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Rubik's Cube has 27 interconnected pieces, 8 corner pieces,12 edge pieces, 6 central pieces defining the six faces and one core piece (not visible). The structure has some wonderful properties which give it, its appeal as a puzzle.

1. There are 10^19 possible arrangements (1 followed by 19 zeroes)
2. Each face rotation moves 8 pieces about a central piece (defining the face)
3. Each of the six faces can rotate to any one of four positions (90,180,270,360)
4. All interactions can be seen as a sequence of these simple face rotations.
5. All of the 10^19 possible arrangements can be produced by a relatively small number of turns. This is similar to a branching tree, where every leaf is only a few steps away from the trunk, yet there can very large numbers of leaves.
6. Returning the cube to order should only take a small number of moves if the optimal pathway is chosen. Mistakes can lead to an excursion into an equally complex or greater distance from order. (another branch, rather than down the trunk)
7. Difficulty is generated by the compound moves or rotations which affect 8 pieces per time. This makes ordering the cube vastly more complicated. A simple 36 one step pathway cannot be generated as we can with an encyclopedia set. We are forced to move 8 pieces every move thus endangering the established order.
8. The insight in solving the cube lies in
   • Utilizing a specific orientation framework, to prevent getting lost. Multiple methods have been developed. For instance, the layer by layer orientation is well known.
   • Assessing the degree of disorder with respect to that orientation
   • Developing a set of movements with known consequences for order/disorder, within that framework. (moving across the canopy /disorder or down the trunk/order)
9. At each step, the set of moves which create further order and the set which creates further disorder will change. In terms of the tree analogy, at each point moving down towards the base, there will be a different set of appropriate movements allowing further progress.
10. Despite the large number of possible ordered states there are 12 times as many further states the cube could be arranged in if individual pieces could be flipped. In other words, someone could construct a slightly different cube by flipping any one of the pieces (by improper damage) leading to a different cube with an equal number of states and equal complexity. The "ordered state" of aligned colours would be unique to each of the 12 changed cube possibilities This is also true of organic structures such as ourselves. In other words, if someone changed the face stickers on a single piece of the cube, the final order would now be different and unique.
11. There are other interesting properties of "solving the cube"
    • Some sequences of moves simply move a number of pieces into a new configuration and return to the original after a number of steps (group closure) Changing patterns of pain or nociception can be associated with a closed cycle of stress preventing any solution.
    • In order to create order in a particular region it is often necessary to temporarily destroy some other ordered region. This is very important concept in the musculoskeletal system. It may be necessary to engage a wider field of anatomy to solve an apparent small area of dysfunction.
    • It is not easy to know the degree of order or disorder. A few rotations of separate side faces will lead to a pattern which has the appearance of complexity, yet remains only a few reverse rotations away from order. Contrastingly a cube with 2 layers ordered, requires many more steps to return the last layer to order.
12. With respect to people, in general, we do not know the final ordered state which is being sought because it is difficult to know of all the damaged components and the nature of the damage. If there is a damaged structure the totality of our possibilities is changed forever. People also have a large number of possible states and movements taking them into those states. It is also possible to damage components changing the ordered set. There is a similar dependency operating as with a cube. There are definite structural arrangements between components allowing a unique set of geometries. We move gracefully between these states through our movement patterns, unless a structure comes under duress. (pain and discomfort) At this point we have to create a pathway back to order. We are also, often forced to re-design the movement patterns which maintain the ordered set, not allowing a return to disorder.
13. For people, as with the cube, it is not generally possible to move one component at a time. This creates the same degree of difficulty. If we could train every joint in the spine separately, life would be considerably easier. The dependency between 25 spinal joints is high and becomes higher with pathology and poor use. There is also the connection to limbs adding to dependency and complexity.
14. When creating order in the spine, the challenges are similar. We need
    • An orientation, such as orthogonal axis. ( x,y,z)
    • An assessment of disorder, such as the region under duress. (The painful area) and the degree of integration with other structures.
    • A sufficient number of movements to enable a return to order. (the movement skills for that region)

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©Copyright 2007 Dr David McGrath. All rights reserved