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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?

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Coke Machine

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design. Coins inserted into the machine slide down a chute into the machine and a drink is duly released. How many more revolutions does the foreign coin make over the 50 pence piece going down the chute? N.B. A 50 pence piece is a 7 sided polygon ABCDEFG with rounded edges, obtained by replacing AB with arc centred at E and radius EA; replacing BC with arc centred at F radius FB ...etc..

Cubic Net

Stage: 4 and 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Although it is very difficult to solve the Cubic Net fully, it is good fun to play with for all ability ranges and a source of many rich tasks.

As a whole class activity teachers may like to try the following activities

Choose a sequence of moves, such as PRP or WWGGB, on a reset net
Before doing the moves, ask questions such as:

  1. Can you choose a corner or edge square on the net which you know will not move?
  2. Choose a square on the net. What colour will it turn into?
  3. What sequence of moves will undo the original sequence?
  4. How many squares on the net will change colour overall?
Scramble the net with 1, 2 or 3 moves
  1. Ask students to discuss how to unscramble the net.
  2. Ask for volunteers to try to unscramble the net. You can press the Restore Last Scramble button to allow others to have a try.
  3. Since the faces only move anti-clockwise you may like to discuss with the class the discovery that a 1 move scramble takes 3 twists to undo and so on.
Secretly record and play a move sequence of between 1 a 5 moves using the 'Record moves to button'
  1. Ask students to guess how many moves they think that the net has been scrambled by
  2. You can play the sequence of moves so that students can count and see the moves in action
  3. Once the sequence of moves is known to the class reset the net. Choose a square on the net which you know will move. Ask the class to visualise the motion of this square throughout the move sequence, making notes if they wish. Play the move sequence asking the class to follow the path of their chosen square. Who correctly predicted the end location?

Black out the corner or edge squares
Ask the students to focus on a particular corner. Tell them that you are going to make a series of moves and that they need to try to follow the path of that square in their head.
  1. Make the moves. Ask for two volunteers to describe their predictions. This should create a discussion about effective ways to describe the squares on the net, such a Red face (2, 1) for second row and first column on the red face.
  2. Once a labellelling system is in place the whole class can play. They write their predictions down and swap with a a partner. Those who make mistakes are eliminated.

Other discussion points
  • Although many people can solve a Rubik's cube from a given position, the problem of the Rubik's cube has not yet been fully understoodby mathematicians due to its enormous complexity.
  • Mathematicians are still actively working on the main problem which is: what is the smallest number of moves with which you can guarantee to solve any Rubiks cube configuration?
  • It has very recently been shown that 26 moves is sufficient to solve any Rubik's cube configuration (see here for a news report on this achievement)

There are many web sites devoted to the Rubik's cube problem. The official site solution page is a good place for the interested to find more information.Why not set an investigative homework on the properties of the Rubik's cube.