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

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

Delight your friends with this cunning trick! Can you explain how it works?

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?

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.

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

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?

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

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

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.

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?

Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .

How many centimetres of rope will I need to make another mat just like the one I have here?

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

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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?

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?

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.

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?

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.

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

Can you work out how to win this game of Nim? Does it matter if you go first or second?

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

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

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

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

Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

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?

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?

Benâ€™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?

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

Can you find all the ways to get 15 at the top of this triangle of numbers?

This task follows on from Build it Up and takes the ideas into three dimensions!

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

Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle 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?

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?

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.