Let me explain:
First of all, admit that any map boundary can be distinguished if only four colors are needed for one side of a 2-D plane. -[Fundamental Theorem]
[Condition 1: Coarse Mobius Circle with Impermeable Color]
Take any small area with any shape in the circle, which can be regarded as a small 2-dimensional unilateral plane area.
We can draw any shape map of this area. According to the Basic Theorem, at least four colors are needed to distinguish the map of this area.
So for the whole Mobius circle, at least four colors are needed.
Then let's consider a few at most.
Because the circle is thick, opaque and has two sides, we can pretend that the whole circle is twisted by paper shells (double layers and one side means that the paper shells are stuck together back to back). Now we separate the two layers of the paper shell and get a paper ring.
If we are ants and walk along the paper circle, we will find that the paper circle is a two-dimensional road, but what is special about it is that I will go back to the original place after walking for a while, and there is nothing special about it.
So this paper circle is completely equivalent to a two-dimensional ring (the middle part of two concentric circles), so it returns to the two-dimensional plane. Since it is a part of the dimensional plane, it naturally satisfies [basic theorem]
So only four colors are needed.
[Condition 2: Mobius Circle of Color Penetration]
Similar to the analysis of condition 1, at least four colors must be met in a small area.
Then in a large area, we can study it better.
We also took apart two layers of paper in Mobius circle.
Due to the color penetration, we will find that half of them are exactly the same as the other half.
So we can cut off half. Of course, in order to meet the boundary conditions, we need to butt the remaining half of the two cuts. After docking, a new two-dimensional small circle will be generated.
The properties of this new 2-dimensional small circle are exactly the same as those of the previous Mobius circle.
So the final conclusion is still that only four colors are needed for cutting.