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.

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?

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?

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

Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?

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?

Can you explain why it is impossible to construct this triangle?

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

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?

In a three-dimensional version of noughts and crosses, how many winning lines can you make?

Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?

Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

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.

A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

A game for 2 players. Given a board of dots in a grid pattern, players take turns drawing a line by connecting 2 adjacent dots. Your goal is to complete more squares than your opponent.

ABC is an equilateral triangle and P is a point in the interior of the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP must be less than 10 cm.

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?

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

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

This article for teachers describes how modelling number properties involving multiplication using an array of objects not only allows children to represent their thinking with concrete materials,. . . .

Here is a solitaire type environment for you to experiment with. Which targets can you reach?

We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

Exchange the positions of the two sets of counters in the least possible number of moves

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

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?

A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .

Which of these dice are right-handed and which are left-handed?

How can you make an angle of 60 degrees by folding a sheet of paper twice?

An extension of noughts and crosses in which the grid is enlarged and the length of the winning line can to altered to 3, 4 or 5.

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

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?

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

The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.

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?

An activity centred around observations of dots and how we visualise number arrangement patterns.

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

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?

How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!

What is the shape of wrapping paper that you would need to completely wrap this model?

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

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.