The following descriptions of the Mathieu M12 finite simple group tantalizingly suggest an elegant (in contest to Oskar Van Deventer's Topsy Turvy and Number Planet which hide why the number 12 is special) implementation as a twisty puzzle, but the descriptions seem unclear. In particular, a 72 degree twist of the top 6 spheres immediately followed by an opposite twist of the bottom 6 is equivalent to just a 144 degree twist of the top 6 spheres.
http://cp4space.wordpress.com/2013/10/14/mathieu-groupoid/
"Twelve ball-bearings touch a central sphere in an icosahedral arrangement. M12 is generated by pairs of clockwise and anticlockwise twists (where we choose an equator orthogonal to a diameter joining opposite spheres, and rotate one of the two hemispheres)."
http://cp4space.wordpress.com/2013/09/12/leech-lattice/
"M12 has a nice construction based on the icosahedron. We place twelve ball-bearings around a central sphere (above), corresponding to the vertices of an icosahedron. An inverted twist is an operation where you choose a ball-bearing and its five neighbours, rotate them by 72°, and reflect the arrangement of ball-bearings in the centre of the central sphere. M12 is the group generated by an even number of inverted twists."
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