Take a perfect cube. While keeping the cube intact and all the pieces together, make as many planar slices as possible through exactly three vertices.

After doing so, separate the resulting pieces. What shapes result and how many are there of each and in total?

To what do these numbers correspond?

Note: Please do NOT punch the problem into a 3D graphics program and then rush to post the solution here so you can be first. This remains my most enjoyable solution to date because I attempted it without pen and paper, computer or polyhedron.

I lay awake thinking about this last nite. Really like doing geometry in my head- once solved a wooden puzzle in my head at school after memorising the pieces at breakfast.

Anyway I first realised that each plane intersects three sides of hte original cube along their diagonals (Hence the intersection of each plane with the cube is an equallateral triangle). At first I thought this led to 24 intersecting planes- DOH, of course I had counted each side three times. So we have eight tringles, three of which meet at each corner. This was a break through as platonic tetrahedra have 3ETs meeting at each corner, hence all the above facts lead me to realise we have two intersecting tetrahedra nestling neatly inside a cube. These breakdown into an octahed in the centre with a tetrahed on each of its faces. I fell asleep at this point.

At breakfast I tried to visualise a cube minus two tetrahedra but was two tired and couldnt resist drawing it to find those anoying skew pyramids around the edges.

Im annoyed with myself for cheating at the end but still very kewl puz. Nice one Charley, more like this please.

Posted by Percy on 2005-08-19 13:56:52