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

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?

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?

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

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.

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?

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

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

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

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

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

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?

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.

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?

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

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?

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.

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

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.

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

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

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?

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?

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?

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?

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

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?

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.

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

Can you find the values at the vertices when you know the values on the edges?

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?

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

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

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

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

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.

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

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

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

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

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?

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

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

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.

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