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?

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

"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 ...?

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

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

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?

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?

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

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

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?

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

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

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

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

Delight your friends with this cunning trick! Can you explain how it works?

How would you count the number of fingers in these pictures?

Find the words hidden inside each of the circles by counting around a certain number of spaces to find each letter in turn.

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

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?

This article for teachers describes a project which explores thepower of storytelling to convey concepts and ideas to children.

Can you deduce the pattern that has been used to lay out these bottle tops?

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

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.

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

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

Sanjay Joshi, age 17, The Perse Boys School, Cambridge followed up the Madrass College class 2YP article with more thoughts on the problem of the number of ways of expressing an integer as the sum. . . .

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?

Draw a pentagon with all the diagonals. This is called a pentagram. How many diagonals are there? How many diagonals are there in a hexagram, heptagram, ... Does any pattern occur when looking at. . . .

A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .

Given a 2 by 2 by 2 skeletal cube with one route `down' the cube. How many routes are there from A to B?

Imagine you have six different colours of paint. You paint a cube using a different colour for each of the six faces. How many different cubes can be painted using the same set of six colours?

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

Class 2YP from Madras College was inspired by the problem in NRICH to work out in how many ways the number 1999 could be expressed as the sum of 3 odd numbers, and this is their solution.

A game for 2 people. Take turns to move the counters 1, 2 or 3 spaces. The player to remove the last counter off the board wins.

How many positive integers less than or equal to 4000 can be written down without using the digits 7, 8 or 9?

How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs?

Ann thought of 5 numbers and told Bob all the sums that could be made by adding the numbers in pairs. The list of sums is 6, 7, 8, 8, 9, 9, 10,10, 11, 12. Help Bob to find out which numbers Ann was. . . .

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?

There are two forms of counting on Vuvv - Zios count in base 3 and Zepts count in base 7. One day four of these creatures, two Zios and two Zepts, sat on the summit of a hill to count the legs of. . . .

Nowadays the calculator is very familiar to many of us. What did people do to save time working out more difficult problems before the calculator existed?