Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

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

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

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

Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own?

If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

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

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

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

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

Strike it Out game for an adult and child. Can you stop your partner from being able to go?

Stop the Clock game for an adult and child. How can you make sure you always win this game?

This task follows on from Build it Up and takes the ideas into three dimensions!

Can you find all the ways to get 15 at the top of this triangle of numbers?

Got It game for an adult and child. How can you play so that you know you will always win?

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

This activity involves rounding four-digit numbers to the nearest thousand.

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

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

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?

Find out what a "fault-free" rectangle is and try to make some of your own.

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

This is a game for two players. Can you find out how to be the first to get to 12 o'clock?

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

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

What happens when you round these three-digit numbers to the nearest 100?

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

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

Are these statements relating to odd and even numbers always true, sometimes true or never true?

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

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.

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.