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

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

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

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. . . .

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.

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

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

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.

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.

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

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

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?

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

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

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?”

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?

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.

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.

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?

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

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 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?

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.

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

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

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

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. . . .

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.

To avoid losing think of another very well known game where the patterns of play are similar.

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

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.

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.

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

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?

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?

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?

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.

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. . . .

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?