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

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

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?

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?

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

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?

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?

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?

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

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

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?

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

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?

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?

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?

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

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

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.

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

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

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?

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.

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

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.

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

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?

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?

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

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.

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.

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

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

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

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

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

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 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?

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?