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I take all my old coop litter and dump it into my chicken run composting system. The paper shreds compost much, much faster than the wood chips did. Last spring, I cleaned out all the old, dirty, winter paper shreds deep bedding litter from the coop, and it was basically finished compost about 2 months later out in the chicken run. Wood chips take maybe 6 months to get to the same level. As you noted, my chicken run is not covered, so the rain keeps the litter moist which aids in composting faster. The chickens are constantly scratching and pecking in the chicken run litter, which turns it over and over and helps to break it down faster.
I am using paper shreds as deep bedding coop litter this winter as well. I just really like using free paper shreds. Paper shreds have no dust, no smell, absorb poo very well, and compost faster than the other litter options I have used.
I see you got a 24-sheet paper shredder

! Very nice! That should shred up just about anything that I can think of that comes into our house. Cardboard shreds are great, too, and I mix them up with my lighter paper shreds from newspapers, junk mail, and office paper.
My biggest paper shredder is rated for 12-sheets. In case you have not read all the previous posts, I will mention again that I bought a nice power cutter from Harbor Freight that I use to cut my cardboard into 2–3-inch-wide strips, and then I feed it down the middle of the shredder slot which on my machine is where you would also feed in something like a credit card. Many companies make these power cutters, so I would suggest looking for the best price if you want to buy one.
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The power cutter just makes it very easy to cut that heavy packing cardboard boxes. I have a big, heavy-duty, manual scissors but my old hands start to cramp up after a short while on cutting heavy cardboard. Since I bought this power cutter, I don't think one cardboard box has been sent to the recycle center from our house. All those Amazon and QVC shipping boxes get turned into chicken coop litter.
With the power cutter, I can easily cut up a cardboard box into strips in no time and shred them up even in my 12-sheet shredder. In fact, after cutting the cardboard into narrow strips, I can even feed them down my 8-sheet shredder. If you listen to motor, you can tell if the paper shredder is bogging down or not. There has been no problem with my paper shredders cutting the heavy cardboard into those 2-3-inch-wide strips.
Have you decided to use paper shreds in your coop? Would love some feedback on how well it works for you. Also, let me suggest you add your geographic area to your member icon. It helps to know where people are from because how I raise my chickens in northern Minnesota might be completely different from someone in the desert of Arizona, for example. As fast as paper shreds turn into compost for me in Minnesota, they might not compost nearly as fast as someone living in a dry, arid, area unless you water them down to keep them moist.
Again, welcome to the BYC forums and hope to hear more from you in this and other threads.