>>31786765The gym puzzle would be trying to figure out which store in the mall the gym leader works at. There'd be clothing stores, books stores, etc. and you have to collect hints from the employees, but you can only do that after you buy their favorite christmas gifts from other stores.
In the end it turns out the gym leader works at the salon, and she is coincidentally also one of the first people you talk to. She is cutting the hair of the Jynx that you battle when you fight her.