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Quick Answer. According to Good Housekeeping, brands of toilet paper that dissolve especially quickly include Scott 1000 one-ply, Target Up & Up two-ply, Seventh Generation two-ply, Scott Naturals two-ply and White Cloud Ultra three-ply. Contrary to expectations, one-ply toilet paper did not dissolve fastest in Good Housekeeping's tests, as of 2014.
It is basically Hydrochloric acid that IMMEDIATELY eats away the hard water deposits (that's what the ring is, by the way, and CLR is not powerful enough to dissolve it).  You pour the sizzle into the overflow tube inside your toilet tank (the tube sticking out of the water), and it will flow down inside the bowl.