# Medal Muddle

## Problem

**If you are a teacher, click here 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 these clues?

**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?

## Getting Started

You may find it useful to print off and cut out these cards. You could arrange the countries randomly and then read through the clues adjusting the order as you go. Here is a set of cards for the extension.

Alternatively, you could begin by figuring out which teams **couldn't** have come first.

If the GeoGebra applet does not load correctly you can save the GeoGebra file and open it using the free to download GeoGebra software.

## Student Solutions

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:

- Turkey / Mexico
- Italy / New Zealand

From hint 2, you can get:

- Portugal
- Venezuela / Mexico / Spain / Romania

From hint 3, you can get:

- Greece / Spain / Serbia / Algeria
- Romania.

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:

- Easy to read and clear
- Enjoyable when arranging the cards
- Pretty quick if you have an idea in your head
- Makes you happy when you complete it!

Cons:

- The writing process is slightly tedious

Tips:

- When writing the notes on the cards write short phrases and clearly so easily read.
- If you write the wrong infomation on the cards then it's not going to be pretty...

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!)

## Teachers' Resources

### Why do this problem?

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.

### Possible approach

"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 this worksheet for students to work on in pairs (or individually at first if they wish).

These cards could be printed and handed out to students so they can manipulate the order as they work their way through the different clues.

Once students have had a chance to discuss the merits of different approaches, hand out this worksheet with the extension challenge, so that they can test how their chosen strategy works on a longer problem with more information to consider. Here is a set of cards for the extension activity.

### Key questions

Which representations or ways of organising your thinking help you to use the information given to solve the problem efficiently?

### Possible extension

Challenge students to create their own versions of the problem, which could be shared on the blog.

### Possible support

The visual representation shown in the hint is a very clear way of seeing the relationship between the different countries.