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

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.

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

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

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs?

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

Place a red counter in the top left corner of a 4x4 array, which is covered by 14 other smaller counters, leaving a gap in the bottom right hand corner (HOME). What is the smallest number of moves. . . .

A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .

How many moves does it take to swap over some red and blue frogs? Do you have a method?

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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?

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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 Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

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

These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

How many different symmetrical shapes can you make by shading triangles or squares?

Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

This is a simple version of an ancient game played all over the world. It is also called Mancala. What tactics will increase your chances of winning?

There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being visible at any one time. Is it possible to reorganise these cubes so that by dipping the large cube into a pot of paint three times you. . . .

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.

Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

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

Can you find a rule which connects consecutive triangular numbers?

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

Show that all pentagonal numbers are one third of a triangular number.

Can you find a way of representing these arrangements of balls?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

Imagine a stack of numbered cards with one on top. Discard the top, put the next card to the bottom and repeat continuously. Can you predict the last card?

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?

How many winning lines can you make in a three-dimensional version of noughts and crosses?

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

Can you describe this route to infinity? Where will the arrows take you next?

Can you find a rule which relates triangular numbers to square numbers?

What is the shape of wrapping paper that you would need to completely wrap this model?

Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?

Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?