Rodolfo Kurchan: Number cards 1 to 12, and arrange them in a circle counterclockwise. To move, choose a card, count
forward clockwise the number on the card, and exchange the card with the card that finishes the count. For example,
if we choose card 4, and finish on card 12, then exchange card 4 with card 12. GOAL: Move the cards to the normal
clock position.
It is worth noting at the start that picking the 1 causes it to swap with the next number ahead of it clockwise.
Repeating this action causes it to move forward around the circle while leaving the other numbers in order, but
shifting them back 1 space. This can be used (a) to position the 1 at any desired position among the other numbers,
and (b) by repeating it in multiples of 11, to shift the entire circle counterclockwise one or more positions. So
the issue of where the original layout starts relative to the finished clock face is irrelevant because once the
numbers are clockwise you can easily shift them to any desired position. Moreover, picking the 11 has a similar
effect, but in the opposite direction.
What is even more worth noting is that in the starting position, any number picked swaps with the 12. So pick any
number other than the 12 or 6. Suppose we pick the 7.
12 11 10 9 8 7 6 5 4 3 2 1 7 11 10 9 8 12 6 5 4 3 2 1
Now place the 7 where we want it to go by picking the 5.
7 11 10 9 8 12 6 5 4 3 2 1 5 11 10 9 8 12 6 7 4 3 2 1
And now put the 5 in its correct place by picking the 5 again.
5 11 10 9 8 12 6 7 4 3 2 1 12 11 10 9 8 5 6 7 4 3 2 1
This mechanism works for swapping each pair (8,4), (9,3), (10,2), (11,1) which needs to be swapped. So the entire
thing is done in 15 moves if we assume the 12 starts in the right place.