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?

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

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

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

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

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?

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.

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?

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

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?

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

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

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

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

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?

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

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.

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

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

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

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?

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

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

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

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?

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

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?

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

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

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.

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

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

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

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

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

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

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?

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

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.

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?

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

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?

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

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

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?

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