Make and prove a conjecture about the cyclic quadrilateral inscribed in a circle of radius r that has the maximum perimeter and the maximum area.

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 incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?

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

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

This article by Alex Goodwin, age 18 of Madras College, St Andrews describes how to find the sum of 1 + 22 + 333 + 4444 + ... to n terms.

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

Can you find the area of a parallelogram defined by two vectors?

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?

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?

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

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

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

An account of some magic squares and their properties and and how to construct them for yourself.

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

What is the total number of squares that can be made on a 5 by 5 geoboard?

Equal touching circles have centres on a line. From a point of this line on a circle, a tangent is drawn to the farthest circle. Find the lengths of chords where the line cuts the other circles.

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

A counter is placed in the bottom right hand corner of a grid. You toss a coin and move the star according to the following rules: ... What is the probability that you end up in the top left-hand. . . .

It would be nice to have a strategy for disentangling any tangled ropes...

Can you find the values at the vertices when you know the values on the edges?

These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.

Charlie has moved between countries and the average income of both has increased. How can this be so?

If you have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg?

Can you see how to build a harmonic triangle? Can you work out the next two rows?

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

What's the largest volume of box you can make from a square of paper?

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.