Our first cat was a kitten found outside. We didn't train her to use the litter box. We just put down a litter box, put her in it... and the rest is history.
Any new addition, we just showed him/her where the litter boxes were located. No problems.
I think cleanliness and location are important factors. We have a box pretty much in every room of the house, excluding the bathrooms. There is always access to a box... wherever a cat may be hanging out.
We currently have 5 boxes and 7 cats. We used to have 6 boxes. I recently tossed one, and and haven't replaced it yet. No one seems to notice we are down one box. lol Obviously, I don't go by the "a box + one for every cat." I think the number of boxes work for us, due to locations and cleaning schedule.
We have covered and uncovered boxes. There doesn't
seem to be a preference. BUT our covered boxes are extra extra large. The litter (Dr. Elsey's) is virtually dust free. It has no perfume-y or pine smells. The cats don't need to plug their noses when using the potty.
I will NEVER (again) use a litter that has ANY kind of perfume-y, pine, etc. smell. A litter box will NOT smell, if it is scooped and washed out on a regular basis.
By the way, when I switched to Dr. Elseys litter (blue bag)...the ones sneezing stopped. If purchased in the 40 pound bag, it is OFTEN cheaper than most other litters. I mention that because, I don't know how many times I have heard..."But it is soooo expensive." I don't know what "they" are buying, but is it totally in-line (price-wise) with other name brands.
Also, the ones that would potty and SHOOT out of the box... stopped. Best of all, everyone takes the time to cover their "deposits." When I used ANY other brand of litter... someone would sneeze, shoot, or not cover.
I personally think the number of boxes and location ...pretty much depends on the cat(s).
If a
healthy cat is eliminating outside of the box (assuming it is kept clean), YOU are doing something wrong. Add boxes, change litter brands, change location, etc.