Rubik's Clock is a puzzle consisting of 18 small clocks, 14 of which are independent, each of which may be set to any 12-hour position. A configuration records only clock-hand
positions, giving 
possible configurations. One move turns a wheel by any nonzero number of hours, with
the four pins selecting which clocks move. Changing the pins costs no move. The God's number (i.e., the graph
diameter of the graph corresponding to Rubik's Clock,
which is the minimum number of moves required to solve it from an arbitrary starting
position-i.e., in the worst case) was shown by Kogler to be 12 (Kogler 2014; cube20.org).
The numbers of positions requiring exactly 
, 1, ..., 12 moves in a shortest solution are 1, 330, 51651,
4947912, 317141342, 14054473232, 428862722294, 8621633953202, 101600180118726, 528107928328516,
613251601892918, 31893880879492, and 39248 (OEIS A256586;
cube20.org), which sum to 
as they must.
