Q:

Famous Problems Where We Only Know the Elementary

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.

A:

The first four perfect numbers have been known for at least 2300 years.  Here are their prime factorizations:
$$\begin{align}
6 & = 2^{\hphantom{2}}\cdot 3 \\
28 & = 2^2\cdot 7 \\
496 & = 2^4\cdot 31 \\
8128 & = 2^6\cdot 127
\end{align}
$$
Any knucklehead, looking at these, or perhaps even just looking at the first two, could make the obvious conjecture that they are all of the form $2^{p−1}(2^p−1)$ for prime $2^p−1$. 
It's easy to show that all numbers of that type are perfect.  (And indeed, Fibonacci's Liber Abaci, published in 1202, makes this observation.)
All known perfect numbers are of that type. Euler proved that all even perfect numbers are of that type. But… 

A:

Is $\pi$ a normal number?

All we know is that rational numbers are not normal, and that $\pi$ is irrational.
You can just as well replace $\pi$ by many other constants, and we don't know much about any of them.

A:

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.

