Navigating a Cube

SydneyRoverP6B

SydneyRoverP6B

Well-Known Member
Staff member
This question like the last is from an Australian Mathematics competition, and like the last it is to be done using pen and paper only, NO calculators allowed.. 8)

Each face of a solid cube is divided into four squares of equal size. Looking at the face of the cube directly in front of you, the top left vertex is labelled P, whilst the bottom right vertex on the rear face is labelled Q. Starting from vertex P, paths can be travelled to vertex Q along connected line segments. If each movement along the path takes one closer to Q, what is the number of possible paths from P to Q?

Ron.
 
Hello Rich,

Glad you like it...I am sure that you will have some fun with it... :D

Are you still on Holidays? How was / is it all going? Have you been having nice weather,...and have you seen any Rovers?

All the best,
Ron.
 
nope - back at work today unfortunately. Had a lovely time in crete and little phoebe loved the sea and seeing all the sights when we went walking... Couldnt have asked for better.

No rovers though i did see a really nice 70s toyota crown... Very rare i think...

Back to work on my rover this weekend! Swap heads etc...
 
Hello Darth,

It can be a bit ricky depending on how you look at it.

I'll post the solution in about 12 hours time.

Ron.
 
SydneyRoverP6B said:
Hello Darth,

It can be a bit ricky depending on how you look at it.

I'll post the solution in about 12 hours time.

Ron.
Click to expand...

Thank you. That has got my grey matter working! :p

If I may ask, where do you get the solutions from?
 
darth sidious wrote,...
If I may ask, where do you get the solutions from?
Click to expand...

Hello Darth,

Of course you can... :D I have some actual exams, but I need to work them out myself, as I do with all questions that I post before doing so.

Ok to the solution. Each face of the cube is divided into 4 squares as is covered in the question. Regardless of which ever corner you start from on the face that faces you, there are 6 paths that can be taken to reach the diagonally opposite corner. This represents the maximum number of ways to navigate the face whilst always moving away from the starting point,...no doubling back.

To reach the middle vertex along the edge of the cube on the same edge as Q, there are another 3 paths that can be taken, and from that point another three that can be taken to reach vertex Q. So 6 X 3 X 3 = 54 ways in which to travel from P to Q.

I applied the rule of products,...Combinatorics....If 2 operations must be performed, and if the first operation can always be performed p1 different ways and the second operation can always be performed p2 different ways, then there are p1p2 different ways that the 2 operations can be performed.

Ron.
 
For some reason, something inside me was saying it was some multiple of 6! As I couldn't substantiate it, I put it aside.

That was a good problem. From how you explained it, it seems I perhaps was overcooking the goose! :?

Oh well, I hope another problem is on its way! :)
 
Junkman said:
There may be 54 ways, but we all know that the answer to all questions is 42.
Click to expand...

LOL

I do believe that 42 is the answer to the ultimate question of life, not necessarily the answer to all questions however.

Even more strange as the question turned out to be, "What do you get if you multiply 9 by 6."

How does that work?

I'm sure someone can work it out, if not I'll tell when I come back from the 60s weekend

Richard
 
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