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A sundial works because the Sun moves across the sky as the Earth rotates. This causes shadows to change position over the course of a day. Sundials use the movement of shadows to tell the time. Because the length of the day and position of the sun in the sky varies from one season to another and with latitude, a sundial needs to be built to take these factors into account. However, there is no reason that you can't build one to work on the equator.
Sure, it will work anywhere if properly installed and aligned. You can even get small, portable ones -- pocketsize, with a peg or unfolding piece or something for the gnomon (the thing that casts a shadow). If you know where north is it will tell the time, or if you know the time it will indicate north (as will your watch, come to that). And sun compasses, similar idea, used to be used by early aviators in very high latitudes where magnetic compasses are no longer helpful.
Incidentally, you may have had the experience of checking a reliable watch against a sundial and finding a few minutes difference (even assuming there's no daylight saving time complication). Neither one is wrong: the sundial is right for the exact longitude where it's installed, and your watch is right for the exact middle of the time zone you're in.
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