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    <p>NOTE! This answer was written before the rotation requirement. Proceed at your own caution :)</p> <p>Here's my initial look at the problem.</p> <p>The first thing to do is calculate the parity of the exterior - +1 if it would fit 'stripes in corners', -1 if it would fit 'solids in corners', +0 if it's the 8 ball. This gives us a range from +12 to -12, and we aim for the extreme we are closer to. (If +0, pick +12 or randomly)</p> <p>For example, this is +1 +1 +1 -1 -1 +1 -1 -1 -1 +1 +0 -1 and thus it's -1 leaning solids in corners:</p> <pre><code>x o x x o x o x o 8 x o o x o </code></pre> <p>The next thing to do is move the 8 ball into the center. <em>If you can do two adjacent swaps with it that moves two balls into position, rather than a single adjacent swap that moves just one ball into position (or a single non-adjacent if it's in a corner), do so.</em></p> <pre><code>x o x x o x 8 x o o x o o x o </code></pre> <p>After we move the 8 ball, all combinations of two adjacent swaps sharing a ball can be produced by a non adjacent swap identically, so we have to consider far less complexity at once.</p> <p>Order all remaining moves by this priority:</p> <p>-A swap between two adjacent balls on the outside is 'worth 4' (2 if it's our last)</p> <p>-A swap between two adjacent balls, one on the outside, is 'worth 2' (1 if it's our last)</p> <p>-A swap between two balls on the outside is 'worth 2'</p> <p>-A swap between two balls, one on the outside, is 'worth 1'</p> <p>And perform them from top to bottom.</p> <p>So we move the o on the top, left (4), the o on the right side in (2), the o on the bottom left in (2) and then swap the x top with the o in the middle (2). We ended up doing five swaps in a 2-2-1 series, so three moves.</p> <pre><code>o x o x o x 8 x x o o o x x o </code></pre> <p>(Notably, this one would have been solved just as fast if we aimed for stripes in corners.)</p> <pre><code>x x o o x o 8 o x o x o x o x </code></pre> <p>I think it's impossible to require 4 turns, but I haven't proven it to myself yet.</p> <p>Another worked example:</p> <p>This has a parity of +1 so we aim for stripes in corners:</p> <pre><code>8 o o o x o o o x o x x x x x </code></pre> <p>swap 8 ball with center x (1-)</p> <pre><code>x o o o x o o o x o 8 x x x x </code></pre> <p>swap two adjacent on outer, 4 points (1-1)</p> <pre><code>x o o o x o o o x x 8 x o x x </code></pre> <p>swap adjacent edge into center, 2 points (1-2-)</p> <pre><code>x o o o x o o x o x 8 x o x x </code></pre> <p>swap edge to edge, 2 points (1-2-1-)</p> <pre><code>x o x o x o o x o x 8 x o o x </code></pre> <p>3 moves.</p> <p>EDIT: This works quite well for the example in the opening post, solving it in two moves:</p> <p>This has a parity of +1 so we aim for stripes in corners:</p> <pre><code>x x o o x o o x o o o 8 x x x </code></pre> <p>Swap 8 with x on edge then with o in center (solving two edges) (2-)</p> <pre><code>x x o o x o o x o o 8 x x o x </code></pre> <p>swap adjacent o and x on top and bottom left (solving four edges) (2-2-)</p> <pre><code>x o x o x o o x o x 8 x o o x </code></pre> <p>2 moves.</p>
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    1. CO4 swaps, 2 time measures - (2-5/3-8)(12-11/4-3)
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    2. COI think this one requires 4 swaps: http://i.imgur.com/dLkQ3H2.png
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    3. COBlenders I solved in 3 moves with a halfie left over: http://pastebin.com/MfRRGshn
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