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60 problems, 83 articles, 25 general resources, 19 Lists, 4 from Stage 1, 9 from Stage 2, 54 from Stage 3, 73 from Stage 4, 118 from Stage 5

By sketching a graph of a continuous increasing function, can you prove a useful result about integrals?

Use the applet to explore the area of a parallelogram and how it relates to vectors.

Can you work through these direct proofs, using our interactive proof sorters?

Learn how to make a simple table using Excel.

In the limit you get the sum of an infinite geometric series. What about an infinite product (1+x)(1+x^2)(1+x^4)... ?

Investigate the molecular masses in this sequence of molecules and deduce which molecule has been analysed in the mass spectrometer.

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

Decipher a simple code based on the rule C=7P+17 (mod 26) where C is the code for the letter P from the alphabet. Rearrange the formula and use the inverse to decipher automatically.

If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .

Investigate polygons with all the vertices on the lattice points of a grid. For each polygon, work out the area A, the number B of points on the boundary and the number of points (I) inside. . . .

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.

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

Learn how to use Excel to create triangular arrays.

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

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

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?

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.

A preview of some of the beam deflection mechanics you will look at in the first year of an engineering degree

What have Fibonacci numbers got to do with Pythagorean triples?

An account of multiplication of vectors, both scalar products and vector products.

A space craft is ten thousand kilometres from the centre of the Earth moving away at 10 km per second. At what distance will it have half that speed?

If you take two integers and look at the difference between the square of each value, there is a nice relationship between the original numbers and that difference. Can you find the pattern using. . . .

Learn how to use advanced pasting techniques to create interactive spreadsheets.

Get to grips with the variables behind the motion of a simple pendulum

A weekly challenge concerning the interpretation of an algorithm to determine the day on which you were born.

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

Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.

Use an interactive Excel spreadsheet to investigate remainders.

A sequence of numbers x1, x2, x3, ... starts with x1 = 2, and, if you know any term xn, you can find the next term xn+1 using the formula: xn+1 = (xn + 3/xn)/2 . Calculate the first six terms of this. . . .

Be playful with graphs and networks, and see what theorems you can discover!

Use Excel to find sets of three numbers so that the sum of the squares of the first two is equal to the square of the third.

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different?. . . .

Derive Euler's buckling formula from first principles.

An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.

Yatir from Israel describes his method for summing a series of triangle numbers.

Learn how to use logic tests to create interactive resources using Excel.

How good are you at finding the formula for a number pattern ?

See how enormously large quantities can cancel out to give a good approximation to the factorial function.

The harmonic triangle is built from fractions with unit numerators using a rule very similar to Pascal's triangle.

An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.

We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.

Find relationships between the polynomials a, b and c which are polynomials in n giving the sums of the first n natural numbers, squares and cubes respectively.

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.

From the measurements and the clue given find the area of the square that is not covered by the triangle and the circle.

What would you do if your teacher asked you add all the numbers from 1 to 100? Find out how Carl Gauss responded when he was asked to do just that.