word problem with distance

[JeffM]

Great research!

I think I just proved that A is also a good answer. In all dimensions (linear, 2d, 3d,...) AND in all possible layouts, route B ≤ A. So the people at Oxford may be turning away good candidates (unless I'm wrong in the following logic )

First, split the routes into individual legs...
A) xa ax xb bc cb bx
B) xa ac cb bx

Put each leg into alphabetical order (eg. write "xa" as "ax"). This doesn't change the distance of each separate leg of the journey.
A) ax ax bx bc bc bx
B) ax ac bc bx

Now "cancel" any pairs that A and B have in common. If A contains "ab" and B also contains "ab" then remove them both. This will reduce the length of both routes by exactly the same amount.
A) (ax) ax bx (bc) bc (bx)
B) (ax) ac (bc) (bx)

A) ax bx bc
B) ac

Clearly, route B which goes direct "ac" is always less than or equal to route A "ax xb bc".



(Yes, I just did the "clearly" thing )

Great research!

I think I just proved that A is also a good answer. In all dimensions (linear, 2d, 3d,...) AND in all possible layouts, route B ≤ A. So the people at Oxford may be turning away good candidates (unless I'm wrong in the following logic )

First, split the routes into individual legs...
A) xa ax xb bc cb bx
B) xa ac cb bx

Put each leg into alphabetical order (eg. write "xa" as "ax"). This doesn't change the distance of each separate leg of the journey.
A) ax ax bx bc bc bx
B) ax ac bc bx

Now "cancel" any pairs that A and B have in common. If A contains "ab" and B also contains "ab" then remove them both. This will reduce the length of both routes by exactly the same amount.
A) (ax) ax bx (bc) bc (bx)
B) (ax) ac (bc) (bx)

A) ax bx bc
B) ac

Clearly, route B which goes direct "ac" is always less than or equal to route A "ax xb bc".



(Yes, I just did the "clearly" thing )

Consider the layout AXBC all collinear

Your answer that XACBX is shorter than XAXBCBX is wrong IN THAT LAYOUT.

Put A at 0, X at 10, B at 30 and C at 55.

Driver goes from X to A and drops off. Distance 10.
Driver returns from A to X. Distance 10.
Driver goes from X to B and drops off. Distance 20.
Driver goes from B to C and drops off. Distance 25.
Driver returns from C to B. Distance 25.
Driver returns from B to X. Distance 20.

Total distance of 110 for XAXBCBX.

Driver goes from X to A and drops off. Distance 10.
Driver goes from A to C (without going through X or B) and drops off. Distance > 55.
Driver goes from C to B. Distance 25.
Driver returns to X. Distance 20.

Total distance > 110 for XACB.

There is no way that XACB is shorter than XAXBCBX. If you travel from A to C without going the straight-line route through X and B, the distance must be greater than 55.


Alternatively, if you are leaving out intermediate points XACB is just a different notation for XAXBCBX.

 

[Cubist]

Driver goes from A to C (without going through X or B) and drops off. Distance > 55.

But the question states that routes between towns are straight, therefore the distance is exactly 55, it isn't greater. My proof is valid UNLESS we introduce the idea that any en-route towns must be stated (see * below)

Alternatively, if you are leaving out intermediate points XACB is just a different notation for XAXBCBX.

No it isn't, one says that the lorry goes straight from X to A, A to C, then C to B
The other states that the lorry goes straight from X to A, then A to X, ... Your quoted statement of equivalent notation would only be true if all points lie on a line in a certain order.

*) There's no place in the question where it says that you must state any en-route destinations. BUT I concede that this IS implied when one looks deeper into the question. My initial interpretation would (probably) lead to multiple correct answers, when they expect only one. Yourself, @lev888 , and @Dr.Peterson have reached this conclusion much quicker than I have! VERY well done! And I only got there with your help. I would not have got this correct if I had sat that exam!

 

[Dr.Peterson]

Alternatively, if you are leaving out intermediate points XACB is just a different notation for XAXBCBX.

I agree, except that it has to start and end at X, so you must mean XACBX. The problem doesn't state (or imply) that the given route must be the only way to describe that route, or that there can't be other routes with the same length (there always will be others, if only the reverse route). It means only that there are no shorter routes.

Your quoted statement of equivalent notation would only be true if all points lie on a line in a certain order.

Yes, that's how I understood JeffM's intent. He is assuming the layout A---X---B---C along a line, so that XC = XB + BC. If it isn't in a line, you have something like what I showed for option C, where XABCX is shorter than XAXBCBX.

There's no place in the question where it states that you must state any en-route destinations.

No one says the route has to be described this one way. But even if you take XACBX as different from XAXBCBX, they are the same length (on this layout), so XAXBCBX is still a shortest route, and that's all it can mean.

But I do agree that it is a tricky problem, and hard to interpret.

 

[JeffM]

@Cubist, @lev888 , @ Dr. Peterson

Yes, I do think that the problem is trickily worded and requires interpretation. In one of my earlier posts. I commented on the weirdness of the right answer in a multiple choice being a wrong answer, which meant that the other wrong answers had to be right answers under a consistent interpretation applied to different situations.

And I should have been explicit in pointing out that "shortest" here has to mean "not longer than any alternative."

 

[lev888]

@Cubist, @lev888 , @ Dr. Peterson

Yes, I do think that the problem is trickily worded and requires interpretation. In one of my earlier posts. I commented on the weirdness of the right answer in a multiple choice being a wrong answer, which meant that the other wrong answers had to be right answers under a consistent interpretation applied to different situations.

And I should have been explicit in pointing out that "shortest" here has to mean "not longer than any alternative."

I don't see "this route is not the longest for any layout" as a wrong answer. I've seen multiple choice questions that sound like "Read the passage and pick the statement that is false based on the text". Is this tricky? I don't think so.

 

[Cubist]

Yes, that's how I understood JeffM's intent. He is assuming the layout A---X---B---C along a line, so that XC = XB + BC. If it isn't in a line, you have something like what I showed for option C, where XABCX is shorter than XAXBCBX.

No one says the route has to be described this one way. But even if you take XACBX as different from XAXBCBX, they are the same length (on this layout), so XAXBCBX is still a shortest route, and that's all it can mean.

...I thought that @JeffM was trying to show that my interpretation of the notation was incorrect by pointing out that XAXBCBX would be equivalent to XACBX (I thought that the lorry would be able to pass through a town without having to explicitly show it in the route notation)...

Alternatively, if you are leaving out intermediate points XACB is just a different notation for XAXBCBX.

Maybe I read too much into this. I was just pointing out that this does not prove that my initial interpretation is wrong or inappropriate. I guess commonsense would suggest that there would be a quick answer, and this is probably sufficient to justify the answer being C. But putting my mathematical hat on, it's not a solid proof that "B" couldn't be the correct answer under my initial interpretation.

One surefire way to prove my initial interpretation invalid would be to find another possible answer, via a proof that route B, C, D or E is always bigger EDIT: is never smaller than one of the other routes (since the multiple choice implies only one valid answer).

EDIT: have to be careful with the phrasing

 

[Cubist]

I've just proved that the length of route B ≤ C in every possible dimension and layout. Therefore both A and C would be correct under my initial interpretation. Therefore this is the wrong way of thinking . Hope that Subhotosh and Jomo don't see this, or I'll be banished to the corner (by the shortest route)

Spoiler: proof

B) xa ac cb bx
C) xa ax xb bc cx

Remove common elements...
B) (ax) ac (bc) (bx)
C) (ax) ax (bx) (bc) cx

B) ac
C) ax cx
But ax xc >= ac, thus C >= B

 

[Dr.Peterson]

Maybe I read too much into this. I was just pointing out that this does not prove that my initial interpretation is wrong or inappropriate. I guess commonsense would suggest that there would be a quick answer, and this is probably sufficient to justify the answer being C. But putting my mathematical hat on, it's not a solid proof that "B" couldn't be the correct answer under my initial interpretation.

One surefire way to prove my initial interpretation invalid would be to find another possible answer, via a proof that route B, C, D or E is always bigger EDIT: is never smaller than one of the other routes (since the multiple choice implies only one valid answer).

Things are getting too hard to follow. What is your initial interpretation, by which you think (B) could be the correct answer? (Or are you saying A?)

 

[Cubist]

What is your initial interpretation,

My initial interpretation was that:-

• the routes needed to be compared with each other, NOT assessed individually
• AND that the notation doesn't require writing any towns that are passed through (obviously this would only ever happen if 3 are co-linear)

But, as stated in #47, I've found two possible correct answers under this interpretation. But it's probably useful to keep my posts in this thread, despite it being the wrong interpretation, just in case anyone else has the same initial thought that I did.

 

[eric beans]

geez, this thread is going on 3 pages now. i don't see how this can be solved in less than 2 minutes when even mathematicians were struggling to answer this and had to reach for the answer key. i don't feel so bad now. it's hard to follow this thread and i'm still in a haze about this. could someone who understands this clearly and succinctly summarize:
1) how to interpret the question firstly (what's it really asking to do? as there seems to be disagreements) and
2) what the process is to figure this out? i understand the triangle inequality bit (AC<ABC) but not sure how it is specifically applied here.

danke schoen.

dr. peterson, i purposefully didn't want to give away the test info b/c i knew you'd go searching for the answer key. i know they have at least once before made an error in their answer key and admitted it a different specimen so i wanted to see if you could find the answer absent the bias of knowing the "answer" and working backwards to fit their answer. i wanted to know how to work the question from the start and see the thinking without working backwards as it were. do you know what i mean? anyway, sorry if i upset you and rest of the folks. wasn't my intent. by the way thanks to all you were trying but i'm still having a hard time understanding this.

 

[lev888]

geez, this thread is going on 3 pages now. i don't see how this can be solved in less than 2 minutes when even mathematicians were struggling to answer this and had to reach for the answer key. i don't feel so bad now. it's hard to follow this thread and i'm still in a haze about this. could someone who understands this clearly and succinctly summarize:
1) how to interpret the question firstly (what's it really asking to do? as there seems to be disagreements) and
2) what the process is to figure this out? i understand the triangle inequality bit (AC<ABC) but not sure how it is specifically applied here.

danke schoen.

I tried, but you ignored my posts.

 

[eric beans]

I tried, but you ignored my posts.

i'm sorry you feel that way. I tried to follow the posts but it was honestly getting too confusing and complicated. i just gave up trying to read through the longish threads. i had a hard time trying to visualize and make sense of the letters and arrangements. i'm a visual learner and so it just was hard to try to see these explanations in picture form. i know some of the posts had pictures but i couldn't follow the paths. what would help is if someone who understood this drew out the pictures and showed how to solve it.

 

[lev888]

I tried to follow the posts but it was honestly getting too confusing and complicated. i just gave up trying to read through the longish threads. i had a hard time trying to visualize and make sense of the letters and arrangements. i'm a visual learner and so it just was hard to try to see these explanations in picture form. i know some of the posts had pictures but i couldn't follow the paths. what would help is if someone who understood this drew out the pictures and showed how to solve it.

I understand how an explanation can be confusing. In such cases it would be appropriate (in my opinion) to thank the helper and ask for a clarification. When I spend time writing paragraphs of text and get crickets in return, I am not exactly motivated to continue helping that particular poster.

 

[eric beans]

I understand how an explanation can be confusing. In such cases it would be appropriate (in my opinion) to thank the helper and ask for a clarification. When I spend time writing paragraphs of text and get crickets in return, I am not exactly motivated to continue helping that particular poster.

like i said, i'm sorry you feel that way.

 

[JeffM]

First, I do not think lev, Dr. Peterson, and I disagreed about the intended meaning of the question though I believe one or more of us did say it was not written as clearly as desirable.

Second, I believe the three of us explained what needed to be done in terms of reasoning to deal with that question. You need to look at each proposed route independently and see if you can imagine any layout that FOR WHICH THAT ROUTE is not longer than any alternative route. The layout is not the same for each route. If you can find such a layout for that route, it is not the one you want.

Third, I believe that a systematic approach will get you through the problem.

The four points are collinear. There are two cases.

Only three of the points are collinear. There is one case.

No three points are collinear. But then you can construct a triangle from any three points. The fourth point will be inside or outside that triangle. There are two cases..

That is, there are five generic maps.

Fourth, I suspect this is a trivial problem in topology, a field I have not studied.

 

[Dr.Peterson]

geez, this thread is going on 3 pages now. i don't see how this can be solved in less than 2 minutes when even mathematicians were struggling to answer this and had to reach for the answer key. i don't feel so bad now. it's hard to follow this thread and i'm still in a haze about this. could someone who understands this clearly and succinctly summarize:
1) how to interpret the question firstly (what's it really asking to do? as there seems to be disagreements) and
2) what the process is to figure this out? i understand the triangle inequality bit (AC<ABC) but not sure how it is specifically applied here.

If the test implies that every problem can be solved that fast, then that is part of what you should have stated in the first place. Does it? When we're given a problem with no context, we don't know what to expect, which makes it harder to do. That's part of the reason our guidelines beg for completeness and for context.

And we didn't have to find the answer key in order to solve it; finding that helped to confirm the interpretation of the problem, because it agreed with what I'd found. Finding the test itself gave us the missing information about what is being tested: creative thinking skills, not some particular math course.

I'm not sure anyone understands the problem fully, so as to make the interpretation obvious. It is not very well written. And, no, I don't think I would have solved it in two minutes if I'd tried. (Or at least, I'd have grudgingly written C by elimination, and gone away frustrated.)

I think we've explained how to think about it (unfortunately, in several contradictory ways). Once you understand that each option is a separate question, asking whether there is any arrangement of the points (that is, any possible map) for which the stated route is the shortest), you need to find such a map, largely by trial and error coupled with a sense of how such routes can vary. And it turns out that straight-line arrangements are the way to go for most of them. I don't know of any theorems, or any specific concepts, to point you to. It's inherently a non-routine problem.

dr. peterson, i purposefully didn't want to give away the test info b/c i knew you'd go searching for the answer key. i know they have at least once before made an error in their answer key and admitted it a different specimen so i wanted to see if you could find the answer absent the bias of knowing the "answer" and working backwards to fit their answer. i wanted to know how to work the question from the start and see the thinking without working backwards as it were. do you know what i mean? anyway, sorry if i upset you and rest of the folks. wasn't my intent. by the way thanks to all you were trying but i'm still having a hard time understanding this.

We know at least as well as you do that answer keys are not always right. But withholding relevant information is counterproductive. We don't work backward from an answer (which as you know is a horrible way to learn anything), but either seeing that a wrong answer might be what is confusing you, or that their answer is incompatible with the natural interpretation of a problem, can be very helpful -- especially in a case like this where the interpretation is the main issue.

EDIT: The main thing I was begging for was not the specific test (so I could look it up, or otherwise), but just some information about what sort of test it was -- for a class in topology, or an intelligence test, or whatever. For many purposes, that is the context that matters most.

Now if you want to understand how to solve it, we'll have to interact yet more. Pick one of our explanations and try again to show your understanding of what you are being told. We can only explain something successfully when we figure out what you are misunderstanding in what we've already said. And, yes, the thread is long enough that this becomes very difficult. (I generally avoid threads with too many people in them, for exactly that reason.)

 

[eric beans]

First, I do not think lev, Dr. Peterson, and I disagreed about the intended meaning of the question though I believe one or more of us did say it was not written as clearly as desirable.

Second, I believe the three of us explained what needed to be done in terms of reasoning to deal with that question. You need to look at each proposed route independently and see if you can imagine any layout that FOR WHICH THAT ROUTE is not longer than any alternative route. The layout is not the same for each route. If you can find such a layout for that route, it is not the one you want.

Third, I believe that a systematic approach will get you through the problem.

The four points are collinear. There are two cases.

Only three of the points are collinear. There is one case.

No three points are collinear. But then you can construct a triangle from any three points. The fourth point will be inside or outside that triangle. There are two cases..

That is, there are five generic maps.

Fourth, I suspect this is a trivial problem in topology, a field I have not studied.

again, i'm having a hard time picturing all this. what points are the "4 points are collinear"? i don't understand why there would be "two cases"? can you draw this out?

 

[Deleted member 4993]

I understand how an explanation can be confusing. In such cases it would be appropriate (in my opinion) to thank the helper and ask for a clarification. When I spend time writing paragraphs of text and get crickets in return, I am not exactly motivated to continue helping that particular poster.

You have to remember there are students who are not good at distant-learning, but in general these students can stare at a glowing screen for a long time.

 

[eric beans]

If the test implies that every problem can be solved that fast, then that is part of what you should have stated in the first place. Does it? When we're given a problem with no context, we don't know what to expect, which makes it harder to do. That's part of the reason our guidelines beg for completeness and for context.

And we didn't have to find the answer key in order to solve it; finding that helped to confirm the interpretation of the problem, because it agreed with what I'd found. Finding the test itself gave us the missing information about what is being tested: creative thinking skills, not some particular math course.

I'm not sure anyone understands the problem fully, so as to make the interpretation obvious. It is not very well written. And, no, I don't think I would have solved it in two minutes if I'd tried. (Or at least, I'd have grudgingly written C by elimination, and gone away frustrated.)

I think we've explained how to think about it (unfortunately, in several contradictory ways). Once you understand that each option is a separate question, asking whether there is any arrangement of the points (that is, any possible map) for which the stated route is the shortest), you need to find such a map, largely by trial and error coupled with a sense of how such routes can vary. And it turns out that straight-line arrangements are the way to go for most of them. I don't know of any theorems, or any specific concepts, to point you to. It's inherently a non-routine problem.


We know at least as well as you do that answer keys are not always right. But withholding relevant information is counterproductive. We don't work backward from an answer (which as you know is a horrible way to learn anything), but either seeing that a wrong answer might be what is confusing you, or that their answer is incompatible with the natural interpretation of a problem, can be very helpful -- especially in a case like this where the interpretation is the main issue.

EDIT: The main thing I was begging for was not the specific test (so I could look it up, or otherwise), but just some information about what sort of test it was -- for a class in topology, or an intelligence test, or whatever. For many purposes, that is the context that matters most.

Now if you want to understand how to solve it, we'll have to interact yet more. Pick one of our explanations and try again to show your understanding of what you are being told. We can only explain something successfully when we figure out what you are misunderstanding in what we've already said. And, yes, the thread is long enough that this becomes very difficult. (I generally avoid threads with too many people in them, for exactly that reason.)

thank you again for clarifying. my head feels like it's all pooped out right now. i'll go back like you said from the beginning and follow your posts and see if i can follow it and the 1 line method to solving this. for me, it's hard to picture a lot of these shapes for some reason. it may take a few days. i got to take a break. i can't make heads or tails out of this right now.

 

[JeffM]

again, i'm having a hard time picturing all this. what points are the "4 points are collinear"? i don't understand why there would be "two cases"? can you draw this out?

You are asked to picture four towns, point on a map.

One way to arrange four point is along a straight line.

There are two variants. One is if X is an end point as in X__P__Q__R.
What is the shortest trip (in the sense that it is not longer than any other route).

Obviously that XPQEQPX. If you see a pattern starting and ending with X and the other points going back and forth in the same order, then you have a map that fits that route.

The other collinear variant is that X is not an endpoint as in P__X__Q__R.
The shortest trip goes XPXQRQX or XQRQCPX. X appears three times in the route. Look for that patern.

And so on.