I had to look up "FizzBuzz" to know what you mean, but you are correct. (I have actually been asked to write programs like this, and this is great even if it is minimal. I think the questions may need to be a bit more complex than this, but at least it is possible to solve in one sitting.) Asking an applicant to write simple programs like this would be good. Or, if the person is a front-end JavaScript developer, you can ask them how they would handle thumbnail mouseovers to display larger versions of the images in a separate div. Things like this are fine and have purpose. Asking a database administrator about the difference between joins makes sense. Another question might be how would you write a class to represent a deck of cards including methods for shuffling the deck. In comparison, sometimes you will have technical questions thrown at you that have no real world applications or literal brain teasers, and I think those types of questions are pointless. There is a difference between asking programming questions and asking programming puzzles, and I wanted to voice the opinion that puzzles should not be misconstrued to the point of suggesting that people cannot program because they cannot solve them.

Real programming projects are best. I think one of the best methods would be to give a person a reasonable assignment that should only take a couple hours to complete (if that long) and ask for the results within a few days. (This will give them plenty of time to clean it up, add surprising features, and create a nice design and front-end after they are finished with it. A day can be used for the promise of completion, and the remaining time can be used to overdeliver.)