Welcome back to this week’s Challenge Problem! As always, the problem and solution below were written by one of our fantastic instructors. Each challenge problem represents a 700+ level question. If you are up for the challenge, however, set your timer for 2 mins and go!
An ant is clinging to one corner of a box in the shape of a cube. The ant wants to get to the most distant corner of the box by crawling only along the edges of the cube and without ever revisiting a place it has been. How many different paths can the ant take to the most distant corner?
First, draw a picture of the situation.
Now, look for patterns as you construct and count paths that don’t retrace or touch themselves at all. One of the first patterns that is useful to spot has to do with the initial choice about which edge to crawl along. The ant can take any one of three initial paths:
These three initial choices are equivalent, because the cube is symmetric. Thus, we can just consider one of the initial paths, count up the ways from that point forward, then multiply by 3.
Let’s take the middle choice:
First, try to go immediately to the goal. We get two similar paths:
If we stopped at this point, we would get 2 × 3 = 6 total paths. This is in fact the number of shortest paths to the goal. However, we can construct additional “zigzag” paths that also satisfy the constraints of the problem.
There are no more possibilities without the path touching itself, so this initial edge leads to 6 possible paths. Since there are 3 initial edges, the total number of possible paths is 6 × 3 = 18.
Alternatively, we can map the corners by their “distance” away from the starting point, measured in edges:
Taking the leftmost edge to start, we can construct the 6 possible paths leading from that edge:
Again, there are 6 × 3 = 18 possible paths.
The correct answer is (C).
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