Having trouble with this one!

A

Argile1845

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How would I go about solving this one?
A counter is initially at square a in the grid below. It is then moved from one square to an adjacent square (horizontally or vertically not diagonally) until it ends up at square B, passing through every square in the grid exactly once. How many different paths could the counter have taken?
 

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Argile1845 said:
How would I go about solving this one?
A counter is initially at square a in the grid below. It is then moved from one square to an adjacent square (horizontally or vertically not diagonally) until it ends up at square B, passing through every square in the grid exactly once. How many different paths could the counter have taken?
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Just start listing possible paths, in an orderly way.

Most likely, there are relatively few paths and you'll discover quickly how to eliminate possibilities so you don't waste time trying dead ends.
 
Dr.Peterson said:
Just start listing possible paths, in an orderly way.

Most likely, there are relatively few paths and you'll discover quickly how to eliminate possibilities so you don't waste time trying dead ends.
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I'm thinking they want me to take a more mathematical approach instead of just trying to write out all the possible paths.
For instance it's a 5x3 pattern, so would that be something like 5!/3! Or am I totally off track? Lol
 
Argile1845 said:
I'm thinking they want me to take a more mathematical approach instead of just trying to write out all the possible paths. For instance it's a 5x3 pattern, so would that be something like 5!/3! Or am I totally off track?
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The wording of the question almost rules that out. The requirement that we must pass through every square in the grid exactly once.
Now start a \(\mathcal{A}\) go two squares down, four additional squares to the right, two additional squares up, three additional squares to the left, one additional squares down, and three additional squares to the right to finish at \(\mathcal{B}\). That is the sort of list Prof. Peterson suggests.
 
Argile1845 said:
I'm thinking they want me to take a more mathematical approach instead of just trying to write out all the possible paths.
For instance it's a 5x3 pattern, so would that be something like 5!/3! Or am I totally off track? Lol
Click to expand...
An orderly list can be just as "mathematical" as a formula; it requires just as much careful thinking.

You're right that other problems involving paths (e.g. only moving right or down everywhere) allow you to calculate easily, but that is because there is a clear pattern, with paths behaving the same everywhere. This problem (using every square) is too global, with what you've done so far changing what you are able to do later. I wouldn't expect to find a nice pattern.

If the problem involved a larger grid, it might be possible to break things into cases and apply formulas, but as it is, each possible path is its own special case!
 
Are these correct paths? If they are correct, it is easy to find all paths. Then, you can add extra column. Find all paths again. Then, compare the number of paths with the number of squares for both drawings, you might find a formula that can solve any drawing. This is just a guess.

10.jpg

11.jpg

12.jpg
 
Correct! I've found six so far but thought my approach was wrong when I saw how others had completed grids similar but not exact to this one. Thank you all fo your help, and thank you for allowing my questions and not being offended. It was not intentional.
 
Dr.Peterson said:
An orderly list can be just as "mathematical" as a formula; it requires just as much careful thinking.

You're right that other problems involving paths (e.g. only moving right or down everywhere) allow you to calculate easily, but that is because there is a clear pattern, with paths behaving the same everywhere. This problem (using every square) is too global, with what you've done so far changing what you are able to do later. I wouldn't expect to find a nice pattern.

If the problem involved a larger grid, it might be possible to break things into cases and apply formulas, but as it is, each possible path is its own special case!
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Thank you for your reply, I really appreciate it!
 
Argile1845 said:
Yes! I've found those as well (6 so far) but was unsure about my approach.
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Yes, nasi112 missed a few, and also doesn't appear to be taking a systematic approach to finding all. I count 10, but without having carefully drawn them all out, that may not be complete. If you show us yours, I may be able to suggest a better technique. On the other hand, I've see no reason to think there will be a general formula.
 
I will draw out a pic in about an hour (all on one paper) to show you my approach if you want to look at it. My mom wants me to go read my assignment.
 
Always pay attention to moms until they become grandmoms. Then you can no longer trust them because they become coconspirators with the grandkids against YOU.
 
I've only found 8 .. can't think of how else I can do it ..
 

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So what I tried to do, even though I didn't really order them in the number in which I did them which I know is probably confusing, is I tried to go one block over one block down one block over one block down and then on the next one I would do two blocks over two blocks down or something like that.
 
Argile1845 said:
I'm thinking they want me to take a more mathematical approach instead of just trying to write out all the possible paths.
For instance it's a 5x3 pattern, so would that be something like 5!/3! Or am I totally off track? Lol
Click to expand...
I think what's intended is an algorithmic approach, not mathematical.
I would try this ordering method.

The 4 possible direction are N, E, S, W.

The program:
If possible go N
else, if possible go E
else, if possible, go S
else, if possible, go W

If we ended up in B after visiting all squares record the sequence of moves.

Go back to the most recent square were a choice was available and go in a different direction turning clockwise (e.g. instead of going W, go S).
 
lev888 said:
I think what's intended is an algorithmic approach, not mathematical.
I would try this ordering method.

The 4 possible direction are N, E, S, W.

The program:
If possible go N
else, if possible go E
else, if possible, go S
else, if possible, go W

If we ended up in B after visiting all squares record the sequence of moves.

Go back to the most recent square were a choice was available and go in a different direction turning clockwise (e.g. instead of going W, go S).
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That's what I was thinking as well! I was thinking it might be set up somewhere like that since online that's how I read other people were doing it which is why I was second-guessing myself.
 
Dr.Peterson said:
Yes, nasi112 missed a few, and also doesn't appear to be taking a systematic approach to finding all. I count 10, but without having carefully drawn them all out, that may not be complete. If you show us yours, I may be able to suggest a better technique. On the other hand, I've see no reason to think there will be a general formula.
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How did you count ten?
 
I got it! It was 8! Thank you all for helping me. You helped me by showing me I was on the right track. Lol
 
Argile1845 said:
How did you count ten?
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Looking again at my quick sketch, I realize I duplicated two paths.

And, yes, what you described in #15 and lev in #16 is the sort of systematic/orderly search I suggested. In addition, you can be looking ahead and realize when a certain choice would leave no way to get to B or to use up all squares.
 
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