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.

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. . . .

Your data is a set of positive numbers. What is the maximum value that the standard deviation can take?

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

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

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

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.

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.

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

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?

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

A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.

Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

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

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .

Can you find the values at the vertices when you know the values on the edges?

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.

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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

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

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 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?”

This article by Alex Goodwin, age 18 of Madras College, St Andrews describes how to find the sum of 1 + 22 + 333 + 4444 + ... to n terms.

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

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

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

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?

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

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

Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.

To avoid losing think of another very well known game where the patterns of play are similar.

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

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

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

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?

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.

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?

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

A game for 2 players with similarities 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?

What's the largest volume of box you can make from a square of paper?