A way of comparing difficulty...? As a method for comparing the difficulty of the various versions of Dioctipoid, expressing the number of possible legal solution permutations against the total number of permutations as a ratio is maybe as good a way as any. The total number of possible permutations of the segments on a Dioctipoid is going to be really large. If I have analysed the mechanism correctly this number K can be calculated as follows. There are 6 central square segments that can be arranged in 6! (factorial) ways, each of which can be arranged in 2 orientations, giving 2 possibilities for each permutation of the centre pieces. Only ½ * of the permutations have the rotations of the central squares in possible orientations. There are 24 triangle segments that behave in quite a counter-intuitive manner. A triangle with the acute apex touching "a pole" or rotation axis will not migrate to the opposing pole for instance and thus these segments form two distinct sets of 12 for analysis purposes. These segments have a single possible orientation so each set has 12! possible permutations. If we discount the first set to 11! positions this provides for a global reference frame by holding one of the pieces fixed. There are 12 diamond segments which can be arranged in 12! ways. By observation, it can be shown that each diamond has a single possible orientation. But in the Dioctipoid, the permutations of the pieces are dependent on the positions of related pieces and only ½ * of these have the correct segment arrangement parity; we can not manipulate a single piece in isolation for instance. This gives: ( 6! · 2 · 11! · 12! · 12! ) (2 · 2) = 105,507,206,707,660,000,000,000,000,000 or 1.055 x 10 possible permutations. * This parity figure of ½ applied to the whole device as per the MIT/Rubik analysis. By way of comparison, a similar calculation** for Rubik’s Cube (K ) can also be performed : (8! · 3 · 12! · 2 · 4 ) (3 · 2 · 2 · 2) = 88,580,102,706,155.200,000,000 or 8.86 x 10 possible permutations. ** factoring in the numbers for the orientation of the centre squares in order to compare like with like, irrelevant in Rubik‘s design. ANALYSIS OF DIOCTIPOIDS 1 & 2 TBD TO BE CONTINUED ..... ---------------------------------------------------- This article has drawn heavily on a paper produced by the Massachusetts Institute of Technology describing the mathematics of Rubik’s Cube. This original work can be found at : Moreover, I made a complete hash of my first attempt at this page. As I still have not solved my own puzzle I took a far too simplistic approach in my analysis of the device. Thanks to bmenrigh, Konrad et al. at the for putting me straight on a number of points. Still not certain that I have this right. The math involved to prove this to an academic standard is beyond my capability but I'll throw it out for comment. Thank you, (Any mistakes herein are still mine). Gary 26th March 2013 |
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