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

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?

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?

I added together some of my neighbours house numbers. Can you explain the patterns I noticed?

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

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?

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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

Does this 'trick' for calculating multiples of 11 always work? Why or why not?

Can you explain the strategy for winning this game with any target?

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

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

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

This challenge asks you to imagine a snake coiling on itself.

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?

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?

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?

Take a look at the video of this trick. Can you perform it yourself? Why is this maths and not magic?

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

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

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

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.

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?

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

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

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?

Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

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

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?

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

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.

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

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

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

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

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

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

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

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

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.

Try out this number trick. What happens with different starting numbers? What do you notice?

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

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?

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

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

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

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?