the end we put 4 in the 'For' column and 2 in the 'Against' column.</p> <p>At one point we worked out the results to each match. These were:</p> <table style="" border=""> <tbody> <tr align="center" valign="top"> <td align="center" valign="middle" style=""><strong>A v B</strong></td> <td align="center" valign="middle" style="">0 - 2</td> <td align="center" valign="middle" style=""><strong>B v C</strong></td> <td align="center" valign="middle" style="">2 - 2</td> <td align="center" valign="middle" style=""><strong>C v D</strong></td> <td align="center" valign="middle" style="">not played</td> </tr> <tr align="center" valign="top"> <td align="center" valign="middle" style=""><strong>A v C</strong></td> <td align="center" valign="middle" style="">2 - 2</td> <td align="center" valign="middle" style=""><strong>B v D</strong></td> <td align="center" valign="middle" style="">1 - 0</td> </tr> <tr align="center" valign="top"> <td align="center" valign="middle" style=""><strong>A v D</strong></td> <td align="center" valign="middle" style="">2 - 0</td> </tr> </tbody> </table> <p>Here is the finished table of results:</p> <table style="" border=""> <tbody> <tr align="center"> <td align="center" valign="left" width="50" style="">Team</td> <td align="center" valign="left" width="50" style="">Played</td> <td align="center" valign="left" width="50" style="">Won</td> <td align="center" valign="left" width="50" style="">Drawn</td> <td align="center" valign="left" width="50" style="">Lost</td> <td align="center" valign="left" width="50" style="">For</td> <td align="center" valign="left" width="50" style="">Against</td> <td align="center" valign="left" width="50" style="">Points</td> </tr> <tr> <td align="center" valign="left" width="50" style="">A</td> <td align="center" valign="left" width="50" style="">3</td> <td align="center" valign="left" width="50" style="">1</td> <td align="center" valign="left" width="50" style="">1</td> <td align="center" valign="left" width="50" style="">1</td> <td align="center" valign="left" width="50" style="">4</td> <td align="center" valign="left" width="50" style="">4</td> <td align="center" valign="left" width="50" style="">3</td> </tr> <tr> <td align="center" valign="left" width="50" style="">B</td> <td align="center" valign="left" width="50" style="">3</td> <td align="center" valign="left" width="50" style="">2</td> <td align="center" valign="left" width="50" style="">1</td> <td align="center" valign="left" width="50" style="">0</td> <td align="center" valign="left" width="50" style="">5</td> <td align="center" valign="left" width="50" style="">2</td> <td align="center" valign="left" width="50" style="">5</td> </tr> <tr> <td align="center" valign="left" width="50" style="">C</td> <td align="center" valign="left" width="50" style="">2</td> <td align="center" valign="left" width="50" style="">0</td> <td align="center" valign="left" width="50" style="">2</td> <td align="center" valign="left" width="50" style="">0</td> <td align="center" valign="left" width="50" style="">4</td> <td align="center" valign="left" width="50" style="">4</td> <td align="center" valign="left" width="50" style="">2</td> </tr> <tr> <td align="center" valign="left" width="50" style="">D</td> <td align="center" valign="left" width="50" style="">2</td> <td align="center" valign="left" width="50" style="">0</td> <td align="center" valign="left" width="50" style="">0</td> <td align="center" valign="left" width="50" style="">2</td> <td align="center" valign="left" width="50" style="">0</td> <td align="center" valign="left" width="50" style="">3</td> <td align="center" valign="left" width="50" style="">0</td> </tr> </tbody> </table> <p><span class="editorial">Correct solutions to the problem were also sent in by:</span><br></br> <span class="editorial">Daniel and David from Archbishop Sancroft High School, Mary from West Flegg Middle School, Ruth from Lynn Grove High School, and students from Necton Middle School and Sheringham High School.</span></p> <br></br> <p> </p></mdoxml> 2 4 0 0 1 1 0 Arsenal Collection: Football Logic Can you use your powers of logic and deduction to work out the missing information in these situations?