Some Cubic Questions

1. You are about to build an elegantly decorated regular tetrahedron which you want to mail to a special friend of Rickπs. The smallest cubic gift box you can find is four inches on a side (inside dimensions). What is the largest possible edge length for your tetrahedron if it is to fit in the box?

2. What fraction of the interior of the box is occupied by the tetrahedron?

3. You have painted one vertex of a cube blue and the antipodal vertex (look that up in your Funk and Wagnallπs) red. All points closer to the blue vertex are coloured blue, points closer to the red vertex are red. Describe the boundary between the blue points and the red points in the interior and on the surface of the cube.

4. You have painted all eight vertices of a cube blue and the centre of the cube yellow. All points are coloured yellow if they are closer to the centre than one of the vertices and are painted blue if they are closer to one of the vertices than to the centre. Describe as completely as possible the boundary between the yellow points and the blue points.

5. A point in three-dimensional space all of whose coordinates are even is called an even point. A point all of whose coordinates are odd is called, wait for it, an odd point. What is the boundary between those points that are closer to a given odd or even point than to any other odd or even point?

6. A cube is being filled with water. Is it possible that at some time the horizontal surface of the water is an equilateral triangle? If yes, how full is the cube?

7. Is it possible that at some time the horizontal surface of the water is a regular hexagon? If yes, how full is the cube?

8. How do the perimeter of the largest possible equilateral triangular water surface and the perimeter of the regular hexagonal surface compare?

9. The diagram below is a possible net of a cube. Draw on the net the waterlines corresponding to the largest equilateral triangular cross-section and the waterline corresponding to the regular hexagonal cross-section. Hmmm.