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.

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

Let's tackle this problem step by step to show why a player who adds the 18th edge wins.

### Step 1: Analyze the Diagram
Consider the triangle with 3 vertices and 6 additional vertices inside it. The total number of vertices is \(9\). The players will be drawing edges between these vertices.

### Step 2: Maximum Number of Edges Without Crossing
To understand the game, it's important to know the maximum number of edges that can be drawn without violating the rules (no crossing edges, no repeated edges between the same pair of vertices). 

For any \(n\) points in general position (i.e., no three points are collinear), the maximum number of non-crossing edges that can be drawn is given by:

\[
\text{Maximum number of non-crossing edges} = 3n - 6
\]

In our case, \(n = 9\) (since we have 9 vertices), so the maximum number of non-crossing edges is:

\[
3(9) - 6 = 27 - 6 = 21
\]

This calculation gives us the upper bound, but remember that some edges are already used up by the original triangle.

### Step 3: Pre-Existing Edges
The problem states that the 3 vertices of the triangle and its 3 edges are already drawn. So we already have 3 edges.

### Step 4: Number of Edges Left to Draw
Given that the maximum number of non-crossing edges is 21 and 3 edges already exist, the players can draw a maximum of:

\[
21 - 3 = 18 \text{ new edges}
\]

### Step 5: Win Condition
The players alternate turns to draw edges. If a player draws the 18th edge, no more edges can be added according to the rules. Thus, the other player has no legal moves left and loses the game. Therefore, the player who adds the 18th edge wins.

### Conclusion
The player who adds the 18th edge wins because this move leaves the opponent without any legal moves to make.

---

Would you like to explore more details about the solution, or do you have any questions about the logic used?

Here are some additional questions to expand on this topic:

1. How would the strategy change if we had a different number of interior vertices?
2. What if the players could only add one edge per turn?
3. What happens if three vertices are collinear inside the triangle?
4. How does the strategy differ if the initial triangle had more than three edges?
5. Can you generalize this strategy to any polygon?

**Tip:** In combinatorial games like this, always calculate the maximum number of possible moves to understand the win/loss conditions clearly.

Explore how players win in a combinatorial game involving a triangle and interior vertices by strategically adding edges. Understand the maximum number of non-crossing edges and game rules. Suitable for advanced high school students studying graph theory.

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