It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .

Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more. . . .

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.

The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make.

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

Re-arrange the pieces of the puzzle to form a rectangle and then to form an equilateral triangle. Calculate the angles and lengths.

One side of a triangle is divided into segments of length a and b by the inscribed circle, with radius r. Prove that the area is: abr(a+b)/ab-r^2

Which is larger cos(sin x) or sin(cos x) ? Does this depend on x ?

What are the shortest distances between the centres of opposite faces of a regular solid dodecahedron on the surface and through the middle of the dodecahedron?

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

Can you prove this formula for finding the area of a quadrilateral from its diagonals?

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

A belt of thin wire, length L, binds together two cylindrical welding rods, whose radii are R and r, by passing all the way around them both. Find L in terms of R and r.

An environment that simulates a protractor carrying a right- angled triangle of unit hypotenuse.

Describe how to construct three circles which have areas in the ratio 1:2:3.

A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height?

Beautiful mathematics. Two 18 year old students gave eight different proofs of one result then generalised it from the 3 by 1 case to the n by 1 case and proved the general result.

The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times?

If you were to set the X weight to 2 what do you think the angle might be?

How far should the roof overhang to shade windows from the mid-day sun?

Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.

From the measurements and the clue given find the area of the square that is not covered by the triangle and the circle.

Trigonometry, circles and triangles combine in this short challenge.

Join some regular octahedra, face touching face and one vertex of each meeting at a point. How many octahedra can you fit around this point?

Can you explain what is happening and account for the values being displayed?

Can you deduce the familiar properties of the sine and cosine functions starting from these three different mathematical representations?

The first of three articles on the History of Trigonometry. This takes us from the Egyptians to early work on trigonometry in China.

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

The second of three articles on the History of Trigonometry.

The length AM can be calculated using trigonometry in two different ways. Create this pair of equivalent calculations for different peg boards, notice a general result, and account for it.

Logo helps us to understand gradients of lines and why Muggles Magic is not magic but mathematics. See the problem Muggles magic.

The third of three articles on the History of Trigonometry.

How can you represent the curvature of a cylinder on a flat piece of paper?

On a nine-point pegboard a band is stretched over 4 pegs in a "figure of 8" arrangement. How many different "figure of 8" arrangements can be made ?

Find the exact values of some trig. ratios from this rectangle in which a cyclic quadrilateral cuts off four right angled triangles.

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different?. . . .

Without using a calculator, computer or tables find the exact values of cos36cos72 and also cos36 - cos72.

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

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

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

The sine of an angle is equal to the cosine of its complement. Can you explain why and does this rule extend beyond angles of 90 degrees?

Prove Pythagoras' Theorem for right-angled spherical triangles.

Two places are diametrically opposite each other on the same line of latitude. Compare the distances between them travelling along the line of latitude and travelling over the nearest pole.