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. . . .

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

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

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.

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.

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

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

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

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?”

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?

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?

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

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

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.

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.

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.

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.

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.

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.

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

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

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

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

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

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?

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?

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?

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

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

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.

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

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?

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.

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

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

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 ...?

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

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

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.

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.

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

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