# Site Swaps: Problems

*A while back somebody posted a request for site swap problems.
I wrote these at that time, and for some reason didn't post them.
They start out relatively easy and get harder. -- Jack Boyce*

1) I am juggling 868671 with clubs (yeah, right). How many do I have?
2) I have a bowling ball that I can only throw as a site swap '3'
throw. Which of the following patterns can I theoretically run, using
the bowling ball in addition to 3 normal balls:
a) 53 b) 6631 c) 633 d) 577131
3) You are doing a 4 ball fountain and decide you want to switch into
741, an excited state site swap. You can't just start throwing:
4444741741... Since the last fountain throw (4) will collide with the
first 1 you do. Some connecting throws are needed. What is the
shortest starting sequence for 741?
4) You want to get back into the fountain, from 741. What is the
shortest connecting (ending) sequence in this direction?
5) Go from the 4 ball fountain to 714. What are the shortest
connecting sequences (both directions)?
6) You are already doing 741 and want to switch directly into 714.
What is the shortest sequence for doing so? [You could just
concatenate the ending sequence found in (4) above and the starting
sequence in (5), but this is not the shortest solution.]
7) Is the trick 66671777161 simple? If not, which portion of the
pattern can be repeated within the larger trick?
Is the trick 6316131 simple? If not, which portion can be repeated?
8) There is 1 ground state 5 ball trick of length 1 (5), 2 of length 2
(55, 64), 6 of length 3 (555, 564, 645, 663, 744, 753), 24 of length
4, 120 of length 5, and 720 of length 6. Clearly the pattern is N =
L!, which is a big hint that L elements are being permuted. What L
things are permuted by ground state site swaps of length L? (Bear in
mind that L is not the number of balls.)
9) There are not 7! = 5040 ground state 5 ball patterns of length 7,
as the above pattern would suggest (the actual number is around
4300). Why does the pattern break down? Can you calculate (not by
brute force!) how many ground state patterns there are for L =
7,8,9,...?