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"?

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

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

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?

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

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?

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.

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

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?

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

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

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?

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.

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?

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

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

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 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?

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.

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

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?

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?

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?

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?

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

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

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

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

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.

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

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

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

How many moves does it take to swap over some red and blue frogs? Do you have a method?

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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

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?

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

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.

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4