From Halopedia, the Halo wiki
An arum is a small device used to teach young Sangheili the value of patience when faced with a difficult problem. Part ornament and part puzzle, they are composed of nested concentric spheres which need to be aligned properly in order to release an object from its center (usually some type of gemlike crystal).
Arums can be designed with varying levels of difficulty, with ones made by the monks of the Ontom region of Sanghelios being regarded as the most difficult to solve. They also come in various styles, ranging from the simple pale spheres of Ontom to highly decorative affairs carved from a single block of material complete with display stand found in the Mdama region.
In addition to being both a diversion and a lesson in character, an arum was said to represent what made Sangheili strong; a perfectly engineered, orderly system that presented a smooth, impenetrable face to the outside world, and each had his appointed place in it. It seems that few Sangheili ever manage to release more than a few crystals from their prisons in their lives.
Professor Evan Phillips is the only human known to have solved an arum. He has also gained quite the reputation on Sanghelios as he has yet to tackle an arum that he is unable to solve in less than an hour. This has impressed and awed many Sangheili; one asked the Professor if all humans were capable of solving the puzzles so quickly. When Phillips answered in the negative the response was: "Good. Then you would be a much more dangerous species." This is confirmed when Serin Osman was given an arum to help her pass the time, and she not only was unable to solve the puzzle, but gave up after countless attempts.
Link with the Forerunners
There is evidence to suggest that the arum is based on a Forerunner design. A Forerunner structure in the state of Ontom featured a lock that strongly resembled an arum, consisting of layered concentric spheres that needed to be aligned; unlike an arum, however, this lock also included holes in the layers. Professor Evan Phillips theorized that puzzle-locks such as these were the inspiration for the arum.
The Forerunner Shield World Requiem, coincidentally or not, also bears a strong resemblance to an arum: the world is composed of several layers, and whereas an arum has at its center a valuable stone, Requiem holds the Didact's Cryptum at its core.
List of appearances