Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?

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. . . .

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.

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

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

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

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

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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

An article which gives an account of some properties of magic squares.

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.

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 Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

An account of some magic squares and their properties and and how to construct them for yourself.

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

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

Can you find the area of a parallelogram defined by two vectors?

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.

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?”

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

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

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

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?

Equal touching circles have centres on a line. From a point of this line on a circle, a tangent is drawn to the farthest circle. Find the lengths of chords where the line cuts the other circles.

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.

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

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.

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?

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

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

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.

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.

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

Here explore some ideas of how the definitions and methods of calculus change if you integrate or differentiate n times when n is not a whole number.

Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?

Can you describe this route to infinity? Where will the arrows take you next?

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

A game for 2 players with similaritlies to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.

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

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

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?