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.
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.
See if you can anticipate successive 'generations' of the two animals shown here.
Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.
Which of the following cubes can be made from these nets?
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.
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?
Imagine you have six different colours of paint. You paint a cube using a different colour for each of the six faces. How many different cubes can be painted using the same set of six colours?
Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology.
Some diagrammatic 'proofs' of algebraic identities and inequalities.
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!
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.
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 picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .
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. . . .
A huge wheel is rolling past your window. What do you see?
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.
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?
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?
Can you use the diagram to prove the AM-GM inequality?
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. . . .
The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.
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?
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.
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?
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
How can you make an angle of 60 degrees by folding a sheet of paper twice?
In how many different ways can I colour the five edges of a pentagon red, blue and green so that no two adjacent edges are the same colour?
Bilbo goes on an adventure, before arriving back home. Using the information given about his journey, can you work out where Bilbo lives?
How many ways can you write the word EUROMATHS by starting at the top left hand corner and taking the next letter by stepping one step down or one step to the right in a 5x5 array?
Can you discover whether this is a fair game?
Can you find a rule which connects consecutive triangular numbers?
Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
How much of the square is coloured blue? How will the pattern continue?
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?
When dice land edge-up, we usually roll again. But what if we didn't...?
How much of the field can the animals graze?
An introduction to bond angle geometry.
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?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
This task depends on groups working collaboratively, discussing and reasoning to agree a final product.
ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.
Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube so that the surface area of the remaining solid is the same as the surface area of the original?
Can you describe this route to infinity? Where will the arrows take you next?
Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?
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?