Imagine a standard QWERTY keyboard on which a vertical line was drawn down, beginning between the digits 5 and 6; thus dividing the keyboard into the left and right sides.
In respect to this division any word can be matched by a couple of integers L and R, specifying the quantity of distinct letters in the left and right side respectively.
Examples: Honolulu >> (0,5); letters >>(4,1); word >>(3,1).

Now, answer this:
1. List all American one-word states that are matched by a couple, containing 0 either as L or R.
2. Same for world cities (replacing “all” by “two or more”.
3. Chose a certain category ( fruits, animals, famous people et al) and provide at least 3 members of this category for which abs(L-R)=2
4. What one-word names of numbers satisfy (L-R)^2=1? Hint: “one” does. Others?