A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

Can you find all the different ways of lining up these Cuisenaire rods?

Can you rearrange the biscuits on the plates so that the three biscuits on each plate are all different and there is no plate with two biscuits the same as two biscuits on another plate?

How many rectangles can you find in this shape? Which ones are differently sized and which are 'similar'?

What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

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

Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.

Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

An investigation that gives you the opportunity to make and justify predictions.

Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

There are lots of different methods to find out what the shapes are worth - how many can you find?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.

Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?

In how many ways can you stack these rods, following the rules?

Rather than using the numbers 1-9, this sudoku uses the nine different letters used to make the words "Advent Calendar".

I was in my car when I noticed a line of four cars on the lane next to me with number plates starting and ending with J, K, L and M. What order were they in?

Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?

Systematically explore the range of symmetric designs that can be created by shading parts of the motif below. Use normal square lattice paper to record your results.

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

You need to find the values of the stars before you can apply normal Sudoku rules.

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

Alice and Brian are snails who live on a wall and can only travel along the cracks. Alice wants to go to see Brian. How far is the shortest route along the cracks? Is there more than one way to go?

On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.?

These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.

Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?

This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?

Find all the different shapes that can be made by joining five equilateral triangles edge to edge.

In this matching game, you have to decide how long different events take.

What is the smallest number of coins needed to make up 12 dollars and 83 cents?

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

A few extra challenges set by some young NRICH members.

You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

Can you draw a square in which the perimeter is numerically equal to the area?

How could you put these three beads into bags? How many different ways can you do it? How could you record what you've done?

When you throw two regular, six-faced dice you have more chance of getting one particular result than any other. What result would that be? Why is this?

These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.