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

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?

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?

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

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?

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?

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

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?

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

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

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

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

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?

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?

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

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?

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

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

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

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

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

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

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?

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

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

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

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

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

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?

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

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

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

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

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?

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 ...?

Nim-7 game for an adult and child. Who will be the one to take the last counter?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

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?

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?

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?

With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?