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The first step in completing an escape room is one of the most important steps. I call it inventory, and it has two parts: search and connect.
Begin by scouring the room for any element that might be a puzzle, hint, tool, or lock. Look under furniture, dummies, props, and rugs. Look in every pocket, drawer, door, window, and case that will open, and check to see if you can move the pictures, books, blinds, or wall panels. Look behind opened doors and on every side of every item in the room. We have struggled many times because we didn't spend enough time searching the room. We've failed to move furniture because we simply assumed they wouldn't move, and we've avoided searching dummies because they made us nervous. Don't cut your search time short. Move methodically around a room and search everything. When the search is complete, make sure everyone in the room knows about everything that was found.
Then it's time to connect puzzles and tools to locks. This sounds simple, but this skill is largely what makes experienced players better than beginners. Unfortunately, some game designers don't do a good job of giving hints that connect puzzles to locks. Those rooms requires a lot of trial and error, but the good rooms provides hints that connect puzzles to locks. You usually follow those hints by looking for symbols, colors, or designs that connect elements to each other.
Did you find a few wooden blocks or dominoes? Compile them then figure out if those blocks fit in anything in the room. This will often reveal whether or not you have all the pieces. Do the pieces connect together, or do you see two or more blocks that have the same symbols on them? If yes, they are likely puzzle pieces. Attempt to put them all together. If they don't cleanly fit together, there's a good chance you don't have all the pieces yet.
Did you find some playing cards or pictures? Look to see if there are elements in the room that have the same symbols as the cards. If they are playing cards, look for black and red elements or elements that have diamonds, hearts, spades (sometimes look like upside-down hearts), or clovers on them. If they are pictures, are there elements or people in the picture that you can count? If so, you can often treat them like play cards. Also check to see if there is anything on the back of the cards. When you put the cards in order, do the backs create a design?
Did you find a sheet (paper, board, metal, etc.) that has holes in it? Check to see if that sheet fits in anything, is the same size (length/width) as something else in the room, or has similar symbols as something else in the room. Place the sheet on objects (usually objects with letters, numbers, or symbols on them) to see if the holes reveal which letters, numbers, or symbols on the object are important. If you were given something that fits into the holes, there's a chance that something else in the room will tell you which holes should be covered.
Good players are always watching for elements that connect to other elements. Even when a room designer intentionally places red herrings (distractions) in a room (a cheap trick that is used to steal your time and money), the game designer will use symbols, colors, or some kind of image to show you which items should be focused on. If you have a key with an elephant tusk keychain on it, look to see if there are any elephants in the room. If you have a pile of severed hands with hooks on them, look to see if there are bloody hooks somewhere to hang them from. If there's a blue ball and a blue shelf, see if they can interact with each other in some way. If there's a paddle with a triangle on it, and a triangle shaped sign, connect them together.
When searching, you also have the option to search for elements that are intended to be surprise elements. If you're willing to remove some of the immersion and intrigue, you can search for trap doors, wires coming out of desk drawers, books shelves that look like they will eventually move, and other creative design elements. Knowing where these elements are will sometimes help you determine what elements you need to interact with.
If you have a key card, and you know the book shelf will eventually move, you might discover that swiping the key card in a box on the shelf will open the shelf. When you know there are red herrings in a room, it's a good idea to start by looking for locks then find keys/codes/solutions for those locks. Otherwise you'll spend time solving puzzles that don't need to be solved. A hidden door is a kind of lock, and knowing where it is and what it is connected to will help you determine which puzzle will open that door.
I wrote another post about some common puzzles to watch for here, but those puzzles include morse code, pigpen, Caesar encryptions, and more.