TITLE: Famous Problems Where We Only Know the Elementary
QUESTION [25 upvotes]: Define a graph with vertex set $\mathbb{R}^2$ and connect two vertices if they are unit distance apart. The famous Hadwiger-Nelson problem is to determine the chromatic number $\chi$ of this graph. For the problem as stated (there are many variations), the best known bounds are $4 \leq \chi \leq 7$. The lower bound comes from the existence of a clever subgraph on just seven vertices that is readily seen to require four colors. The upper bound comes from a straightforward tiling of the plane by monochromatic hexagons that admits a proper $7$-coloring.
I like to show this problem to young students because they are always fascinated by the fact that we can cover "everything" that is known about it in just a few minutes. What are some other problems of this type?

What are some famous problems for which the best known results are fairly obvious or elementary?

Update: Aubrey de Grey has recently improved the lower bound to 5 using an elaborate subgraph, so we now know slightly more than the elementary.

REPLY [10 votes]: Let $S$ be a set of integers, each greater than 1, and $m(S)$ be the largest positive integer that is not expressible as a sum of elements of $S$, if it exists. (Elements of $S$ may be used more than once, of course, so that $7 = 2+2+3$ is considered to be a sum of 2's and 3's.)  A well-known example is $m(\{6, 9 ,20\}) = 43$.
J.J. Sylvester showed (1884) that $m(S)$ exists if and only if $\gcd(S) = 1$, and also that $m(\{s_1, s_2\})$, when it exists, is equal to $$s_1s_2 - s_1 - s_2.$$ Both of these could certainly be conjectured, and probably proved, in an hour by a bright high-school student.
But for $|S| > 2$ the problem is wide open.