A regular octahedron is made up of eight equilateral triangles, each with side length one unit, as shown below. An ant starts at the top vertex, walks along the edges of the triangles without ever traversing the same edge twice, and ends at the top vertex. If she did not pass through the top vertex at any other point in her walk, how many units are in the maximum distance she could walk?

Question

[asy]
import three;

size(150);

currentprojection=orthographic(1/4,-1,1/2);

triple A, B, C, D, E, F;

A = (0,0,0.5);
B = (0.5,0,0);
C = (0,0.5,0);
D = (-0.5,0,0);
E = (0,-0.5,0);
F = (0,0,-0.5);

draw(A--B--F--D--cycle);
draw(B--E--D);
draw(B--C--D,dashed);
draw(A--E--F);
draw(A--C--F,dashed);

[/asy]

Answer 1

To solve the problem, we first need to analyze the structure of the regular octahedron and the paths the ant can take.

The octahedron consists of 6 vertices and 12 edges, all of which are of equal length, specifically 1 unit. The vertices of the octahedron can be labeled as \( V_0 \) (top vertex), \( V_1, V_2, V_3, V_4 \) (the four vertices in the middle), and \( V_5 \) (bottom vertex). The connections between these vertices are as follows:

• \( V_0 \) is connected to \( V_1, V_2, V_3, V_4 \).
• Each middle vertex \( V_i \) (for \( i = 1, 2, 3, 4 \)) is connected to \( V_0 \) and also to \( V_5 \).
• Each middle vertex \( V_i \) is connected to its neighboring middle vertices as follows: \( V_1 \) connects to \( V_2 \), \( V_2 \) connects to \( V_3 \), \( V_3 \) connects to \( V_4 \), and \( V_4 \) connects to \( V_1 \) forming a cycle.

To avoid revisiting edges, the ant can only traverse each edge once. The goal is to determine the maximum length of the continuous path starting and ending at \( V_0 \) without passing through \( V_0 \) except at the beginning and the end.

Finding Non-Returning Paths

1. Understanding the structure: The ant can start at \( V_0 \), move to any of the middle vertices, and then must navigate toward \( V_5 \) without revisiting any edges. After reaching \( V_5 \), it must return to \( V_0 \).

2. Path exploration:

   • The ant can choose one of the four edges leading from \( V_0 \) to \( V_1, V_2, V_3, \) or \( V_4 \).
   • A full traversal can include the following pathway:
     • From \( V_0 \) to any \( V_i \) (1 unit)
     • From \( V_i \) to \( V_5 \) (also 1 unit)
     • From \( V_5 \) to any \( V_j \) (1 unit), where \( j \neq i \)
     • From \( V_j \) back to \( V_0 \) (1 unit).

3. Count each unique edge once:

   • Since the ant is required to different middle vertices, it should continue to traverse edges leading between middle vertices. The complete path can be understood to cover all unique edges. If traversing every vertex (except the ones in between) once provides maximum travel without crossing edges.

4. Total Path Length Calculation:

   • Starting at \( V_0 \): 1 unit
   • To \( V_5 \): 1 unit
   • To all possible four edges between \( V_1, V_2, V_3, V_4 \) covering the edges between them using \( 4 \) additional edges (which measure \( 1 \) unit each).
   • Lastly, returning to \( V_0\): another 1 unit.

The ant cannot exceed the maximum of 8 edges used without returning or traversing the same back again which conclude as follows:

\[ \text{Total path equals } 1 + 1 + 4 + 1 + 1 = 8 \text{ units.} \]

Conclusively, the ant can walk a maximum distance of \( \boxed{8} \) units.