In a league of 5 football teams which play in a round robin tournament show that it is possible for all five teams to be league leaders.

Written for teachers, this article discusses mathematical representations and takes, in the second part of the article, examples of reception children's own representations.

Looking at the 2008 Olympic Medal table, can you see how the data is organised? Could the results be presented differently to give another nation the top place?

A manager of a forestry company has to decide which trees to plant. What strategy for planting and felling would you recommend to the manager in order to maximise the profit?

Skippy and Anna are locked in a room in a large castle. The key to that room, and all the other rooms, is a number. The numbers are locked away in a problem. Can you help them to get out?

Can you find which shapes you need to put into the grid to make the totals at the end of each row and the bottom of each column?

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

Can you use the information to find out which cards I have used?

This problem is designed to help children to learn, and to use, the two and three times tables.

My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?

In how many ways can a pound (value 100 pence) be changed into some combination of 1, 2, 5, 10, 20 and 50 pence coins?

You may like to read the article on Morse code before attempting this question. Morse's letter analysis was done over 150 years ago, so might there be a better allocation of symbols today?

There are exactly 3 ways to add 4 odd numbers to get 10. Find all the ways of adding 8 odd numbers to get 20. To be sure of getting all the solutions you will need to be systematic. What about. . . .

This Sudoku, based on differences. Using the one clue number can you find the solution?

A group of children are discussing the height of a tall tree. How would you go about finding out its height?

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

A random ramble for teachers through some resources that might add a little life to a statistics class.

Invent a scoring system for a 'guess the weight' competition.

You are organising a school trip and you need to write a letter to parents to let them know about the day. Use the cards to gather all the information you need.

This activity challenges you to decide on the 'best' number to use in each statement. You may need to do some estimating, some calculating and some research.

The machine I use to produce Braille messages is faulty and one of the pins that makes a raised dot is not working. I typed a short message in Braille. Can you work out what it really says?

In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...

Three teams have each played two matches. The table gives the total number points and goals scored for and against each team. Fill in the table and find the scores in the three matches.

Baker, Cooper, Jones and Smith are four people whose occupations are teacher, welder, mechanic and programmer, but not necessarily in that order. What is each person’s occupation?

Did you know that ancient traditional mazes often tell a story? Remembering the story helps you to draw the maze.

After some matches were played, most of the information in the table containing the results of the games was accidentally deleted. What was the score in each match played?

Investigate polygons with all the vertices on the lattice points of a grid. For each polygon, work out the area A, the number B of points on the boundary and the number of points (I) inside. . . .

Blue Flibbins are so jealous of their red partners that they will not leave them on their own with any other bue Flibbin. What is the quickest way of getting the five pairs of Flibbins safely to. . . .

How many ways can you write the word EUROMATHS by starting at the top left hand corner and taking the next letter by stepping one step down or one step to the right in a 5x5 array?

Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

Imagine a machine with four coloured lights which respond to different rules. Can you find the smallest possible number which will make all four colours light up?

This investigation is about happy numbers in the World of the Octopus where all numbers are written in base 8 ... Find all the fixed points and cycles for the happy number sequences in base 8.

A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.

Using the 8 dominoes make a square where each of the columns and rows adds up to 8

A train leaves on time. After it has gone 8 miles (at 33mph) the driver looks at his watch and sees that the hour hand is exactly over the minute hand. When did the train leave the station?

Can you use your powers of logic and deduction to work out the missing information in these sporty situations?

Design your own scoring system and play Trumps with these Olympic Sport cards.

A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article.

Suppose you are a bellringer. Can you find the changes so that, starting and ending with a round, all the 24 possible permutations are rung once each and only once?

In this article for teachers, Bernard uses some problems to suggest that once a numerical pattern has been spotted from a practical starting point, going back to the practical can help explain. . . .

Use the differences to find the solution to this Sudoku.

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?

The graph represents a salesman’s area of activity with the shops that the salesman must visit each day. What route around the shops has the minimum total distance?

Five children went into the sweet shop after school. There were choco bars, chews, mini eggs and lollypops, all costing under 50p. Suggest a way in which Nathan could spend all his money.

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

Four small numbers give the clue to the contents of the four surrounding cells.

If you'd like to know more about Primary Maths Masterclasses, this is the package to read! Find out about current groups in your region or how to set up your own.

Can you each work out the number on your card? What do you notice? How could you sort the cards?