Euro 2016 maths puzzle

With the twenty four best football teams in Europe battling it out to be number one - put your maths skills to the test to see if you really understand how the competition works.

This is a maths quiz for kids and big kids aged 5 to 11, but whatever your age simply start at the beginning and keep going!

• The first round of the European Championships is called the group stage. The teams are drawn in six groups, each of which has four teams. Here are all the teams.

• In every group, each of the teams play each other team once.

• Group-stage matches last 90 minutes.

• If a team wins a game, they get three points and their opponents get none. If there's a draw, both sides get one point.

• If two teams have the same number of points, goal difference is used as a tie-breaker. You work this out as (total goals for) - (total goals against).

• If two teams have the same number of points and the same goal difference, the team that scored more goals finishes higher in the table.

• The top two teams in each group qualify for the second round, as well as four of the teams who finish third in their groups.

• The second round, and all the rounds after it, are called the 'knock-out stages': the winner of each match qualifies for the next round, and the loser is out of the tournament. Here's a planner showing all the competition stages.

• Knock-out matches also last 90 minutes, unless there's a draw! If the scores are level after 90 minutes, the teams play another 30 minutes of extra time to try to find a winner.  If the scores are still level after extra time, there's a penalty shoot-out, but we won't talk about that here!

The Puzzle

First Round (ages 5-6)

1. How many teams are there altogether?
2. How many games does each team play in the group stage?
3. What's the most points one team can get in a group?
4. How many teams qualify for the second round?
5. How many teams qualify for the third round?
6. How many games must the winning team play altogether?

The Knock out round (ages 7-8)

Here are the scores from all the games in Wales and England's group.

• Wales 2 - 1 Slovakia
• England 1 - 1 Russia
• England 2 - 1 Wales
• Slovakia 2 - 1 Russia
• England 0 - 0 Slovakia
• Wales 3 - 0 Russia

• How many points did each team get?
• Fill out the group table below
• How many goals did each team score?
• How many goals did each team let in?
• What is each team's goal difference?
• What order did the teams finish in?

The Finals (ages 9-11)

1. How many games are played in each group? (Be careful!)
2. How many games are played altogether in the group stages?
3. How many games are played altogether in the knock-out stages?
4. How many games are played in the whole tournament?
5. What's the smallest total number of minutes the winning team can play?
6. What's the greatest total number of minutes the winning team can play?

The challenge question - a hard one!

1. How could a team qualify for the second round with only one point?

I hope you enjoyed this puzzle!

Colin.

I'm Colin Beveridge, friend of Komodo and author of "Basic Maths for Dummies" and "Cracking Mathematics". I worked as a researcher on NASA’s Living With A Star program, using data from satellites and telescopes to study the structure of the Sun’s magnetic field, but left the USA in 2008 to focus on writing and tutoring. I now live in Dorset where I run long distance and look after my young family.

About Komodo - Komodo is a fun and effective way to boost primary maths skills. Designed for 5 to 11 year olds to use in the home, Komodo uses a little and often approach to learning maths (15 minutes, three to five times per week) that fits into the busy routine. Komodo users develop fluency and confidence in maths - without keeping them at the screen for long.

1. How many teams are there altogether? 6 × 4 = 24
2. How many games does each team play in the group stage? Three
3. What's the most points one team can get in a group? If a team wins all three games, it gets nine points.
4. How many teams qualify for the second round? 16 -- the top two in each group make 6 × 2 = 12 teams, and when you add on the four third-place teams, you have 16 altogether.
5. How many teams qualify for the third round? 8
6. How many games must the winning team play altogether? 7 (Three group-stage games, the second round, quarter-final, semi-final and final.)

Here's the completed table:

If you used the real scores you'll find the table here.

1. How many games are played in each group? Six
2. How many games are played altogether in the group stages? 6×6=36
3. How many games are played altogether in the knock-out stages? There are 8 + 4 + 2 + 1 = 15 games in the knock-out stages. (You can also reason that each knock-out game gets rid one team, and you start with 16 -- you need 15 games to find a winner!)
4. How many games are played in the whole tournament? 36 + 15 = 51.
5. What's the smallest total number of minutes the winning team can play? If they win all of their games without extra-time, they would play 7×90= 630 minutes.
6. What's the greatest total number of minutes the winning team can play? If all of their knock-out games go to extra time, they would play an extra 4×30=120 minutes, making 750 minutes altogether.

Challenge question

How could a team qualify for the second round with only one point?

Let's call our team C. To get one point, C must have lost two games (against teams A and B) and drawn one (against D). A and B have at least three points, so must finish higher than C (with one point) -- so it's not possible for C to be in the top two. Could they qualify in third? Team D have at least one point (from drawing with C), but they could finish below C if they also lost to A and B, and ended up with a worse goal difference than C.

In addition, at least two teams in other groups would have to finish third with one point and a worse goal difference. It's not likely, but it's possible!

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