Medal Muddle

661 /www/nrich/html/content/99/09/six2/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">If you are a teacher, click <a href="http://nrich.maths.org/661/note/">here</a> for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on... Thirteen nations competed in a sports tournament. Unfortunately, we do not have the final medal table, but we have the following pieces of information: 1. Turkey and Mexico both finished above Italy and New Zealand. 2. Portugal finished above Venezuela, Mexico, Spain and Romania. 3. Romania finished below Algeria, Greece, Spain and Serbia. 4. Serbia finished above Turkey and Portugal, both of whom finished below Algeria and Russia. 5. Russia finished above France and Algeria. 6. Algeria finished below France but above Serbia and Spain. 7. Italy finished below Greece and Venezuela, but above New Zealand. 8. Venezuela finished above New Zealand but below Greece. 9. Greece finished below Turkey, who came below France. 10. Portugal finished below Greece and France. 11. France finished above Serbia, who came above Mexico. 12. Venezuela finished below Mexico, and New Zealand came above Spain. Can you recreate the medal table from this information? Can you describe an efficient strategy for solving problems like this? Extension The following year, twice as many teams entered the tournament. Can you use your strategy to sort out the medal table from <a href="/content/99/09/six2/Medal%20Muddle%20long.pdf">these clues</a>? Perhaps you might like to try creating a similar problem of your own. You will need to consider the following: Although there are twelve statements above, there are more than twelve pieces of information, because some sentences compare more than one pair of teams. What is the minimum number of pieces of information needed to order the teams? Which information, if any, is redundant? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Thank you to everyone who participated! The correct answers, which almost everyone got, were: 1. Russia 2. France 3. Algeria 4. Serbia 5. Turkey 6. Greece 7. Portugal 8. Mexico 9. Venezuela 10. Italy 11. New Zealand 12. Spain 13. Romania Congratulations if you got the order right! Let's have a look at some of the ways of doing it. Alex, from Winnersh Primary School, had the following interesting idea to find the countries in order, one at a time: My method was to choose a random country and then go through the clues until I found a country that was higher up. I carried on until I found a country where I could go all the way through the clues without finding another one that was higher up. I then put that country (Russia) in 1st place. I would then do the same but ignoring Russia, and found the 2nd, then 3rd, etc. Rebecca, from Woodchurch, had a similar idea: Try to count how many times one country came above each other country. Then repeat this thirteen times. Then put your answers in order. Ta-da! Daniel, from Wilson's School, wrote down at each step what he got from the hints: From hint 1, you can get: <ul> <li>Turkey / Mexico</li> <li>Italy / New Zealand</li> </ul> From hint 2, you can get: <ul> <li>Portugal</li> <li>Venezuela / Mexico / Spain / Romania</li> </ul> From hint 3, you can get: <ul> <li>Greece / Spain / Serbia / Algeria</li> <li>Romania.</li> </ul> etc. Many people thought it was a good idea to write the names of countries on bits of paper or card and swap them round - this saves a lot of writing! For example, Michelle, from Globe Academy, wrote: I started by putting the country names in a random order. Then I read through the clues and started swapping around the countries. When I got to the end of the clues I went back through the clues and checked again. Mrs. McGuire's class at Lakewood Catholic Academy were another one of many who followed this approach - they say it took them about 45 minutes and lots of trial and error. Could it have been speeded up, do you think? Jade, at Oakmeeds, sent us the following comments on the card idea: I wrote some of the infomation to do with the country on the country's card, e.g. "above Spain and Algeria and below New Zealand". Pros: <ul> <li>Easy to read and clear</li> <li>Enjoyable when arranging the cards</li> <li>Pretty quick if you have an idea in your head</li> <li>Makes you happy when you complete it!</li> </ul> Cons: <ul> <li>The writing process is slightly tedious</li> </ul> Tips: <ul> <li>When writing the notes on the cards write short phrases and clearly so easily read.</li> <li>If you write the wrong infomation on the cards then it's not going to be pretty...</li> </ul> Charlie, from Wentworth Primary, had this interesting idea: I used a mathematical method allocating points for each one above and subtracting a point for below, to eventually work out where each country should go by adding up the points I had allocated. Mrs. Gale's class, from Churchill Academy, had a trick to speed things up slightly: Colour coding the countries to make them stand out more easily. This made it clear there was most information about France. On a similar note, Alastair from Richmond CoE sent us lots of flags that he printed out and cut up while constructing his solution. Nice! A few people moved onto the extension problem using the same sorts of techniques as above. The correct answer was: Sri Lanka, Great Britain, Brazil, Spain, Turkey, Austria, Romania, Finland, Mexico, Germany, Serbia, Italy, Canada, Algeria, New Zealand, Australia, Norway, France, Portugal, Greece, Japan, Sweden, Venezuela, USA, Russia, Denmark. Thanks to Brain Academy at St. Peters CEVC Primary, Mrs. C's class at Court Moor School, and Ms. Troup's class at Prior's Field School for sending in their answers to the extension problem - this one was tough! (Finally, Stefan from Afghanistan said: "this is so cool"! Thanks, Stefan!)</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><h3>Why do this problem?</h3> This problem is an exercise in strategic thinking, accessible to lower Stage 3 students but hinting at work on sorting algorithms that they might meet at Stage 5 in Decision Maths. <h3>Possible approach</h3> "I'm going to give you a problem to solve, and while you work on it, I'd like you to think about the strategies you are using. Imagine you had to solve lots of problems like this one. How would you ensure that you found the correct answer accurately and efficiently?" Hand out <a href="/content/99/09/six2/Medal%20Muddle.pdf">this worksheet</a> for students to work on in pairs (or individually at first if they wish). Once they have had time to make progress (but not necessarily solve the problem completely), bring the class together to discuss the strategies they have tried. We have included a few ideas in the <a href="/661/clue">hint</a> that you could share with your class if they don't come up with suggestions of their own. Once students have had a chance to discuss the merits of different approaches, hand out <a href="/content/99/09/six2/Medal%20Muddle%20long.pdf">this worksheet</a> with the extension challenge, so that they can test how their chosen strategy works on a longer problem with more information to consider. <h3>Key questions</h3> Which representations or ways of organising your thinking help you to use the information given to solve the problem efficiently? <h3>Possible extension</h3> Challenge students to create their own versions of the problem, which could be shared on the blog. <h3>Possible support</h3> <a href="/content/99/09/six2/Medal%20Muddle%20Cards.pdf">These cards</a> could be printed and handed out to students so they can manipulate the order as they work their way through the different clues. The visual representation shown in the hint is a very clear way of seeing the relationship between the different countries.</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">You may find it useful to print off and cut out these <a href="/content/99/09/six2/Medal%20Muddle%20Cards.pdf">cards</a>. You could arrange the countries randomly and then read through the clues adjusting the order as you go. <a href="/content/99/09/six2/Medal%20Muddle%20Cards%20long.pdf">Here</a> is a set of cards for the extension. Alternatively, you could begin by figuring out which teams couldn't have come first. <div>One way of working is to take each clue and turn it into a set of relations between pairs of countries:</div> <div>"Russia finished above France and Algeria" could be turned into "Russia above France" and "Russia above Algeria".</div> <div> </div> <div>The diagram below can help you draw vectors between pairs of countries, with arrows used to indicate the order in which they finished. (Be patient; the applet may take some time to load.)</div> <div> </div> <mdo:applet height="500" width="550" code="geogebra.GeoGebraApplet" archive="http://jars.geogebra.org/webstart/4.0/geogebra.jar" datafile=""><param name="filename" value="http://nrich.maths.org/content/99/09/six2/representation.ggb" ></param><param name="framePossible" value="true" ></param><param name="showResetIcon" value="true" ></param><param name="enableRightClick" value="true" ></param><param name="showMenuBar" value="false" ></param><param name="showToolBar" value="true" ></param><param name="showToolBarHelp" value="true" ></param><param name="showAlgebraInput" value="false" ></param></mdo:applet> If the GeoGebra applet does not load correctly you can save the <a href="http://nrich.maths.org/content/99/09/six2/representation.ggb">GeoGebra file</a> and open it using the free to download <a href="http://www.geogebra.org">GeoGebra</a> software. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">New solution <ol> <li>Russia</li> <li>France</li> <li>Algeria</li> <li>Serbia</li> <li>Turkey</li> <li>Greece</li> <li>Portugal</li> <li>Mexico</li> <li>Venezuela</li> <li>Italy</li> <li>New Zealand</li> <li>Spain</li> <li>Romania</li> </ol> Original version: In the Olympics, countries from across the world competed against each other. In what order did they finish? <ol> <li>The Australian team came before the team from Norway and after the team from Turkey and Finland, both of which came before the teams from Algeria, Germany and Yugoslavia.</li> <li>Denmark's team came after the teams from Algeria, Japan, Portugal, Russia and Venezuela.</li> <li>The team from Venezuela came before the team from Russia and after the team from Portugal and Japan came before teams from USA, Sweden and Russia.</li> <li>The team from Turkey came before Portugal.</li> <li>Greece came before Japan, but Algeria and Canada came before both Portugal and Japan.</li> <li>Japan came after France, Sri Lanka, Portugal and Finland.</li> <li>The team from Romania came before the teams from Mexico, Greece, Finland, Italy and Canada.</li> <li>Canada came before the teams from Algeria, France and Greece, but after the teams from Brazil, Austria, Italy and Spain.</li> <li>Spain came before Turkey and Romania, both of which came after the teams from Brazil and Sri Lanka.</li> <li>The team from Sri Lanka came before Great Britain and Brazil, the latter of which came after Great Britain but before teams from Spain, Italy and France.</li> <li>France came before Portugal, who came before Greece.</li> <li>Germany came after Austria and Mexico, but before Yugoslavia.</li> <li>The Mexican team came before Yugoslavia.</li> <li>Algeria came before teams from New Zealand and Norway, as did Italy. Denmark and New Zealand, and Austria came after Turkey.</li> <li>France came after New Zealand and Norway and the team from New Zealand came before Australia.</li> <li>Romania came after Austria and Great Britain, the latter of which came before Spain.</li> <li>Sweden came before Venezuela, who came before the team from Russia.</li> <li>The team from Mexico came after Finland and the team from Yugoslavia came before the team from Italy.</li> </ol> Original solution: Congratualtions to Rachel Holland, age 14, John Henry Newman School, Stevenage; to Josh Tattersall, age 10 from Ampthill, Bedfordshire; and to Kerry Skilbeck and Gary Gourlay for solving this one. One method used was to write each team on a separate bit of paper and to go through the clues putting the bits of paper in order. There is a some ambiguity here as the team from the USA could come in 22nd, 23rd or 24th place but all the other teams must be in the given order. This is Rachel's solution. It looks much harder than it actually is. The method I used to solve this logic puzzle is: <ul> <li>Work out what the clues tell you and what they actually mean. (I got confused a few times but only because I hadn't read and or understood the question fully.)</li> <li>Choose one of the countries and see which ones come before it and which ones come after it.</li> <li>Piece together the clues or use the trial and improvement method to put the countries into a logical order.</li> <li>Check through thoroughly to see if your order matches all of the clues, and even if there is only a small error, start from the beginning checking again. <table border="1"> <tbody> <tr> <td>1. Sri Lanka</td> <td>14. Algeria</td> </tr> <tr> <td>2. Great Britain</td> <td>15. New Zealand</td> </tr> <tr> <td>3. Brazil</td> <td>16. Australia</td> </tr> <tr> <td>4. Spain</td> <td>17. Norway</td> </tr> <tr> <td>5. Turkey</td> <td>18. France</td> </tr> <tr> <td>6. Austria</td> <td>19. Portugal</td> </tr> <tr> <td>7. Romania</td> <td>20. Greece</td> </tr> <tr> <td>8. Finland</td> <td>21. Japan</td> </tr> <tr> <td>9. Mexico</td> <td>22. Sweden</td> </tr> <tr> <td>10. Germany</td> <td>23. Venezuela</td> </tr> <tr> <td>11. Serbia</td> <td>24. USA</td> </tr> <tr> <td>12. Italy</td> <td>25. Russia</td> </tr> <tr> <td>13. Canada</td> <td>26. Denmark</td> </tr> </tbody> </table> </li> </ul> </mdoxml> 2 3 0 0 1 0 0 Medal Muddle Countries from across the world competed in a sports tournament. Can you devise an efficient strategy to work out the order in which they finished? Using, Applying and Reasoning about Mathematics Working systematically Decision Mathematics and Combinatorics Algorithms Using, Applying and Reasoning about Mathematics Selecting and using information Applications sport