CutCube

A cube is hanging from string attached to a vertex, and then it is cut perfectly in half with a horizontal plane right through the middle. Describe the cross-section.

It’s a great problem because, though it’s not obvious at first, there are some simple symmetry arguments that will make the answer clear.  The problem delivers structure and beauty.

But for me, it’s the extension of this argument to 4 dimensions that really completes the experience. We played with it at the 2026 RabbitMath summer camp and the kids were pretty excited, even those that needed help at many points. The feeling of tangling with 4-space is heady stuff.

Here’s the 4-D version. Take a hypercube, hang it by a string attached to a vertex and then cut it perfectly in half with a horizontal hyperplane. Describe the cross-section. Of course, in this case, it will be a 3-dimensional object, a polyhedron in fact. What does it look like?

This activity has everything we’d ever want in a problem, intuition, right-brain, left-brain, geometry, symmetry––structure on steroids!