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....?

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

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?

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?

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

Add or subtract the two numbers on the spinners and try to complete a row of three. Are there some numbers that are good to aim for?

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?

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

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

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

Different combinations of the weights available allow you to make different totals. Which totals can you make?

A few extra challenges set by some young NRICH members.

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

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

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.

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?

Tim's class collected data about all their pets. Can you put the animal names under each column in the block graph using the information?

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

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

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

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

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?

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?

Try out the lottery that is played in a far-away land. What is the chance of winning?

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.

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

What could the half time scores have been in these Olympic hockey matches?

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.

Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

What is the best way to shunt these carriages so that each train can continue its journey?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

The Zargoes use almost the same alphabet as English. What does this birthday message say?

What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?

The clues for this Sudoku are the product of the numbers in adjacent squares.

Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.

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.

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.