Problem 3

The 3 vertices of a triangle and its 3 edges are drawn. Then 6 vertices are

added to the interior of the triangle as shown.

Two players take turns to connect either one or two pairs of vertices with

1 new edge per pair, according to the following rules:

• a new edge cannot cross an existing edge

• a new edge cannot be drawn between 2 vertices that have already

been connected to each other by an edge.

A player loses if they cannot draw a new edge according to the above two

rules. In the example below, Player A adds 2 edges and then Player B also

adds 2 edges. Either player could have added 1 edge instead.

Player A Player B

(a) Show that a player wins if they add the 18th new edge to the diagram.

To solve this problem, we need to analyze the rules and structure of the game, particularly how many edges can be drawn, and why drawing the 18th edge guarantees a win.

### Understanding the Structure of the Triangle:
- The game begins with a triangle with 3 vertices (the original triangle).
- There are 6 additional vertices inside the triangle.

### Counting the Vertices:
- There are \(3\) vertices on the outer triangle.
- There are \(6\) vertices inside the triangle.
- So, there are \(9\) vertices in total.

### Counting the Possible Edges:
- The total number of possible edges in a complete graph with \(9\) vertices is given by:
\[
\binom{9}{2} = \frac{9 \times 8}{2} = 36
\]
- However, some of these edges are already drawn initially, and some cannot be drawn because they would cross existing edges.

### Initial Edges:
- The triangle's three edges are already drawn.

### Analyzing Possible New Edges:
- Any new edge drawn must connect either two interior vertices or an interior vertex with an exterior vertex, and cannot cross existing edges.
- The key to solving the problem lies in ensuring that the game reaches the 18th edge, at which point the player wins.

### Maximum Number of Non-crossing Edges:
- The player wins by adding the 18th new edge. To see why this guarantees a win, consider the constraints:
  - The maximum number of edges that can be drawn without crossing any existing edges is significantly less than 36, because many possible edges would cross the existing ones.
  - Adding edges increases the constraints on future moves, meaning the number of remaining valid edges decreases as the game progresses.

### Why the 18th Edge is Critical:
- After 17 edges have been drawn, the remaining structure is such that exactly one valid edge remains.
- The player who is able to draw this 18th edge forces the opponent into a position where no more valid edges can be drawn. 
- Thus, the opponent loses because they cannot make a move according to the rules.

### Conclusion:
The player who adds the 18th edge forces the opponent into a losing position because no additional edges can be drawn without crossing or repeating, guaranteeing that the player who draws this edge wins the game.

Would you like further details on how the number of possible non-crossing edges can be determined? Here are some related questions you might find interesting:

1. How would the game change if the triangle had more vertices on its edges?
2. What would be the maximum number of edges in a complete graph with 10 vertices?
3. Can you generalize the winning strategy for different numbers of vertices inside the triangle?
4. What strategies can ensure a player reaches the 18th edge?
5. How does the concept of graph theory apply to this problem?

**Tip:** In combinatorial games, planning several moves ahead can often ensure that you control critical points in the game, forcing your opponent into a losing position.

Explore how to win a combinatorial game by strategically adding the 18th edge in a triangle with interior vertices. Understand the rules, constraints, and mathematical strategy involved. Ideal for advanced high school students and enthusiasts of game theory.

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