Polycube Snakes

Whenever I present an unsolved problem to you I feel like I should lead with that fact to avoid people feeling cheated of a nice neat answer, but to me, the fact that there are still unsolved problems which are easy to state, but hard to prove shows just how deep mathematics is.

The polycube snake problem comes from the veteran puzzle designer Scott Kim, although I came across it in an article about Kim from Martin Gardner. A polycube snake is a series of unit cubes which are connected to each other such that each cube along the snake only has faces in common with the two cubes either side of it (apart from the two end ones). However you may have the snake twist and turn such that the edges and the vertices of the snake may touch earlier parts of itself. Each snake can be infinite in length if you like, however if you want it to have one end (with the other end shooting off to infinity) or even two ends then that is fine too.

The question is: how many snakes does it take to fill the whole of 3D space?

The 2D version of these problem can be solved with just two snakes in a spiralling pattern as below:

Picture from The Colossal Book of mathematics by Martin Gardner

But the 3D problem has proven harder to do. Kim himself has found a 4 snake solution with each having its head near the centre and then spiralling outwards in a helix forever. However no amount of searching on the internet has been able to provide me an image of this solution. In the article Gardner simply says: "the method is too complicated to explain in a limited space; you will have to take my word that it can be done" which is a little unsatisfying. If any of you can out google me, or prove it yourself I would be grateful if you could help me out.

However, while 1 snake is definitely ruled out, Kim was not able to prove that it couldn't be done in 3 snakes or even 2 snakes. If there is a 2 snake solution then it would be an astounding ying-yang like structure.

I'm going to leave you with a sign off from Gardner, because it is one of my heroes talking about meeting another one of my heroes: "I once had the pleasure of explaining the polycube-snake problem to John Horton Conway, the Cambridge mathematician now at Princeton University. When I concluded by saying that Kim had not yet shown that two snakes could not tile three-dimensional space, Conway instantly said, "But it's obvious that--" Here checked himself mid-sentence, stared into three-space for a minute or two, then exclaimed, "It's not obvious!" I have no idea what passed through Conway’s mind. I can only say that if the impossibility of filling three-space with two snakes is not obvious to Conway or to Kim, it probably is not obvious to anyone else."