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

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?

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

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

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

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

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. . . .

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

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.

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

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

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.

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

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.

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

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?

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.

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

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

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?

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.

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

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?”

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.

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.

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.

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

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.

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

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

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

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

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?

A counter is placed in the bottom right hand corner of a grid. You toss a coin and move the star according to the following rules: ... What is the probability that you end up in the top left-hand. . . .

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

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 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?

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.

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.

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. . . .

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.