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.
 A: One such example in PDEs is Kirchhoff's formula for the solution to the initial value problem for the wave equation:
\begin{equation}
\begin{cases}
\partial^2_t u - \Delta u = 0, & x \in \mathbb{R}^n,\ t \in \mathbb{R}, \\
u(0,x) = g(x), \\
\partial_t u (0, x) = h(x).
\end{cases}
\end{equation}
In space dimension $n=3$ it is relatively easy to derive the formula
\begin{equation}\tag{1}
u(t,x)=\frac{1}{4\pi t^2} \int_{\partial B(x;t)}\!\! \big[ 
t\cdot h(y) + g(y) + \nabla g(y)\cdot (y-x) \big]\, dS(y),
\end{equation}
which expresses the solution in terms of the initial data. $^{[1]}$ The same cannot be done directly for dimension $n=2$, though. The usual method to recover a formula analogous to (1) in the two-dimensional case is called method of descent; it works by embedding the two dimensional equation into a three dimensional space and then using (1).

$^{[1]}$ One can either exploit symmetries or use the Fourier transform. The first method is known as the “method of spherical means”; see e.g. Evans, Partial differential equations, chapter 2. For the latter method, see e.g. Folland's Real Analysis, chapter "Topics in Fourier analysis".
A: 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.
A: Counting regular convex polytopes
In 2D, there are infinitely many regular polygons.
In 3D, there are five regular convex polyhedra.
In 4D, there are six regular convex polytopes.
In 5D and above, there are only three.
I was quite surprised when I first learned this.
List of regular polytopes and compounds (Wikipedia)
