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.

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

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?

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.

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

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?

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

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

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?

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

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.

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.

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

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?

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?

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?

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

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?

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?

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

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

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?

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

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

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

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?

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

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

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

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

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

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.

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

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.

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

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

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.

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

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

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