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

A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn.

Challenge Level

ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?

Challenge Level

These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?

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

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

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

What is the minimum number of squares a 13 by 13 square can be dissected into?

Challenge Level

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?

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

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

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

The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?

Challenge Level

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

Challenge Level

A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?

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?

Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?

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

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. . . .

Challenge Level

Given a 2 by 2 by 2 skeletal cube with one route `down' the cube. How many routes are there from A to B?

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

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

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

If you move the tiles around, can you make squares with different coloured edges?

Challenge Level

A huge wheel is rolling past your window. What do you see?

Challenge Level

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

Challenge Level

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

Challenge Level

Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.

Challenge Level

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

Challenge Level

What can you see? What do you notice? What questions can you ask?

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

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

Challenge Level

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

Challenge Level

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

Challenge Level

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

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

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

Challenge Level

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

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

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

Can you mark 4 points on a flat surface so that there are only two different distances between them?

Challenge Level

Some treasure has been hidden in a three-dimensional grid! Can you work out a strategy to find it as efficiently as possible?

Challenge Level

Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

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

Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?