Walking a grid

@joe-shmo said
"It's not as easy as that"

Actually venda, it is that easy.

You have a 6 letter string

ALL possible ending points are counted by the multiplication principle of possible directions for each step.

S1,S2,S3,S4,S5,S6

4*4*4*4*4*4 = 4^6 possible strings and hence 4^6 journey ending points.

Sorry, just to clarify. There aren't 4^6 journey end points, there are 4^6 paths, but that is all we need to know for this question...how many paths there are.

If you are interested, I will pose another question along these lines?

@joe-shmo said
Sorry, just to clarify. There aren't 4^6 journey end points, there are 4^6 paths, but that is all we need to know for this question...how many paths there are.

If you are interested, I will pose another question along these lines?

Thanks for the puzzle Jo.
Seems I "overthought" it
I will always look at any puzzle to see if it interests me.
Whether I can solve it or not is another matter but it puts me to sleep at night thinking on it!

@joe-shmo said
400 is correct for the number of ways to get back to "P" in 6 steps!

ATY is correct about ALL possible paths = 4^6

thus;

P = 400/4^6 = 400/4096 = (20/64)^2 = (5/16)^2

I'll give you both partial credit!

Thank you -good fun!

@venda said
Thanks for the puzzle Jo.
Seems I "overthought" it
I will always look at any puzzle to see if it interests me.
Whether I can solve it or not is another matter but it puts me to sleep at night thinking on it!

You had 90% of the work done. Not seeing that last 10% was just a normal case of not seeing the forest through the trees! Good Work!

Ok...same thing, more steps.

Starting at "P": In exactly 11 steps , how many ways can you get to a location 3 units North and 1 unit East of "P"?

@joe-shmo said
Ok...same thing, more steps.

Starting at "P": In exactly 11 steps , how many ways can you get to a location 3 units North and 1 unit East of "P"?

Ok.I'll look later

@joe-shmo said
Ok...same thing, more steps.

Starting at "P": In exactly 11 steps , how many ways can you get to a location 3 units North and 1 unit East of "P"?

Either I have misread the question,or the answer is none!
You can reach a location 3 steps north and 1 step east of P in 4 steps.
Therefore,you need to take 7 steps to walk from P to P or an equivalent position.
This is the same as the much publicised "bridge" problem of old.
It can't be done!

@venda said
Either I have misread the question,or the answer is none!
You can reach a location 3 steps north and 1 step east of P in 4 steps.
Therefore,you need to take 7 steps to walk from P to P or an equivalent position.
This is the same as the much publicised "bridge" problem of old.
It can't be done!

Correct! Well done.

Ok...Next. Probably a bit more challenging.

Given exactly "k" steps in any cardinal direction from "P", how many positions can be reached?

@joe-shmo said
Ok...Next. Probably a bit more challenging.

Given exactly "k" steps in any cardinal direction from "P", how many positions can be reached?

Let the positions = "r"
Let steps -="k"
r = 4k
I'm trying not to "overthink" it!!

@venda said
Let the positions = "r"
Let steps -="k"
r = 4k
I'm trying not to "overthink" it!!

Nope...

@joe-shmo said
Nope...

Ok.I'll think about it later

3K?

@venda said
3K?

Afraid not. If you haven't yet; start with k = 1 step and work your way up. The pattern should jump out at you within the first few "k". Proving it in general for all "k" isn't quite so simple.

I've tried all that,with little dots on a piece of paper.
First, I went up to 7 "dots" in a straight line east and put a dot at north and south at every step.
I then added up the dots and came up with the answer 3k+1(1 representing an 8th step east)
I then tried with random directions (e.g 2 steps east 1 step north etc) but could only count 3k this way.
I suspect I need a different approach or a bigger piece of paper!!