An ant must walk from one vertex of a cube to the "opposite" vertex (that is, the vertex that is farthest from the starting vertex) and back again to its starting position. It may only walk along the edges of the cube. For the entire trip its path must traverse exactly six edges and it may travel the same edge twice. How many different 6 edge paths can the ant choose from?
36 possible combinations
To find the number of different 6-edge paths the ant can choose from, we can consider the possible ways the ant can move on the cube.
The ant starts at one vertex of the cube. From there, it can move to any of the adjacent vertices, which are connected by an edge. There are three edges adjacent to each vertex, so the ant has three possible choices for the first edge.
Now, the ant is at a new vertex on the cube. It can again choose any of the three adjacent edges for the second edge.
For the third edge, the ant is at a third vertex on the cube. It can choose any of the three adjacent edges.
At this point, the ant has traversed three edges and is at a fourth vertex on the cube. It has three possible choices for the fourth edge.
For the fifth edge, the ant is at a fifth vertex on the cube. Again, it has three possible choices.
Finally, the ant is at the sixth vertex on the cube and has three possible choices for the sixth edge, which will take it back to the starting vertex.
To find the total number of different 6-edge paths, we can multiply the number of choices at each step: 3 * 3 * 3 * 3 * 3 * 3 = 729.
Therefore, there are 729 different 6-edge paths that the ant can choose from.
To find the number of different 6-edge paths the ant can choose from, let's break it down step by step:
Step 1: Determine the number of possible starting points for the ant. There are 8 vertices in a cube, so there are 8 possible starting points.
Step 2: Consider all possible edges the ant can choose for each step. At each vertex, there are exactly 3 edges connected to it. Since the ant can travel the same edge twice, there are 3 choices for each step.
Step 3: Calculate the number of possible paths. Since the ant must traverse exactly 6 edges, and for each edge, there are 3 choices, the total number of possible paths is 3 multiplied by itself 6 times (3^6).
So, the ant can choose from a total of 3^6 = 729 different 6-edge paths.