Problem Solving

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?

A. 6
B. 12
C. 18
D. 24
E. 30

The OA is C.

Please, can anyone assist me with this PS question? I'm confused. Thanks in advance.

Hello BTGmoderatorLU.

I will give you an answer using the image of a cube with the vertex A, B, C, D, E, F, G, H.



Let's suppose the ant starts in the vertex A and it wants to arrive to the vertex G.

Now, the ways the ant can go are:

In 3 steps:
1. A-B-C-G
2. A-D-C-G
3. A-B-F-G
4. A-D-H-G
5. A-E-F-G
6. A-E-H-G

In 5 steps:
7. A-B-F-E-H-G
8. A-D-H-E-F-G
9. A-B-C-D-H-G
10. A-D-C-B-F-G
11. A-E-F-B-C-G
12. A-E-H-D-C-G

In 7 steps:
13. A-B-C-D-H-E-F-G
14. A-D-C-B-F-E-H-G
15. A-E-F-B-C--D-H-G
16. A-E-H-D-C--B-F-G
17. A-B-F-E-H-D-C-G
18. A-D-H-E-F-B-C-G

These are all the possible paths the ant can make to go from vertex A to vertex G. Therefore, the correct answer is the option [spoiler]C=18[/spoiler]

I hope it can help you.

I'd like to see an algebraic solution. <i class="em em-smiley"></i>

Let the initial vertex = A, implying that the ant must travel to G.
Let's divide the journey into 3 stages and count the number of options for each stage.

First stage;
Since the ant can travel from A to B, D, or E, the number of options for the first stage = 3.

Second stage:
Whether the ant is at B, D or E, it has 2 options for the next stage.
If the ant is at B, it can travel next to F or C.
If the ant is at D, it can travel next to C or H.
If the ant is at E, it can travel next to F or H.
Thus:
The number of options for the second stage = 2.

Third stage:
Wherever the ant ends in the second stage, it can then travel to G by a short route, a medium route, or a long route, for a total 3 options.
For example:
If the ant travels A-B-F in the first 2 stages, it can then travel to G as follows:
Short route = F-G.
Medium route = F-E-H-G.
Long route = F-E-H-D-C-G.
Thus:
The number of options for the third stage = 3.

To combine the options for each stage, we multiply:
3*2*3 = 18.

The correct answer is C.