This is not to say that AES should not be trusted. AES is a fine cipher, it is efficient, and unless someone can show us a practical attack the heuristic evidence is pretty strong.
Now, as for Merkle puzzles, that system is not considered secure by cryptographic standards. A cryptographic construction is not secure unless it requires the adversary to do work that is exponential in some parameter in the system (the security parameter), while parties that know some secret (such a key) only do work that is polynomial in all parameters of the system. In the case of RSA, for example, parties that are aware of the secret key must do work that is cubic in the security parameter, while the adversary must do work that is exponential in the cube root of the security parameter. Whether such systems actually exist is still an open question, as it turns out; a positive answer to this question would imply that P does not equal NP. Cryptographers generally assume certain truths about complexity theory, beyond the P vs. NP problem, and cryptography research has actually opened new areas of complexity theory that are based on such assumptions (such as the notion of knowledge complexity, which emerged from the work on zero knowledge proof systems).
The reason I brought up Merkle puzzles is because they depend on a symmetric cipher or one-way hash function, giving them one foot in that first category.