I only have a partial answer, as everyone else has covered this nicely. Here are just a couple more things to consider, and probably integrate with other answers.
The planet tumbles chaotically
There are a few moons in our solar system (one around Saturn, the rest around Pluto) that instead of rotating consistently around one axis in a periodic way, tumble and rotate every which way due to the gravitation of nearby bodies. This depends on the planet you're on being more potato shaped than round (and defying the IAU definition of a "planet" in the process), but its worth considering. Here is a source discussing how this applies to at least two of Pluto's moons:
He’d been looking at how Nix and Hydra, the two largest of Pluto’s small moons, got darker and brighter as they traveled through their orbits. With the help of some new computer simulations, he realized that these two oblong moons were chaotically rotating—tumbling rather than spinning like tops.
“If you lived on Nix or Hydra, you would not know if the sun is coming up tomorrow,” <PERSON> says. “Or it could rise in the west and set in the north.”
Only one other moon, Saturn’s moon Hyperion, is known to perform such an off-kilter jig in space. In the Plutonian system, Nix and Hydra tumble about because they orbit both Pluto and its abnormally large moon Charon, which together form a gravitational dumbbell that tugs and nudges the moons in random ways. Styx and Kerberos also orbit the binary system formed by Pluto and Charon, so they are probably tumbling as well, <PERSON> says.
The planet isn't round, but instead a torus
Amazingly, a planet shaped like a torus would be stable. It is extremely unlikely to form naturally, but if it did there is nothing about the shape that would cause it to be torn apart or crush itself into a ball. 196
Here is a source where someone confirmed the math based on earlier models. There is all sorts of cool stuff that happens to a planet with these characteristics, depending on where the torus you are and how it is tilted relative to the plane of its orbit around the star. You can even get all sorts of stable moon orbits passing through the hole in the middle. If you want to keep it a little closer to your original idea, you can have it be a circumbinary planet (it orbits two stars, which orbit each other. It goes in a circle around the pair, not a figure 8 between them). A few moons can reflect light on other parts of the planet too (+1 to the answer that mentioned tons of moons). Check out the link above. I've included a couple of the pictures from that page, but the full article mentions a ton of cool properties of a toroidal planet, including how it would be lit by its sun.
To tie it back to your original question, if the "day" of the toroidal planet and the "year" aren't a multiple of each other (Earth isn't either, but its close enough to fool people without accurate measurements), you could make predicting when a region gets sunlight very difficult for a culture that doesn't have a strong grasp of trigonometry.
Light being cast on a toroidal planet as it orbits a star.
An example of a crazy moon orbit around a toroidal planet. This is one of many stable orbits. 393