An alternate strategy is just to band just the large "ring" sidechains. There are two phenylalanine and two tyrosine sidechains in the puzzle, both featuring six-sided rings. Banding just these four rings can be enough to solve the puzzle. Hence the LOTR or Lord of the Rings strategy, although there are only four rings here, not nine.
You'll see also a couple of sort-of rings, histidine and proline. Banding histidine and proline may help if banding the four true rings isn't enough. (The protein in this puzzle doesn't have any tryptophan, the biggest ring of them all.)
When you start or reset the puzzle, you should be able to see the two phenylalanine and the two tyrosine sidechains. The phenylalanine sidechains are easy, since they're near the ends of the protein. One tyrosine is near a phenylalanine, and the other is on a sheet near the helix.
Try banding the phenylalanine first. Hover over the first "nub" of the sidechain (the beta carbon), and shift-click to start drawing the band. Just make a short band to start with, and keep it away from the rest of the protein.
When you hover over a sidechain, the corresponding sidechain opens up on the guide. Drag on the round end of the band to move the end toward the corresponding beta carbon of the guide. You'll need to keep rotating the protein, about 90 degrees each time, in order to get the band to the right spot.
It's easy to either delete the band your dragging, or accidentally band to part of the protein instead of the guide. If this happens, use Undo or the control-z shortcut to get back to the previous step. Holding down the alt key while dragging on the band end may also help avoid accidentally banding to the protein.
Another problem is that bands may become disabled if you accidentally hit the D keyboard shortcut. Toggling D a couple of times until you see "Disable Bands" on the icon will make sure all your bands are pulling.
After banding the four rings, use a fairly long Wiggle, followed by a Shake. Let the wiggle run until the counter in the upper left starts spinning. For the shake, a count of 2 is always enough. Repeat wiggle and shake one more time. This may be enough to solve the puzzle, but if not, disable the bands using the D shortcut and wiggle again.
Getting the band to the right spot still takes a bit of practice, but banding just four sidechains is a lot faster than trying to everything. If you're still having problems, try banding more sidechains. You can try the histidine and the prolines, and there are also some long lysine and arginine sidechains to tackle. Disabling or deleting the bands at the end will often to solve the puzzle when it's being stubborn.