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Cut Cube

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!



The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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