If you made them dice instead of balls, if they were numbered, if you had stated that we had sacks or cups or such that we already knew were of equal weight, if we had a pen or other marking device that would not make a detectable change in weight on the balls, or if ...

A typical (if not downright the most typical) way to render this problem is as a search for a counterfeit coin of unknown weight in a collection of fair coins; clearly, with coins, it is not difficult to keep track of the individual coins from one weighing to the next. (BTW, this "coin version" of the puzzle strikes me as realistic enough to suggest the possibility that the puzzle originally arose as an embellishment of a relatively common everyday task.)