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.

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

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

How many trains can you make which are the same length as Matt's and Katie's, using rods that are identical?

How will you go about finding all the jigsaw pieces that have one peg and one hole?

Here are some rods that are different colours. How could I make a yellow rod using white and red rods?

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

What happens when you try and fit the triomino pieces into these two grids?

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

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

Design an arrangement of display boards in the school hall which fits the requirements of different people.

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

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

Can you work out how to balance this equaliser? You can put more than one weight on a hook.

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

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

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?

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

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

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?

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.

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?

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

This challenge is about finding the difference between numbers which have the same tens digit.

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

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?

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

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.

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

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

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

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

In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

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?

There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.

El Crico the cricket has to cross a square patio to get home. He can jump the length of one tile, two tiles and three tiles. Can you find a path that would get El Crico home in three jumps?

This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.

What two-digit numbers can you make with these two dice? What can't you make?

George and Jim want to buy a chocolate bar. George needs 2p more and Jim need 50p more to buy it. How much is the chocolate bar?

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.

Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?

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

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