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!
To avoid losing think of another very well known game where the patterns of play are similar.
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.
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. . . .
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 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. . . .
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.
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?
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.
A game for 2 players
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. . . .
This task depends on groups working collaboratively, discussing and reasoning to agree a final product.
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
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.
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?
Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
A bicycle passes along a path and leaves some tracks. Is it possible to say which track was made by the front wheel and which by the back wheel?
Can you discover whether this is a fair game?
How can visual patterns be used to prove sums of series?
Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?
A game for 2 players. Can be played online. One player has 1 red counter, the other has 4 blue. The red counter needs to reach the other side, and the blue needs to trap the red.
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?
Can you find a rule which connects consecutive triangular numbers?
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
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?
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?
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?
Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?
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?
On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?
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?
Every day at noon a boat leaves Le Havre for New York while another boat leaves New York for Le Havre. The ocean crossing takes seven days. How many boats will each boat cross during their journey?
On a clock the three hands - the second, minute and hour hands - are on the same axis. How often in a 24 hour day will the second hand be parallel to either of the two other hands?
This article is based on some of the ideas that emerged during the production of a book which takes visualising as its focus. We began to identify problems which helped us to take a structured view. . . .
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?
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
What is the minimum number of squares a 13 by 13 square can be dissected into?
In the game of Noughts and Crosses there are 8 distinct winning lines. How many distinct winning lines are there in a game played on a 3 by 3 by 3 board, with 27 cells?
Find all the ways to cut out a 'net' of six squares that can be folded into a cube.
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.
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?
You can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area.
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?
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. . . .
Can you mentally fit the 7 SOMA pieces together to make a cube? Can you do it in more than one way?