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

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

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

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?

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

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

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

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

Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?

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

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?

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

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?

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?

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

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

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?

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

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

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

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

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

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?

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

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

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.

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

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

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 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?

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?

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

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

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

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

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?

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

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

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

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?

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.

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.

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?

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?

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