Helen Joyce interviews the neuropsychologist Brian Butterworth whose research has shown that we are all born with a "built-in" sense of cardinal number.

While musing about the difficulties children face in comprehending number structure, notation, etc., it occured to the author that there is a vast array of occasions when numbers and signs are used. . . .

In this problem, we're investigating the number of steps we would climb up or down to get out of or into the swimming pool. How could you number the steps below the water?

Lee was writing all the counting numbers from 1 to 20. She stopped for a rest after writing seventeen digits. What was the last number she wrote?

Buzzy Bee was building a honeycomb. She decided to decorate the honeycomb with a pattern using numbers. Can you discover Buzzy's pattern and fill in the empty cells for her?

Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?

What could these drawings, found in a cave in Spain, represent?

Daisy and Akram were making number patterns. Daisy was using beads that looked like flowers and Akram was using cube bricks. First they were counting in twos.

Can you find a path from a number at the top of this network to the bottom which goes through each number from 1 to 9 once and once only?

Can you use the numbers on the dice to reach your end of the number line before your partner beats you?

How many legs do each of these creatures have? How many pairs is that?

Nim-7 game for an adult and child. Who will be the one to take the last counter?

Incey Wincey Spider game for an adult and child. Will Incey get to the top of the drainpipe?

How could you estimate the number of pencils/pens in these pictures?

You'll need two dice to play this game against a partner. Will Incey Wincey make it to the top of the drain pipe or the bottom of the drain pipe first?

There are three versions of this challenge. The idea is to change the colour of all the spots on the grid. Can you do it in fewer throws of the dice?

This is a game in which your counters move in a spiral round the snail's shell. It is about understanding tens and units.

This problem challenges you to find out how many odd numbers there are between pairs of numbers. Can you find a pair of numbers that has four odds between them?

Here is a version of the game 'Happy Families' for you to make and play.

Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.

In this game, you can add, subtract, multiply or divide the numbers on the dice. Which will you do so that you get to the end of the number line first?

25 students are queuing in a straight line. How many are there between Julia and Jenny?

Investigate the different ways these aliens count in this challenge. You could start by thinking about how each of them would write our number 7.

Some children were playing a game. Make a graph or picture to show how many ladybirds each child had.

Dotty Six game for an adult and child. Will you be the first to have three sixes in a straight line?

In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?

Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?

Dotty Six is a simple dice game that you can adapt in many ways.

In this problem it is not the squares that jump, you do the jumping! The idea is to go round the track in as few jumps as possible.

Can you design a new shape for the twenty-eight squares and arrange the numbers in a logical way? What patterns do you notice?

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

Solve this Sudoku puzzle whose clues are in the form of sums of the numbers which should appear in diagonal opposite cells.

An activity centred around observations of dots and how we visualise number arrangement patterns.

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

Use five steps to count forwards or backwards in 1s or 10s to get to 50. What strategies did you use?

In how many ways could Mrs Beeswax put ten coins into her three puddings so that each pudding ended up with at least two coins?

"Ip dip sky blue! Who's 'it'? It's you!" Where would you position yourself so that you are 'it' if there are two players? Three players ...?

A game for two people, who take turns to move the counters. The player to remove the last counter from the board wins.

Can you work out how to win this game of Nim? Does it matter if you go first or second?

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

Exactly 195 digits have been used to number the pages in a book. How many pages does the book have?

If you have ten counters numbered 1 to 10, how many can you put into pairs that add to 10? Which ones do you have to leave out? Why?