Question

the following graph is planar. (trust me.) if it is drawn with no intersecting edges, into how many regions (\faces\) does it cut up the plane?
2
3
faces =
1
4
8
5
7
6
submit question

Explanation:

Step1: Identify Vertices (V), Edges (E)

Vertices (V): 8 (points 1,2,3,4,5,6,7,8).
Edges (E): Let's count. Each vertex connects:
1: 2,8,3,6 (4)
2: 1,3,4 (3)
3: 1,2,4,6,8 (5)
4: 2,3,5,6 (4)
5: 4,6,7 (3)
6: 1,3,4,5,7,8 (6)
7: 5,6,8 (3)
8: 1,3,6,7 (4)
Total edges: \(\frac{4 + 3 + 5 + 4 + 3 + 6 + 3 + 4}{2} = \frac{32}{2} = 16\).

Step2: Apply Euler's Formula for Planar Graphs

Euler's formula: \(V - E + F = 2\) (for connected planar graphs, which this is).
We have \(V = 8\), \(E = 16\).
Solve for \(F\): \(8 - 16 + F = 2\) → \(F = 2 + 16 - 8 = 10\). Wait, no—wait, Euler's formula is \(V - E + F = 2\), so \(F = E - V + 2\). Wait, \(8 - 16 + F = 2\) → \(F = 2 + 16 - 8 = 10\)? Wait, no, let's recalculate: \(V - E + F = 2\) → \(F = E - V + 2\). So \(16 - 8 + 2 = 10\)? Wait, but maybe I miscounted edges. Let's check again. Alternatively, maybe the graph is a complete graph? No, 8 vertices, but edges: let's list all edges:
1-2, 1-8, 1-3, 1-6,
2-3, 2-4,
3-4, 3-6, 3-8,
4-5, 4-6,
5-6, 5-7,
6-7, 6-8,
7-8.
Wait, that's 4 (1) + 2 (2) + 3 (3) + 2 (4) + 2 (5) + 3 (6) + 2 (7) + 1 (8)? No, better to count:
1-2, 1-8, 1-3, 1-6 (4)
2-3, 2-4 (2) → total 6
3-4, 3-6, 3-8 (3) → total 9
4-5, 4-6 (2) → total 11
5-6, 5-7 (2) → total 13
6-7, 6-8 (2) → total 15
7-8 (1) → total 16. Yes, 16 edges.

So \(V = 8\), \(E = 16\). Euler's formula: \(8 - 16 + F = 2\) → \(F = 10\). Wait, but let's confirm. Alternatively, maybe the graph is connected, so Euler's formula applies. So \(F = E - V + 2 = 16 - 8 + 2 = 10\). Wait, but let's check with a smaller graph. For a triangle (3 vertices, 3 edges), \(F = 3 - 3 + 2 = 2\) (inside and outside), which is correct. For a square (4 vertices, 4 edges), \(F = 4 - 4 + 2 = 2\)? No, wait, square with a diagonal: 4 vertices, 5 edges. Then \(F = 5 - 4 + 2 = 3\) (inside, outside, and the triangle). So formula works. So here, \(V=8\), \(E=16\), so \(F=16 - 8 + 2 = 10\). Wait, but maybe I made a mistake. Wait, the problem says "trust me, it's planar". So Euler's formula: \(V - E + F = 2\). So \(F = E - V + 2\). So 16 - 8 + 2 = 10. So the number of faces is 10? Wait, no, wait: the "outer face" is included. So yes, Euler's formula counts all faces, including the outer one. So the answer should be 10? Wait, let's check again. Wait, maybe my edge count is wrong. Let's list all edges:

1 connected to 2,8,3,6 (4)
2 connected to 1,3,4 (3) → edges 1-2, 2-3, 2-4
3 connected to 1,2,4,6,8 (5) → edges 3-1, 3-2, 3-4, 3-6, 3-8
4 connected to 2,3,5,6 (4) → edges 4-2, 4-3, 4-5, 4-6
5 connected to 4,6,7 (3) → edges 5-4, 5-6, 5-7
6 connected to 1,3,4,5,7,8 (6) → edges 6-1, 6-3, 6-4, 6-5, 6-7, 6-8
7 connected to 5,6,8 (3) → edges 7-5, 7-6, 7-8
8 connected to 1,3,6,7 (4) → edges 8-1, 8-3, 8-6, 8-7

Now, count unique edges:
1-2, 1-8, 1-3, 1-6,
2-3, 2-4,
3-4, 3-6, 3-8,
4-5, 4-6,
5-6, 5-7,
6-7, 6-8,
7-8.

That's 4 (1) + 2 (2) + 3 (3) + 2 (4) + 2 (5) + 2 (6) + 1 (7) + 1 (8)? Wait, no, let's count each edge once:

1-2, 1-8, 1-3, 1-6,
2-3, 2-4,
3-4, 3-6, 3-8,
4-5, 4-6,
5-6, 5-7,
6-7, 6-8,
7-8.

Let's count:

1. 1-2
2. 1-8
3. 1-3
4. 1-6
5. 2-3
6. 2-4
7. 3-4
8. 3-6
9. 3-8
10. 4-5
11. 4-6
12. 5-6
13. 5-7
14. 6-7
15. 6-8
16. 7-8

Yes, 16 edges. So \(V=8\), \(E=16\). Then \(F = E - V + 2 = 16 - 8 + 2 = 10\). So the number of faces is 10.

Answer:

10