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?

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

Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

Prove Pythagoras' Theorem using enlargements and scale factors.

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

A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?

A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle?

The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

What fractions can you divide the diagonal of a square into by simple folding?

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

This is the second article on right-angled triangles whose edge lengths are whole numbers.

Can you find the areas of the trapezia in this sequence?

The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!

Can you make sense of these three proofs of Pythagoras' Theorem?

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

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.

Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

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

An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.

If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .

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.

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

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?

Can you make sense of the three methods to work out the area of the kite in the square?

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. . . .

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?

Here are some examples of 'cons', and see if you can figure out where the trick is.

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.