If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

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?

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.

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

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?

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. . . .

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

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

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?

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.

Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

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. . . .

Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?

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

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

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

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.

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?

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. . . .

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

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

This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only returning to the vertex you started at.

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

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?

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

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

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

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

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?

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

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?

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

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

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.

When dice land edge-up, we usually roll again. But what if we didn't...?

A Hamiltonian circuit is a continuous path in a graph that passes through each of the vertices exactly once and returns to the start. How many Hamiltonian circuits can you find in these graphs?

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

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

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

This game for two, was played in ancient Egypt as far back as 1400 BC. The game was taken by the Moors to Spain, where it is mentioned in 13th century manuscripts, and the Spanish name Alquerque. . . .

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?

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