how do you answer :a bug lives on the corner of a cube and is allowed to travel only on the edges of the cube. in how many ways can the bug visit each of the other seven corners once and only once returning to its home corner only at the end of the trip?
I have not done an analytical solution, but just by intuition, I seem to get 12 as an answer. You can check if you can get more (or less).
Here's how I approach it.
Let the cube be represented by superimposing two squares, ABCD and EFGH where vertex A is above vertex E, and so on.
The bug lives at A and has to visit BCDEFGH and return to A in any order.
Let N=number of choices
For the first trip, it has three choices, AB,AD and AE. N=3 at this point.
For the route AB, on reaching B, it has two choices, BC and BF. By symmetry, it has now a total of N=3*2=6 routes.
If it chooses the route ABC, on reaching C, it has yet two more choices, CD and CG. So far, N=3*2*2=12 routes.
If it chooses ABCD, it will be forced to continue on to HGFEA if it wishes to visit the remaining vertices without repeating. If it chooses ABCG, he will have to continue with FEHDA for the same reason.
So the total number of routes it can take is 12.
To solve this problem, we can use a simple approach known as the "Hamiltonian cycle" technique. Let's break it down step by step.
1. Visualize the cube: Start by visualizing a cube in your mind or draw one on a piece of paper. Label each corner with a unique number to help keep track of them.
2. Identify the starting corner: As the problem states, the bug starts at one of the corners. Let's call this corner "A" for now.
3. Determine the neighboring corners: Look at corner A and identify its neighboring corners. In the case of a cube, each corner is connected to three other corners by edges. Label these neighboring corners "B," "C," and "D."
4. Choose the first neighboring corner: Choose one of the neighboring corners, let's say "B," as the bug's first move.
5. Determine the neighboring corners of the chosen corner: Look at corner B and identify its neighboring corners (excluding corner A, as we already visited it). Label these neighboring corners "E," "F," and "G."
6. Continue this process: Repeat steps 4 and 5, choosing the next neighboring corner from each previously chosen corner until you have visited each corner once and only once.
7. Returning home: Once you have visited each of the other seven corners, you need to return to the starting corner (A). Look at the last corner you visited and find its neighboring corners. Choose one of these neighboring corners as the final move to return to corner A.
8. Count the total number of possible paths: To determine the number of ways the bug can visit each corner once and return home, count the total number of possible paths you obtained from step 7.
Note: You might notice that there are multiple paths that lead to the same result. For example, you might obtain different paths but still visit all corners and return home. However, since the question asks for the number of ways, you need to count each unique path separately.
Remember to consider all possible paths in order to get an accurate count.