Florence, Ethan and Alma have each added together two 'next-door' numbers. What is the same about their answers?

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?

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?

Are these statements 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.

This problem challenges you to find out how many odd numbers there are between pairs of numbers. Can you find a pair of numbers that has four odds between them?

Here are two kinds of spirals for you to explore. What do you notice?

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?

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

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

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

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?

Are these statements relating to calculation and properties of shapes always true, sometimes true or never true?

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.

Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?

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

Watch the video of Fran re-ordering these number cards. What do you notice? Try it for yourself. What happens?

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

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

Take a look at the video of this trick. Can you perform it yourself? Why is this maths and not magic?

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

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

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

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

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

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

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

Watch this animation. What do you see? Can you explain why this happens?

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

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

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

Try out this number trick. What happens with different starting numbers? What do you notice?

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

This challenge asks you to imagine a snake coiling on itself.

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.

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

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?

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

A game for 2 players with similarities to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.

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

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

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

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

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

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?