While I agree that puzzles and mind games are silly ways to appraise coding skills, they do give an insight about a person's raw intelligence, or knowledge, or potential. As CS is an application of math and programming is an application of CS, being good in math does not necessarily mean proficiency in it's application; same goes for CS. IMO, a good programmer must, at least, have:

+ knowledge - generally mastery of math/CS concepts and can be thought of as the potential

+ application skills - modeling a real world problem into a theoretical, computable, and (ultimately) programmable form

+ execution skills - implementation (coding) of a solution including the ability to utilize requisite tools/technologies such programming languages, DBs, OS, and so on

That said, hiring process should cover each of these areas and programmers should work on all these as well.