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?

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

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

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

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

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"?

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

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

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?

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.

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?

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

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?

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

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

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

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?

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

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 three-digit numbers to the nearest 100?

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

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 challenge focuses on finding the sum and difference of pairs of two-digit numbers.

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

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

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

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?

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?

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?

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

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

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

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

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

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

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

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

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

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.

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

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

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

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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

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

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?