Experiments with Neodymium magnets

I bought a set of neodymium to play with. When you are bored, without a computer, they come in handy.

The set consisted of 216 small 4-5mm balls of magnets. The magnets came in a 6x6x6 shaped cube. That shape was very unstable and stuck to the metallic box. On trying to break apart the cube the cube disintegrated into chains.

Trying to put back the magnets into a cube is not very easy. I can say that, because my friend, a physicist, also failed at the first attempt. Here's my approach and the difficulty I face along with the tricks involved.

My approach was to create a stack of 6 rings of the spheres each ring consisting of twelve balls.
Three stacks are required to make a cube. Here what the stacks looked like.

Now I flattened out all the stacks and formed sheets of 6x6x2,

Now I just stuck the stacks together and I had my cube.

The problem

The problem was to get a stack to align properly. Actually, there are two possible ways,

Proper (left) and improper (right) alignment

The one on the left is the desired one.

Proper (top) and improper (bottom) alignment

Also all the three stacks must align properly, although you don't have to combine those stacks. Because otherwise the 6x6x2 sheets would not stick together to form a cube. One of the sheets would slide over the other similar to the bottom chains in the above figure.

The trick

Suppose you are trying to join two stacks of rings and just would not align properly. The trick is to flip one stack and then try again. There's a catch here. If both of the stacks consist of even number of rings, flipping won't work. So adding rings one by one to a stack always works.

Suppose now you have formed three stacks of rings, all properly aligned. But all the three stacks do not align properly and so you cannot stick the respective sheets together. Here, atleast two stacks must be aligning properly and the third one would not align with any of the other two. So instead of  trying to re-create the third stack ad hoc, if all the rings of the third stack are inverted, it works. 

Here's the underlying diagrams.

Magnets aligning to form rings

The spheres colored regions inside the spheres represent the north and south poles of the magnet. So if two different configurations are superimposed they align properly, otherwise flipping one of them works. 

But suppose instead of a ring we have stacks of even number of rings ( both aligned properly ), flipping does not change the configurations of the stack and hence they will always align wrong. So inverting a ring, as follows, is required.

Inverting a ring (or stack of rings)

Here are some more configurations. Interestingly, the hexagon is more stable than the cube.