Question 1199921: Mr.Greenstamps posted a problem here:
https://www.algebra.com/algebra/homework/Length-and-distance/Length-and-distance.faq.question.1134175.html
Also, the original image link has expired, so here it is again for your reference: https://ibb.co/yNfgt93
I would like some help understanding Mr.Greenstamps' solution.
Firstly, I don't really understand the 4 units forward ("F"), 4 units left (L"), and 4 units down ("D"). The "4 units down" in particular.
Here's a diagram I made:
https://ibb.co/qNnw2Bn
Clearly, you can see that from A, you can only move UP, LEFT, or FORWARD. Perhaps my intuition here is wrong, but I'm not sure. Please let me know.
Thanks!!
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13198)   (Show Source):
You can put this solution on YOUR website!
Our two figures are not the same; we are moving different directions on the cube.
But how we draw the figure does not change the answer. We need to move 4 units in each of three directions; it doesn't matter how we name those three directions. The number of different ways to do that is the number of ways of arranging the symbols AAAABBBBCCCC; that number is
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Since the links to the original figure and the original problem have been lost, I don't know what the original problem was.
The solution above allows paths through the interior of the cube; if the paths must be on the surface of the cube, then the solution above is of course incorrect.
Answer by ikleyn(52778)  (Show Source):
You can put this solution on YOUR website! .
The solution by @greenestamps is obviously incorrect,
since it allows the paths INSIDE the cube,
while, according to the condition, only the paths on the cube surface are allowed.
For a correct solution, see this link
https://www.maa.org/sites/default/files/pdf/pubs/Rubiks6.pdf
or this link
https://math.stackexchange.com/questions/740960/paths-on-a-rubiks-cube
The answer is 384 paths.
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