What problems are easier to solve in a higher dimension, i.e. 3D vs 2D?

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.

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).

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$^{[1]}$ 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".