Have you ever hated puzzles in a video game, not because you don't like puzzles but because it got in the way of your reason for playing in the first place? This situation was something that I experienced regularly, and at first, my solution was to remove puzzles from games that didn't explicitly focus on puzzle-solving. However, after some more thinking, I decided that the problem wasn't the puzzles in non-puzzle games but rather the puzzles having vastly different game mechanics.
Puzzles designed for non-puzzle games should not serve just as a change of pace from the core game mechanics but also complement them. An excellent example of this is the bombs in Bomberman 64. While bombs in Bomberman 64 damage enemies and destroy the terrain, they also can be used as a springboard to get to higher or farther areas. Applying bombs to different contexts means that the same variables that affect combat also affect puzzles, such as the size, strength, and type of bomb the player uses. Employing the same game mechanics for puzzle-solving as everything else prevents puzzle-solving from feeling like a separate part of the game. Another excellent example of using the same core game mechanics for puzzles is "VFX Powers" in Viewtiful Joe. VFX Powers are abilities that affect the titular character's attacks and the environment. For example, the VFX Power "Slow" slows enemy attacks and lowers flying platforms by slowing down their propellers. Puzzles that utilize the core game mechanics are even more intuitive because players are already familiar with the tools to solve them.
Up to this point, I have been discussing non-puzzle games, but also worth mentioning are instances when puzzles are isolated from the story. Even well-designed puzzles can feel like a completely different game if they're separate from the narrative. I experienced this dissonance between puzzle solving and story in Professor Layton and the Curious Village. Professor Layton and the Curious Village is a puzzle adventure game that links story progression with solving brain teasers. The issue lies with the brain teasers in Professor Layton that often have no context within the story, making them nothing more than arbitrary puzzles. An excellent example of a puzzle adventure game with its story and puzzles in sync is Ghost Trick: Phantom Detective. In Ghost Trick, you play as a ghost that can save lives by traveling in time four minutes before a victim's death. However, before the player can attempt to save a life, they are forced to watch the events that led to the victim's death. This replay provides clues that help the player figure out how to save victims and reveals details about the plot simultaneously.
That doesn't mean the only way for puzzles to complement the story of a game is through its mechanics. Catherine is a puzzle adventure game with distinct puzzle-solving and social simulation segments. However, Catherine doesn't feel disjointed because the puzzles and the story are linked thematically. While socializing in a bar sounds unrelated to climbing blocks in a nightmare, the latter represents the events of the former. This results in story beats at the bar, raising expectations for the nightmares and vice versa.