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.

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)
