Knuth's Art of Computer Programming, vol 2 [1], not surprisingly, gives a thorough discussion of the balanced ternary system.
The solution for a nice brainteaser can be found quickly once one thinks about balanced trinary, here it is: "Using a balance scale, what is the minimum number of wheights needed to weigh any whole number of grams up to 40g?"
I often wonder if there is some notion of a basis of computation in mathematics. You can do stuff in binary, trinary, what about further out systems? What about working with functions/mappings which take more than two inputs. What can be said about the expressive power of these different ways of computing? Any one know where I should be looking for this kind of stuff?
If you are interested in "functions/mappings" then you can look at Lambda Calculus and work your way right up to modern functional programming languages:
I'm reasonably well versed in these topics, I found them unsatisfying, they don't capture the essence for me. I don't really know what I'm looking for I just know I haven't seen it yet.
