What problems are easier to solve in a higher dimension, i.e. 3D vs 2D? I'd be interested in knowing if there are any problems that are easier to solve in a higher dimension, i.e. using solutions in a higher dimension that don't have an equally optimal counterpart in a lower dimension, particularly common (or uncommon) geometry and discrete math problems.

The kissing number problem asks how many unit spheres can simultaneously touch a certain  other  unit sphere, in $n$ dimensions.
The $n=2$ case is easy; the $n=3$ case was a famous open problem for 300 years; the $n=4$ case was only resolved a few years ago, and the problem is still open for $n>4$…  except for $n=8$ and $n=24$.  The $n=24$ case is (relatively) simple because of the existence of the 24-dimensional Leech lattice, which  owes its existence to the miraculous fact that $$\sum_{i=1}^{\color{red}{24}} i^2 = 70^2 .$$  The Leech lattice has a particularly symmetrical 8-dimensional sublattice, the $E_8$ lattice and this accounts for the problem being solved for $n=8$.
There are a lot of similar kinds of packing problems that are unsolved except in 8 and 24 dimensions, for similar  reasons.
