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. . . .

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

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

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?

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

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.

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

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

A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

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. . . .

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. . . .

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?

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

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

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

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

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

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?”

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?

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.

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

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?

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

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

A game for 2 players with similarities to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.

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?

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

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 ...?

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

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

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.

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

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?

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

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

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

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.

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

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

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