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.

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

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?

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?

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?

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?

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

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

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?

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?

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

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

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

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

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?

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

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

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

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?

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

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.

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?

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.

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

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

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

It would be nice to have a strategy for disentangling any tangled ropes...

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

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.

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

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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

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?

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

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.

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .

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.

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

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

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

Got It game for an adult and child. How can you play so that you know you will always win?