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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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# Cubic Net

##### Stage: 4 and 5 Challenge Level:

This activity involves a Rubik's cube which has been opened out into a net. With this activity it will help if you have already spent some time playing with an ordinary 3-dimensional Rubik's cube. It provides excellent training of your visualisation skills.

To play the net you simply click on the centre of each face to make it twist anti-clockwise.

You may like to try these activities:
1. Click on the White face1, 2, 3 and 4 times. Try to understand how it is moving squares on the other faces.
2. Repeat for the other faces to try to understand how the net is connected up.
3.Click on the 'Scramble' button for 1 move. The effect of this is to make one randomly chosen anticlockwise twist of a face. Can you spot which face must have been twisted? Can you work out how to undo the effect of this twist?
3. Now try scrambling by 2 moves. Can you spot which moves were randomly chosen? How would you undo the effects of these moves?
4. You may like to try scrambling by 3 or 4 moves.If you make a mistake trying to undo the scramble you can reset the last scrambled state to try again.

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Once a net is highly scrambled (by, say 15 moves) it is impossible to see how the net became scrambled.
A much more difficult challenge is to try to solve a highly scrambled net so that an entire face is completed.

Once the entire face is completed you can try to solve so that the squares adjacent to the completed face are all in the correct place, as in the following picture:

Progressing beyond this level of solution is very difficult without a systematic approach. To explore this idea please see the problem Solving the Net

Other ideas for playing with the net are:

1) Experiment to find sequences of your own which have nice properties on the net. For example, can you find the12 twists which create this net configuration?

2) If you click on a square on the net then it will be blacked out. Try blacking out the corners, making a few twists and then predict the colour of the blacked out squares. Click on them to see if you were correct. This is a great way of exercising your visual mind power!

Rubik's CubeĀ® is a registered trademark of Seven Towns, Ltd.