Think of a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I know?

Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?

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 ?

If a sum invested gains 10% each year how long before it has doubled its value?

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

Water freezes at 0°Celsius (32°Fahrenheit) and boils at 100°C (212°Fahrenheit). Is there a temperature at which Celsius and Fahrenheit readings are the same?

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?”

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

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

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

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.

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

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

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

Prove that 3 times the sum of 3 squares is the sum of 4 squares. Rather easier, can you prove that twice the sum of two squares always gives the sum of two squares?

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.

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?

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

Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.

Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is $1+\sqrt 2+\sqrt 3$.

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.

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

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.

Create some shapes by combining two or more rectangles. What can you say about the areas and perimeters of the shapes you can make?

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

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

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

Several graphs of the sort occurring commonly in biology are given. How many processes can you map to each graph?

Derive an equation which describes satellite dynamics.

Can you hit the target functions using a set of input functions and a little calculus and algebra?

What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?

However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?

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?

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

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

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

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.

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

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

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

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

Two cyclists, practising on a track, pass each other at the starting line and go at constant speeds... Can you find lap times that are such that the cyclists will meet exactly half way round the. . . .