I'm writing a Earth history time travel story, and the main character needs to solve puzzles from each time period he visits. I'm going very far back in history and I'm concerned about being able to design puzzles for the Mesopotamian and Mayan eras. Where can I take inspiration where history is lacking? I'm trying to be historically accurate as possible. (it's for a game, I need to actually create these puzzles too)
I would suggest researching known puzzles from the time periods you are writing about. There are a lot of materials on Google about puzzles from different time periods. You might also consider asking around on Stack Exchange's main Puzzle site or in it's chat room to see if any puzzle aficionados have any good ideas.
This book Ancient Puzzles covers 10 centuries of puzzles!
"Comprising the entire brainteasing spectrum from the silly and quirky to the clever and profound and drawing on classic wisdom from around the world and from as far back as 7,000 B.C., this collection features puzzles via Hebrew mystics, medieval Muslims, Benjamin Franklin, Sun Tsu, and more."
In order for your puzzles to be historically accurate, they need to
- be made using technology readily available at the time
- not require knowledge unheard of at the time
Word puzzles, such as riddles or cryptography, being puzzles made from words only, can exist at any time or place--no extra technology required. Some word puzzles require creative thinking (lateral thinking or thinking "outside of the box") such as the Riddle of the Sphinx. Other word puzzles, such as Knights and Knaves, require logical, deductive reasoning to solve.
Mechanical puzzles, however, require some level of technology and so are more time sensitive. The oldest known mechanical puzzle supposedly comes from Greece and appeared in the 3rd century BC. The game consists of a square divided into 14 parts, and the aim was to create different shapes from these pieces. This could be made from simple materials like clay or wood. Some puzzles required more advanced technology to make. For example, the Gordian Knot was a puzzle made out of rope, so rope was required. The Baguenaudier was a puzzle made from metal and rope, so metal was required.
Some things revolving around poorly understood or not widely understood scientific principles might not seem like puzzles immediately, but could possibly be considered puzzling enough to early cultures. Math, chemistry, logic, etc. could make for some good puzzles by asking characters about a certain technological/scientific principle not widely known at the time. For example, if you research the history of calculus, you can find it's date of origin or dates different principles were known to exist, and then search for known, existing puzzles formulated around those principles.
Our modern-day knowledge of ancient languages is, to put it lightly, incomplete. Even for languages as famous and prolific as Latin, there are words scholars don't fully understand. Ancus appears only once in all known classical texts, describing weapons that aren't being raised...but what does it actually mean? And this was the lingua franca of Europe for centuries. The Hebrew Bible contains over 1500 words which aren't attested anywhere else, and their meaning is still debated. After all, there aren't many ancient Hebrews around to explain them.
Depending when you arrive in Mesopotamia, for instance, cuneiform writing will be your best resource. The most important variants have been deciphered, and a time traveler could be taught to read before departing (and given some pre-written messages, just in case). Let's say you bring along a clay tablet saying "I am an emissary from a distant land, and must speak to the governor as soon as possible".
But finding someone who can read won't be trivial. While the writing system has been deciphered, we still don't know precisely how it was spoken. For example, there are at least seven different symbols which were probably pronounced gu, but aren't interchangeable. Were there tone differences, like in Mandarin? Or subtly different vowel sounds, or nasalization? Or were these actually just homophones, like English "to" and "too"? We don't know. The usual convention is just to transcribe them as gu, gú, gù...and leave it at that.
This would be an opportunity for a larger puzzle. You can't speak the language, only write it, and most of the people around you can't read. So you need to not be arrested until you can find someone who can communicate in writing, and hopefully teach you enough spoken words to get by. Luckily for you as the author, literacy levels varied greatly over time, so you can justify whatever level of difficulty makes the puzzle interesting. Alternately, you could create cryptograms or a decoding puzzle from (a small subset of) the syllabic characters, or require players to learn the correlation between sound and writing by listening to someone reading an announcement out loud.
Hopefully this helps. If you provide more details about your game, or what types of puzzles you're looking for, that would be useful.
Here are some thoughts:
Some Mesopotamian cultures used cylinder seals, which show puzzle potential. (Perhaps having to figure out how an image will look once reversed and flattened? Or maybe something akin to one of those cylindrical combination puzzles?)
Many Mesoamerican peoples played various forms of a ball game, including the Maya. This was already mentioned in comments, but I highlight it bc it was very culturally significant.
Mayans kept cyclic 365-day and 260-day calendars which we well understand today. However, I've always felt the concentric circle design itself resembled a sort of puzzle. See here or here, for example.