Can you find a rule which relates triangular numbers to square numbers?

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Can you find a rule which connects consecutive triangular numbers?

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

A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

Show that all pentagonal numbers are one third of a triangular number.

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.

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?

Prove that the product of the sum of n positive numbers with the sum of their reciprocals is not less than n^2.

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?

Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?

To break down an algebraic fraction into partial fractions in which all the denominators are linear and all the numerators are constants you sometimes need complex numbers.

Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.

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

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

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

Show there are exactly 12 magic labellings of the Magic W using the numbers 1 to 9. Prove that for every labelling with a magic total T there is a corresponding labelling with a magic total 30-T.

This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?

Can you find the lap times of the two cyclists travelling at constant speeds?

A task which depends on members of the group noticing the needs of others and responding.

Attach weights of 1, 2, 4, and 8 units to the four attachment points on the bar. Move the bar from side to side until you find a balance point. Is it possible to predict that position?

Five equations... five unknowns... can you solve the system?

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. . . .

An algebra task which depends on members of the group noticing the needs of others and responding.

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

By proving these particular identities, prove the existence of general cases.

A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

The heptathlon is an athletics competition consisting of 7 events. Can you make sense of the scoring system in order to advise a heptathlete on the best way to reach her target?

Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

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.

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

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?

My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the. . . .

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

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.

The sums of the squares of three related numbers is also a perfect square - can you explain why?

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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

Can you find the value of this function involving algebraic fractions for x=2000?