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?

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

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

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 always true, sometimes true or never true?

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

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

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?

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?

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?

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?

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

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

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?

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

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?

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

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?

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

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

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?

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

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

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.

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

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

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

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

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

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

Can you find a way of counting the spheres in these arrangements?

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

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

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.

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

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?

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

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

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?

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?

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