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?

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

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?

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.

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

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

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?

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?

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

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

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.

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?

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

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

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

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

Are these statements relating to odd and even numbers always true, sometimes true or never true?

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.

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.

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

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

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?

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

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

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

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

Can you work out how to win this game of Nim? Does it matter if you go first or second?

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 find all the ways to get 15 at the top of this triangle of numbers?

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

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?

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?

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

Can you describe this route to infinity? Where will the arrows take you next?

It starts quite simple but great opportunities for number discoveries and patterns!

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?

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

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?

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

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?

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?