How many ways are there to move from one corner of a cube to the diagonally opposite corner, in three moves, each of which travels along one edge of the cube?

looks like 6 to me

there are 3 ways to make the first move.
Then from the new corner, there are 2 ways to continue.
Then there is a single last move.

Yes, and you have to multiply to get the answer.

Why did the cube go to therapy? Because it had too many unresolved corners!

But let me crunch some numbers here. In a cube, there are 12 edges. So for each move, you have 12 possible options. Since you want to reach the diagonally opposite corner in three moves, the total number of ways would be 12 × 12 × 12, which is 1,728 different ways!

Now, that's a lot of wiggling and squiggling. I hope you've got some comfortable shoes for your cube's adventure!

To move from one corner of a cube to the diagonally opposite corner in three moves, each of which travels along one edge of the cube, we can use the following steps:

Step 1: Select the first move: There are 3 edges connected to each corner of the cube, so we have 3 choices for the first move.

Step 2: Select the second move: After making the first move, we are left with 2 edges connected to the current position. We cannot choose the edge we just came from, so we have 2 choices for the second move.

Step 3: Select the third move: After making the second move, we are left with only 1 edge connected to the current position. Again, we cannot choose the edge we just came from, so we have 1 choice for the third move.

Therefore, the total number of ways to move from one corner of the cube to the diagonally opposite corner in three moves, each along one edge, is calculated by multiplying the choices for each step: 3 x 2 x 1 = 6.

So, there are 6 different ways to make these three moves.

To find the number of ways to move from one corner of a cube to the diagonally opposite corner in three moves, each of which travels along one edge of the cube, we can break down the problem into smaller steps.

Step 1: Understand the cube
A cube has 8 corners and 12 edges. Each edge connects two corners.

Step 2: Identify the starting and ending corners
In this case, we need to find the ways to move from one corner to the diagonally opposite corner. Let's call the starting corner "A" and the diagonally opposite corner "B".

Step 3: Determine the first move
Since we have three moves, the first move can be any of the edges connected to corner A. There are three edges connected to each corner, so there are 3 possible choices for the first move.

Step 4: Determine the second move
After the first move, we need to consider the possible edges connected to the corner we land on. Let's say our first move leads us to corner C. There are three edges connected to corner C, but one of them leads back to corner A, which would be a backtrack. Therefore, we have 2 choices for the second move.

Step 5: Determine the third move
Again, we consider the possible edges connected to the corner we land on after the second move. Let's say our second move leads us to corner D. Like before, we need to eliminate any edge that leads back to the corners we've already visited (A and C). That leaves us with only one choice for the third move, which is the edge connecting D and B.

Step 6: Multiply the choices
To calculate the total number of ways, we multiply the number of choices at each step together. So the total number of ways to move from one corner of a cube to the diagonally opposite corner in three moves, each along one edge, is:

Number of ways = 3 (choices for the first move) * 2 (choices for the second move) * 1 (choice for the third move) = 6

Therefore, there are 6 ways to move from one corner to the diagonally opposite corner of a cube in three moves, each of which travels along one edge of the cube.