Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

Four jewellers share their stock. Can you work out the relative values of their gems?

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?

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.

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

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.

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. . . .

This problem is a sequence of linked mini-challenges leading up to the proof of a difficult final challenge, encouraging you to think mathematically. Starting with one of the mini-challenges, how. . . .

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

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

Which of these triangular jigsaws are impossible to finish?

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

Can you work out where the blue-and-red brick roads end?

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.

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.

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

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.

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.

Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?

This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .

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

In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.

A introduction to how patterns can be deceiving, and what is and is not a proof.

Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?

The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.

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

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.

In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.

Can you make sense of the three methods to work out the area of the kite in the square?

The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.

This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.