The Online Encyclopedia of Integer Sequences is an online database of integer-valued sequences that anyone can contribute to. It's got a lot of interesting problem statements, many with only partial solutions. I've compiled a short list of problems I've been interested in, with graphics included if possible.
A261752 (Knight's Domination): "Minimum number of knights on an n X n chessboard such that every square is attacked."
Examples of optimal tilings:
Other OEIS Problems I Find Interesting But Don't Have Any Graphics For
A181746: "List of dimensions for which there exist several non-isomorphic irreducible representations of E8.."
A343266 (proposed): "List of dimensions for which there exist 8 or more non-isomorphic irreducible representations of E6."