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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?

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

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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?

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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?

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

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Draw all the possible distinct triangles on a 4 x 4 dotty grid. Convince me that you have all possible triangles.

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

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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?

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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?

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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

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?

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Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

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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. . . .

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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?

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

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The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...

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A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .

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

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

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

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?

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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?

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

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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?

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A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

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Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

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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. . . .

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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.

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

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

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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.

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

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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?

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What 3D shapes occur in nature. How efficiently can you pack these shapes together?

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

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

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

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Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?

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

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

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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?

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Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

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Triangles are formed by joining the vertices of a skeletal cube. How many different types of triangle are there? How many triangles altogether?

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How many winning lines can you make in a three-dimensional version of noughts and crosses?

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A cube is made from smaller cubes, 5 by 5 by 5, then some of those cubes are removed. Can you make the specified shapes, and what is the most and least number of cubes required ?

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

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