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

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.

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. . . .

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

Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.

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

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

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

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. . . .

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 triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

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. . . .

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

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?

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

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?

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

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 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?”

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.

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

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?

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.

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

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?

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.

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 ...?

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

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

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?

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

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

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

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

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

Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.

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

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

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Here explore some ideas of how the definitions and methods of calculus change if you integrate or differentiate n times when n is not a whole number.

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.