What are some unsolved problems in math that seem easy at first glance (e.g., the Collatz conjecture)?
1. Here is an unsolved problem that is probably the easiest on this list to state in a formal language whose foundational object is the set:Suppose you have a collection of sets where for every two sets A and B in the collection, A∪B is also in the collection. (This is called the "union-closed" property.) If your collection is finite and non-empty, is it always true that there is some element that is present in at least half of the sets in the collection?If so, the "Union-closed Sets Conjecture" would be true, but this has been unresolved for more than 34 years. It's been verified for collections built from at most 12 elements, which I thought could be improved with a little dose of modern GPU power, but even that is hard.Amazingly, it's not even known if there must exist an element that is present in at least, say, 1% of the sets.2. As a bonus, here is a geometric problem: Is it possible to slice a cake into congruent pieces (more than 1) so that the center of the cake is in the interior of one of the pieces?By "cake" I mean a two dimensional disc, of course. Also, pieces are congruent if their interiors are identical after translation, rotation, and/or reflection.I first saw this problem at Unusual Cake Slicing, which also shows a solution to a related question: Can a cake be sliced into congruent pieces such that the center of the cake doesn't touch some of the pieces? (By "touch" it means that the center is in the closure of the piece.) It's also mentioned at the bottom of page 87 in the 1991 printing of "Unsolved Problems in Geometry" by Croft, Falconer, and Guy.