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

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. . . .

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?

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

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

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

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?

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

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.

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?

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.

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

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.

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?

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

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

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

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

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?

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

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?

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

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?

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

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

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.

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?

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

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

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

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

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

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

Can you find the values at the vertices when you know the values on the edges?

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

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?

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

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

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

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

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?

Find out what a "fault-free" rectangle is and try to make some of your own.

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

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

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.