If you have 4 points on a plane you can join them all with non-crossing paths, its possible. But if you try 5 points it seems like you cant do it. Prove that it doesn't work using Euler's equation. There's 5 vertices, 9 edges and 6 faces.

To prove that it is not possible to join 5 points on a plane with non-crossing paths, we can indeed use Euler's equation, also known as Euler's formula. Euler's formula states that for any planar graph, the number of vertices (V), the number of edges (E), and the number of faces (F) are related by the equation V - E + F = 2.

Let's apply Euler's formula to the case of 5 points on a plane. We have 5 vertices (V = 5), and we need to determine the number of edges and faces.

Each edge connects two vertices, so in order to calculate the number of edges, we need to consider how many possible connections there are among the 5 points. With 5 points, each vertex has 4 potential connections (to the other 4 points), resulting in a total of 20 possible edges. However, since we are looking for non-crossing paths, we must only count the edges that do not intersect with each other. For a graph to be planar, there can be at most 3 edges connecting any 2 points without crossing. Consequently, the maximum number of edges for 5 points is 3 * (5 choose 2), which equals 3 * 10 = 30.

Now, let's determine the number of faces. For our purposes, we consider unbounded regions as faces. For a planar graph, the number of faces can be calculated by adding 1 to the number of edges (F = E + 1).

Plugging the values into Euler's formula, we have:

V - E + F = 2
5 - E + (E + 1) = 2
5 - E + E + 1 = 2
6 = E

From this, we can conclude that E (edges) should be equal to 6. However, we established earlier that the maximum number of edges for 5 points on a plane is 30. Since 6 is less than 30, it indicates that it is not possible to join all 5 points with non-crossing paths.

Therefore, using Euler's equation, we have proven that connecting 5 points on a plane with non-crossing paths is not possible.