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

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

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

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

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

What happens when you round these numbers to the nearest whole number?

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

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?

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

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?

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

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?

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 see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

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

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

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 ...?

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

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

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

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?

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

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

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?

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

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

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

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?

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?

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

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?

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

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

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.

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

A game for 2 players with similaritlies 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.

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

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

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

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?

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.

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