https://www.speedsolving.com/wiki/api.php?action=feedcontributions&feedformat=atom&user=Bootmii 2026-10-27T02:52:57Z User contributions MediaWiki 1.43.8 https://www.speedsolving.com/wiki/index.php?title=Laws_of_the_cube&diff=8886 2010-07-24T23:15:36Z

<p>Bootmii: also unsolvable if 3 edges to flip</p> <hr /> <div>Only a 12&lt;sup&gt;th&lt;/sup&gt; of all states reachable by assembling the cube piece-by-piece can actually be reached by twisting the 6 faces (from the solved state). Here's why:<br /> <br /> <br /> # '''Only an even number of cubie swaps is possible''' (&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt; of all permutations) &lt;sup&gt;[1]&lt;/sup&gt;<br /> # '''Only an even number of edges can be in a flipped state''' (&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt; of all edge orientations) &lt;sup&gt;[2]&lt;/sup&gt; <br /> # '''The total number of corner twists must be divisible by 3''' (&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;3&lt;/sub&gt; of all corner orientations) &lt;sup&gt;[3]&lt;/sup&gt;<br /> <br /> <br /> So combined together, these laws mean that:&lt;br/&gt;<br /> &lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt; * &lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt; * &lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;3&lt;/sub&gt; = &lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;12&lt;/sub&gt; of all states are reachable.<br /> <br /> <br /> '''Footnotes'''<br /> * [1] Assuming permutations are hypothetically solved using a sequence cubie swaps.<br /> * [2] An edge is defined as 'flipped' if cannot be solved using only &amp;lt;U, D, L R, F2, B2&amp;gt;. <br /> * [3] If an oriented corner is defined as one which can be solved using only &amp;lt;U, D, R2, L2, F2, B2&amp;gt;, the number of corner twists can be counted as the number of counter clockwise 120° twists required to orient it.<br /> <br /> <br /> == Recognising an Unsolvable Cube ==<br /> As a consequence of the laws above, if any of the following the end of a solve, cases appear at you can be certain the cube is not solvable with legal moves:<br /> <br /> <br /> # '''Only two pieces to swap'''<br /> # '''An odd number of edges to flip'''<br /> # '''Only a single corner to twist'''<br /> <br /> <br /> This also holds true for combinations of the above. For example, everything solved apart from a single flipped edge and a single twisted corner.<br /> <br /> == See Also ==<br /> * [[Random State Scramble]]<br /> * [[Permutations]]<br /> <br /> == External Links ==<br /> * [http://www.ryanheise.com/cube/cube_laws.html Laws of the Cube] (by Ryan Heise)<br /> * Wikipedia: [http://en.wikipedia.org/wiki/Parity_of_a_permutation Parity of a Permutation]<br /> * Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=14383 The laws of a Rubik's Cube]<br /> * Speedsolving.com: [http://www.speedsolving.com/forum/showthread.php?t=20782 Unreachable positions]<br /> <br /> [[Category:Puzzle Theory]]</div>

Bootmii