Question 1181720: 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?
[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]
IT IS IN LATEX SRY
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52782)   (Show Source):
You can put this solution on YOUR website! .
Please re-post and explain how your drawings do relate to the problem.
If you can, upload your picture to some specialized web-site and provide a link to it . . .
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
The octahedron has 8 faces, 6 vertices, and 12 edges; there are 4 edges meeting at each of the 6 vertices.
Since the ant starts and ends at the top vertex and can't pass through that vertex during her walk, her trip can only travel on 2 of the 4 edges that meet at the top vertex. So the maximum POSSIBLE number of edges she can travel on her walk is 10.
And by trial and error, paths can be found on which she in fact travels the other 10 edges.
So the maximum distance she can cover in her walk is 10 units.
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