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. . . .

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

Take a number, add its digits then multiply the digits together, then multiply these two results. If you get the same number it is an SP number.

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}.

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

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?

A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.

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. . . .

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

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

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

What have Fibonacci numbers got to do with Pythagorean triples?

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.

What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?

Yatir from Israel wrote this article on numbers that can be written as $ 2^n-n $ where n is a positive integer.

Find the maximum value of n to the power 1/n and prove that it is a maximum.

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?

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

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 find a rule which relates triangular numbers to square numbers?

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.

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

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

Evaluate these powers of 67. What do you notice? Can you convince someone what the answer would be to (a million sixes followed by a 7) squared?

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

Join in this ongoing research. Build squares on the sides of a triangle, join the outer vertices forming hexagons, build further rings of squares and quadrilaterals, investigate.

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

Can you find some Pythagorean Triples where the two smaller numbers differ by 1?

Four rods of equal length are hinged at their endpoints to form a rhombus. The diagonals meet at X. One edge is fixed, the opposite edge is allowed to move in the plane. Describe the locus of. . . .

Change the squares in this diagram and spot the property that stays the same for the triangles. Explain...

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Alison has created two mappings. Can you figure out what they do? What questions do they prompt you to ask?

Explore the relationship between quadratic functions and their graphs.

Drawing a triangle is not always as easy as you might think!

This group tasks allows you to search for arithmetic progressions in the prime numbers. How many of the challenges will you discover for yourself?

How many different colours of paint would be needed to paint these pictures by numbers?

Steve has created two mappings. Can you figure out what they do? What questions do they prompt you to ask?

Make a conjecture about the curved track taken by the complex roots of a quadratic equation and use complex conjugates to prove your conjecture.

The points P, Q, R and S are the midpoints of the edges of a non-convex quadrilateral.What do you notice about the quadrilateral PQRS and its area?

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?

The points P, Q, R and S are the midpoints of the edges of a convex quadrilateral. What do you notice about the quadrilateral PQRS as the convex quadrilateral changes?

Points D, E and F are on the the sides of triangle ABC. Circumcircles are drawn to the triangles ADE, BEF and CFD respectively. What do you notice about these three circumcircles?

Find the sum, f(n), of the first n terms of the sequence: 0, 1, 1, 2, 2, 3, 3........p, p, p +1, p + 1,..... Prove that f(a + b) - f(a - b) = ab.

Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

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

If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?

Can you find a rule which connects consecutive triangular numbers?

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