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.

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

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

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. . . .

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

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

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

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

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

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?”

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

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

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?

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?

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.

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.

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.

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

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

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.

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

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

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.

Can you see how to build a harmonic triangle? Can you work out the next two rows?

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.

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.

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.

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

Charlie has moved between countries and the average income of both has increased. How can this be so?

You can differentiate and integrate n times but what if n is not a whole number? This generalisation of calculus was introduced and discussed on askNRICH by some school students.

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?

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 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?

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 ...?

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.

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

These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.

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

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

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 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?

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.

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.