To find the number of regions (or "faces") created when this planar graph is drawn without intersecting edges, we can use Euler's formula for planar graphs:

$V - E + F = 2$

where:

• $V$ is the number of vertices,
• $E$ is the number of edges,
• $F$ is the number of faces.

Step 1: Count the vertices ($V$)

The graph has 8 labeled vertices (1 through 8), so $V = 8$.

Step 2: Count the edges ($E$)

Each vertex is connected to every other vertex, forming a complete graph, $K_8$. For a complete graph $K_n$, the number of edges $E$ is given by:

$E = \frac{n(n-1)}{2}$

For $n = 8$:

$E = \frac{8 \times 7}{2} = 28$

Step 3: Apply Euler’s formula

Using Euler's formula:

$8 - 28 + F = 2$

Solving for $F$:

$F = 2 + 28 - 8 = 22$

Answer

The number of faces ($F$) is 22.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How would Euler's formula change if the graph were not planar?
2. What is the significance of Euler's formula in graph theory?
3. How many faces would a complete graph $K_7$ have if it were planar?
4. What happens to the number of faces if an additional vertex is added to this graph?
5. How can we visually confirm the number of regions in a planar graph?

Tip: Euler’s formula is a powerful tool for analyzing planar graphs and can simplify complex calculations by connecting vertices, edges, and faces in a straightforward relationship.