3D Puzzle Series 2, #2

I’ll call this one Cube. This is more of a brain teaser than a 3D wood puzzle. The pieces are all held together by a length of elastic cord that runs through holes in the blocks to create only one solution for forming a 3×3 cube (actually, two solutions, but they’re mirror images of each other). I have to admit that I was not able to remake the cube again once I got it out of the plastic packaging. So, it is a challenging puzzle, but I’ve said before that I’m not that good at these kinds of puzzles.

On the other hand, when I tried getting the next puzzle from the capsule dispenser, I got two extra copies of Cube, so it’s a simple matter now of videoing the extra Cube being opened up, and just using that to reverse-engineer this one. I will say right now, though, that Cube looks better as a Cube while it’s still in the packaging. When sitting on its own, the elastic cord causes it to self-disassemble.

Take 2: Ok, a few days after I wrote the above description, I had a little spare time, so I took my camera and one of the extra cubes, and recorded the process of unraveling it after removing the plastic. The result was absolute and total failure. Even with the photos in hand, I could only get halfway through the solution before running into the same wall – the first half of the assembly (the “outer shell”) was in the way for my rotating the rest of the puzzle (the “inner core”) 180 degrees in order to put the core within the shell. The problem is that it’s easy to figure out how to start at the “head end” of the snake and make the 3x3x3 outer shell. But if you try from the “tail” end, there’s no recognizable structure for approaching the core.

(The outer shell portion complete, leaving the “inner core” part unfinishable.)

In a way, once the outer shell is formed, what you have is positive and negative space. The positive space is the shell, and the negative space is what the core is supposed to look like if you move the pieces around the right way. But, if you take the remaining pieces and make them look like the negative space, you can’t easily rotate them 180 degrees so that the core slips inside the shell (I say “easily”, because if you do it right, you can stretch the walls of the shell apart enough to swing the middle part of the core into the gap of the U, and you can finish the puzzle that way. However, that’s kind of damaging the cord and increasing the chance it will snap some time later.)

(The inner core mostly assembled and ready to be rotated 180 degrees into place. Shame it’s not designed to rotate like this…)

One of the books I’d gotten for Christmas is Shing-Tung Yau’s “The Shape of Inner Space“. Yau is a mathematician that co-developed the Calabi-Yau manifolds that are central to String and M(Membrane)-Theory. The book is part autobiography, part history of String Theory development, and lots and lots of talk about the math revolving around Calabi-Yau manifolds, and how they relate to astrophysics. In it, Yau frequently says that the ability to switch between different domains is central to solving complex problems. Having a transformation (like mirroring) lets you move from one domain and then back. And that a problem that is almost intractable in one domain may be really easy to solve in the other.

(Starting over mid-way, with the core rotated into position correctly, but the outer shell partly unraveled.)

I was thinking about the book as I was walking to work, and about the Cube puzzle, and it struck me that the Cube is kind of an example of approaching a problem from both directions, and meeting in the middle to massage the two halves together to make a finished whole.

(View from the other side, where the inner core joins with the outer shell.)

So. Take the snake as before and start from the head. Create the U shape, and keep bending the pieces to get the outer 3x3x3 shell. There’s really only one direction the pieces can turn to form the shell.

(And, rewrapping the shell around the core, working backwards this time.)

When you get to the stage where you have to stop, look at the gap in the middle top of the shell. There’s going to be a double-back, with one segment that is 3 pieces long, folded back under an adjacent segment that is also 3 pieces long. This is the heart of your core. And this is what the outer shell is going to wrap around. Get the double-back ready, hold it in the orientation it needs to face to fill the existing negative space within the shell. Then, let go of the shell and shake it out a bit.

You have the connecting piece on the side of the core in the location it’s supposed to be in. All that’s left is to build the U-shaped shell backwards around the core, and finishing with the head of the snake to complete that half of the Cube.

(Almost done now.)

To make things a bit simpler, I left the last few pieces of the tail to the end. Turn the cube around so the tail is facing you, and it should be pretty obvious how they need to be routed to wrap up the puzzle. After you solve the puzzle a few times, you’ll know how to fold the tail so it’s in the right place relative to the core double-back, as you’re building up the rest of the shell.

(The “tail” of the snake to the left, the “head” – and very last portion of the outer shell – to the right.)

Anyway, the point is that this kind of puzzle is a lot easier if you start from the end of the snake that forms a 3×3 U square, continuing up to where you’re about to form the core, make the core so that it will fit the negative space of the cube, and then shake out the shell and start again. The second time, though, you’re building the shell backwards around the core, and part of the U shapes will still be in place in the snake, so you’ll be able to see more easily where they go. Trying to start at the tail end, and make the core at the outset is possible, but it’s much more difficult to get it right. The first few times you try to solve this puzzle, that is. Eventually, I’ve noticed that I can begin at the tail end, identify where the double-back occurs, and then continue through the full solution. Oddly, even having solved it 10 times, I’ll still suddenly feel my mind go blank and nothing will make sense. I’ll go through several false starts before I figure out the problem. Strange.

One other thing I consider interesting about Cube, after having thought about it for a while, is that this design is a variant on the traveling salesman problem. Draw a 3x3x3 matrix, putting little spheres at the line intersections. Now, draw a minimum-distance path that connects the dots such that you go through each dot once and only once, the paths don’t cross, and that the path only connects neighboring dots (that is, you can’t jump directly from one corner of the cube to the opposite one. You can only connect dots that are next to each other). When you’re done, the spheres will represent the centers of the cube block pieces, and the path will be the elastic cord. Straight-through paths will turn into holes that go straight through a wooden block, and paths that take right-angle turns will be right-angle holes in the blocks. (The ends of the cord are held in place in the end blocks by pounding small wooden plugs into the holes.) Armed with this traveling salesman solution, making and solving a real-world Cube like this will be relatively easy.

(All that’s left is to rotate the last of the tail into place.)


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