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

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

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

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

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

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

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?

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

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?

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.

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?

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

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

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

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.

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?

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

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

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

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

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

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?

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

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?

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

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

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

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.

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

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 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?

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?

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

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?

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.

This article for primary teachers discusses how we can help learners generalise and prove, using NRICH tasks as examples.

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.

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

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

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

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

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

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

It starts quite simple but great opportunities for number discoveries and patterns!

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

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.