Challenge Level

Can you work out how these polygon pictures were drawn, and use that to figure out their angles?

Challenge Level

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

Challenge Level

Join pentagons together edge to edge. Will they form a ring?

Challenge Level

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

Challenge Level

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

Challenge Level

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

Challenge Level

A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?

Challenge Level

A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?

Challenge Level

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.

Challenge Level

A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't. . . .

Challenge Level

The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . .

Challenge Level

See if you can anticipate successive 'generations' of the two animals shown here.

Challenge Level

On a clock the three hands - the second, minute and hour hands - are on the same axis. How often in a 24 hour day will the second hand be parallel to either of the two other hands?

Challenge Level

A cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels?

Challenge Level

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

Challenge Level

A right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates?

Challenge Level

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?

Challenge Level

Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?

Challenge Level

A half-cube is cut into two pieces by a plane through the long diagonal and at right angles to it. Can you draw a net of these pieces? Are they identical?

Challenge Level

This is a simple version of an ancient game played all over the world. It is also called Mancala. What tactics will increase your chances of winning?

Challenge Level

There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being visible at any one time. Is it possible to reorganise these cubes so that by dipping the large cube into a pot of paint three times you. . . .

Challenge Level

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

Challenge Level

Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .

This game for two, was played in ancient Egypt as far back as 1400 BC. The game was taken by the Moors to Spain, where it is mentioned in 13th century manuscripts, and the Spanish name Alquerque. . . .

Challenge Level

The image in this problem is part of a piece of equipment found in the playground of a school. How would you describe it to someone over the phone?

Challenge Level

How many different symmetrical shapes can you make by shading triangles or squares?

Challenge Level

This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only returning to the vertex you started at.

Challenge Level

A 10x10x10 cube is made from 27 2x2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner.

Challenge Level

In how many ways can you fit all three pieces together to make shapes with line symmetry?

Challenge Level

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?

Challenge Level

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

Challenge Level

ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.

Challenge Level

Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?

Challenge Level

An irregular tetrahedron has two opposite sides the same length a and the line joining their midpoints is perpendicular to these two edges and is of length b. What is the volume of the tetrahedron?

This is the first article in a series which aim to provide some insight into the way spatial thinking develops in children, and draw on a range of reported research. The focus of this article is the. . . .

Challenge Level

The farmers want to redraw their field boundary but keep the area the same. Can you advise them?

Challenge Level

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

Challenge Level

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

Challenge Level

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

Challenge Level

How much of the square is coloured blue? How will the pattern continue?

Challenge Level

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

Challenge Level

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

Challenge Level

Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?