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?
Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.
The reader is invited to investigate changes (or permutations) in the ringing of church bells, illustrated by braid diagrams showing the order in which the bells are rung.
Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.
Can you make a tetrahedron whose faces all have the same perimeter?
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.
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
A game for 2 players
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . .
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 game for 2 people. Take turns joining two dots, until your opponent is unable to move.
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?
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.
See if you can anticipate successive 'generations' of the two animals shown here.
To avoid losing think of another very well known game where the patterns of play are similar.
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
Given a 2 by 2 by 2 skeletal cube with one route `down' the cube. How many routes are there from A to B?
Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?
This task depends on groups working collaboratively, discussing and reasoning to agree a final product.
A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .
Which of the following cubes can be made from these nets?
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?
Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?
Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?
What is the shape of wrapping paper that you would need to completely wrap this model?
Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface. . . .
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. . . .
When dice land edge-up, we usually roll again. But what if we didn't...?
Have a go at this 3D extension to the Pebbles problem.
Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .
This is an interactive net of a Rubik's cube. Twists of the 3D cube become mixes of the squares on the 2D net. Have a play and see how many scrambles you can undo!
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
I found these clocks in the Arts Centre at the University of Warwick intriguing - do they really need four clocks and what times would be ambiguous with only two or three of them?
An irregular tetrahedron has two opposite sides the same length a and the line joining their midpoints is perpendicular to these two edges and is of length b. What is the volume of the tetrahedron?
Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology.
A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn.
Can you find a rule which relates triangular numbers to square numbers?
Can you mark 4 points on a flat surface so that there are only two different distances between them?
Here is a solitaire type environment for you to experiment with. Which targets can you reach?
A bus route has a total duration of 40 minutes. Every 10 minutes, two buses set out, one from each end. How many buses will one bus meet on its way from one end to the other end?
Bilbo goes on an adventure, before arriving back home. Using the information given about his journey, can you work out where Bilbo lives?
A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?
Two angles ABC and PQR are floating in a box so that AB//PQ and BC//QR. Prove that the two angles are equal.
What can you see? What do you notice? What questions can you ask?
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?