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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

In mathematics, projections are often used to help understand the properties of 3 dimensional objects. Formally if an object is created from a set of points (x_n, y_n, z_n) then the projection in the z-direction, for example, will be the same set of points with the z coordinates squashed to zero (x_n, y_n, 0).

An obvious use of projections is in computer games where the 3D virtual world is shown on a flat screen.

It would be an interesting activity to attempt to construct real 3D models of the interpenetrated objects. Drinking straws would enable the creation of a framework and then the surfaces could be tiled with coloured paper cut to an appropriate size. It would be very interesting to receive photos of any such constructions!