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

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.

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

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

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?

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

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

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’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?

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

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

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

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

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?

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

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

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

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

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

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

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

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

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

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?

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

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

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?

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

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

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

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

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.

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?

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

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

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.

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

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

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.

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?

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

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

Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?

Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

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

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

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