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Find the shape and symmetries of the two pieces of this cut cube.

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Cubic Spin

Prove that the graph of f(x) = x^3 - 6x^2 +9x +1 has rotational symmetry. Do graphs of all cubics have rotational symmetry?

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Middle Man

Mark a point P inside a closed curve. Is it always possible to find two points that lie on the curve, such that P is the mid point of the line joining these two points?

Interpenetrating Solids

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

For the first part of this problem try to view the cubes from above to turn it into a simpler 2D problem to begin with.

The second part of this problem will be greatly simplfied if you work with a physical model of a cube, preferably wire framed which you can see through (such as with polydron) and the bigger the better. You may wish to attempt to create frames using wires or straws to help to understand the problem.

Instead of trying to visualise the entire solid 3 dimensional cube, you may like to consider the cube as defined by its 8 corners. If you work out where the corners go then the rest of the cube will follow by joining up with straight lines and planes.Visualising the objects by looking directly at one of the original faces will help you to understand the problem.

Finally, don't forget that if you can work out the view of the interpenetrated object in the direction of one of the faces, then the view in the direction of the other faces may follow by symmetry.