The
4x4x4 cube has an exciting feature so-called 'parity' caused by
possibility of dynamically changing facets location. In that type of
the cube one can put each facet's center in an arbitrary position.
That can be done for any even type of the Rubic's cube: 4x4x4, 6x6x6,
8x8x8 etc. By definition, a 'parity' is a situation when the cube
being passed to 3x3x3 configuration, i.e. four internal elements
forming the centers are in their right positions and all edge
elements are doubled, cannot be solved as 3x3x3 cube thereafter. One
of such parity cases is shown in the last photo of the post 'Again
about the cube Rubic' by 08.04.2010. This parity is not the only
one: there is a second, shown in the first photo. A proof that they are
different is simple: the second type keeps right edge elements
orientation whereas the first does not.
From
the other hand, the second parity's type is also expressed by the
three angle elements dislocation (second photo).
One can easily transform
it to the edge parity case by methods mentioned e.g. here:
http://youcandothecube.com
What
is the nature of the 4x4x4 cube's parity?
Well, whereas 3x3x3 cube possesses 9 degrees of freedom (DF), 4x4x4 one has 12. Additional 3 DFs let making inversion of the edge elements as shown in the third photo. Knowing the mechanism that forms these combinations we may conclude - there are two and only two types of the 4x4x4 cube's parity - all other combinations of the edge elements dislocation either lead to an ordinary cube's state meaning that no parity at all or lead to the known 2 situations shown above.
Well, whereas 3x3x3 cube possesses 9 degrees of freedom (DF), 4x4x4 one has 12. Additional 3 DFs let making inversion of the edge elements as shown in the third photo. Knowing the mechanism that forms these combinations we may conclude - there are two and only two types of the 4x4x4 cube's parity - all other combinations of the edge elements dislocation either lead to an ordinary cube's state meaning that no parity at all or lead to the known 2 situations shown above.
I
thank my colleague Rafael Lago for the cube's combinations discussion and agreeably passed time around table tennis matches.
P.S. Recently we discussed with Rafael a topic that might be titled as 'Double parity means no parity?'. Indeed, if we double the first or the second cube's parity situation does it mean that we come to a proper cube's state? Well, we experimentally checked that this is truth.
P.S. Recently we discussed with Rafael a topic that might be titled as 'Double parity means no parity?'. Indeed, if we double the first or the second cube's parity situation does it mean that we come to a proper cube's state? Well, we experimentally checked that this is truth.
