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

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

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

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

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?

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

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.

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

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.

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.

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.

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.

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.

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?

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

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

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

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

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

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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

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?

Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .

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

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

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?”

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

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?

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

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

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

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

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

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

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

Can you work out how to win this game of Nim? Does it matter if you go first or second?

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

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

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?

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

It starts quite simple but great opportunities for number discoveries and patterns!

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

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

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. . . .