TITLE: Open math problems which high school students can understand
QUESTION [49 upvotes]: I request people to list some moderately and/or very famous open problems which high school students,perhaps with enough contest math background, can understand, classified by categories as on arxiv.org. Please include statement of the theorems,if possible, and if there are specific terms, please state what they mean.
Thank you.I am quite inquisitive to know about them and I asked this question after seeing how Andrew J.Wiles was fascinated by Fermat's last theorem back in high school.

REPLY [4 votes]: An open problem I find surprising, the PAC (Perimeter to Area Conjecture) due to Keleti (1998):

Conjecture: The perimeter to area ratio of the union of finitely many unit squares in the plane does not exceed 4.

See for example Bounded - Yes, but 4? and references therein.