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. . . .

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

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.

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

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

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

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

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

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?

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?”

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 area of a parallelogram defined by two vectors?

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

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. . . .

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.

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?

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.

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

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

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 ...?

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.

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.

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

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

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

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.

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

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

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.

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

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.

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

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?

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.

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

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

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.