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

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. . . .

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 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.

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?

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

Looking at the 2012 Olympic Medal table, can you see how the data is organised? Could the results be presented differently to give another nation the top place?

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.

When the number x 1 x x x is multiplied by 417 this gives the answer 9 x x x 0 5 7. Find the missing digits, each of which is represented by an "x" .

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?

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. . . .

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.

I am thinking of three sets of numbers less than 101. They are the red set, the green set and the blue set. Can you find all the numbers in the sets from these clues?

Nearly all of us have made table patterns on hundred squares, that is 10 by 10 grids. This problem looks at the patterns on differently sized square grids.

I am thinking of three sets of numbers less than 101. Can you find all the numbers in each set from these clues?

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?

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

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?

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

How many numbers less than 1000 are NOT divisible by either: a) 2 or 5; or b) 2, 5 or 7?

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?

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. . . .

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

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?

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

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

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

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

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

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

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?

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?

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

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. . . .

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.

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?

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

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

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

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?

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.

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?

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?

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?

Use the differences to find the solution to this Sudoku.

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.

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?