First you really should have at least one litter box per cat (one extra beyond that is also a better idea).

There is more info here on litterboxes and cat behaviour than I can copy/paste. You'll need to sit down and read it for yourself. http://www.catinfo.org/litterbox.htm

For catching the culprit you could try a motion activated web cam pointed at the couch when you're out of the room or gone.

Otherwise try to get urine samples from all the cats and take that in, maybe one of them will show an infection and you can get that one treated and the problem may stop. If nothing else maybe elevated levels in the test may indicate which cat needs to go in for an exam.

Getting urine samples from cats isn't all that hard, I've done it myself and I have 7 cats.

Just put all of them in a room with you with only one litterbox, keep a container handy (may ask the vet for one) and the minute you see a cat get in the box get ready. Just slide the container in the urine stream, label that container with the cat's name, put it in the fridge, and let that cat out of the room. It may take a bit of time to get samples from all of them.

If that fails to produce samples from all the cats try shutting the cats in with you without a litterbox. Periodically take one cat at a time to a litterbox, having the container ready. If they don't go for you, back in the no litterbox room they go. Keep doing this until you have samples from all the cats. It's easier if you know which cats tend to go when. (In my case I have a few that like to go right after eating breakfast).

You don't want your urine samples to sit for more than 8-10 hours so timing may be important, check your vet for hours he/she is available to do a urinalysis. If you can only get a couple cat samples at a time then go ahead and have those run and continue to try to get the rest. It took me two days to get all 7 cats when I did it because of time available I had to do it.

Yes it will take a bit of time to get samples, and yes there's a good chance you might get your hand wet but TRUST ME, this method works, besides hands are washable.

Good luck