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At some point, and for a variety of reasons, most of our cats will refuse to use the litter box. Get your patience in check because these problems are often difficult to solve, especially if they are rooted non medical reasons. First things first.... have your veterinarian do a complete medical workup. This may include urinalysis, complete blood count, biochemical testing, and a thyroid test. It is important to rule out diabetes, urinary tract infection, and a variety of other causes. If no medical problems are detected it's time to turn attention to social and behavioral problems. Social and behavioral problems are not the same. Social problems involve those which derive from a cats normal behavior in their natural outdoor environment. Cats will spray, rub their scent and use bowel movements to announce their claim on a territory. Behavioral problems could be related to an aversion to litter type, too small a litter box, litter box covers, or unclean litter boxes. Cleanliness is one the most important factors, single cats should have their litter box scooped several times daily. Change the box at least every other day, then wash and dry several times weekly. Avoid scented litter and litter liners if possible. Use the largest litter box possible. Large boxes do not make the cat feel confined like a small box will. Cats are more likely to enter the box routinely if they do not feed cramped (avoid covered boxes).
Recent studies by Dr Jacqueline Neilson (Diplomate American College of Veterinary Behaviorists)
Revealed several basic items about cats and litter boxes.

1) Cats prefer finely granular sandlike material, also known as clumping or scoopable litter with minimal dust.
2) Cats probably prefer unscented materials.
3) Cats prefer bleach, fish, and cedar scents in the scented materials
4) Cats least prefer floral and citrus scents, so avoid these types of products.
5) Cats prefer lower odor boxes over higher odor boxes, so methods other than just scooping feces daily may be required to
solve litter box aversion problems.
6) Cats will preferentially use litter boxes with activated charcoal litter over those without.
7) Cats
significantly prefer odor control litters with activated charcoal over those with sodium bicarbonate (baking soda).
8) Cats seem to prefer larger boxes over smaller boxes although the difference was not statistically significant.

My final thoughts on Kitty cats and Litter box Problems
Most importantly ....... Do not forget that.......
Cats are cats. They are not dogs, horses or any other such creature. They have a mind of their own and are not necessarily about pleasing you. If you can accept this, then you are on your way to solving your problems. The above mentioned items are guidelines and not written in cement. Cats tend to be less scripted than other animals. You may have a cat that does not fit any of the molds above.

1) Do not underestimate the value of multiple litter boxes even if you have only one cat.
2) Do not be afraid to try variety. Different litters , boxes and box locations.
3) Odor control may be one of the most important factors in controlling litter box aversion problems.
4) I believe that the single most important act is thorough cleaning and rinsing of your cats' litter box as frequently as possible