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?

Are these statements always true, sometimes true or never true?

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

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?

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.

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?

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

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

Find the sum of all three-digit numbers each of whose digits is odd.

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

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

These tasks give learners chance to generalise, which involves identifying an underlying structure.

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

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

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?

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

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

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

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

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?

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

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

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

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

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

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

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 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?

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

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

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? Many opportunities to work in different ways.

Investigate the different ways that fifteen schools could have given money in a charity fundraiser.

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

Does this 'trick' for calculating multiples of 11 always work? Why or why not?

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.

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?

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.

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

Ben and his mum are planting garlic. Can you find out how many cloves of garlic they might have had?

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

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.

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?

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

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.