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.

This isn't "famous", but it's long been of interest to me:

What is the Pythagorean Theorem for right-corner simplices in hyperbolic $d$-space?



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*For $d=2$, we have ... $$\cosh a \cosh b = \cosh c$$ for a triangle with legs of length $a$, $b$, and hypotenuse of length $c$.

*For $d=3$, we have ... $$\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2} - \sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2} = \cos\frac{D}{2}$$ for a tetrahedron with right-triangular leg-faces of area $A$, $B$, $C$, and hypotenuse-face of area $D$.

*For $d\geq 4$, we have ... $$\text{No Idea. (So far as I know.)}$$ 


Three(!) years ago, I posted some thoughts about the $d=4$ case to MathOverflow. (The series stuff is probably mis-guided.) I've made no progress since then.
