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# In the x-y plane, the square region bound by (0,0), (10, 0)

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Magoosh GMAT Instructor
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In the x-y plane, the square region bound by (0,0), (10, 0) [#permalink]

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21 Nov 2012, 16:34
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In the x-y plane, the square region bound by (0,0), (10, 0), (10, 10) and (0, 10) is isolated. A boundary point is any of the 40 points on the edge of this region for which both coordinates are integers; boundary points are indicated as purple in the diagram. Square J, bound by the points (4, 4), (4, 6), (6, 6), and (6, 4), are shown in green. If two boundary points are selected at random, and the line segment connecting these two is drawn, what is the probability that this line segment touches or passes through Square J?
Attachment:

10 x 10 region with Square J.JPG [ 31.62 KiB | Viewed 3924 times ]

This is a very challenging question. Answer will follow after some discussion ....
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Last edited by mikemcgarry on 26 Nov 2012, 11:43, edited 2 times in total.

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Re: hard probability problem, lines in the x-y plane [#permalink]

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21 Nov 2012, 18:23
mikemcgarry wrote:
In the x-y plane, the square region bound by (0,0), (10, 0), (10, 10) and (0, 10) is isolated. A boundary point is any of the 36 points on the edge of this region for which both coordinates are integers; boundary points are indicated as purple in the diagram. Square J, bound by the points (4, 4), (4, 6), (6, 6), and (6, 4), are shown in green. If two boundary points are selected at random, and the line segment connecting these two is drawn, what is the probability that this line segment touches or passes through Square J?
Attachment:
10 x 10 region with Square J.JPG

This is a very challenging question. Answer will follow after some discussion ....

Where are the options?

My ans is 1/25.
Can explain once the OA is provided.
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Re: hard probability problem, lines in the x-y plane [#permalink]

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21 Nov 2012, 22:37
I am getting 1/63. Not very sure though. Will explain if correct.

Edit. In view of Vips's comment below., if there are 40 points... My answer would be 8/39

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Last edited by MacFauz on 22 Nov 2012, 04:34, edited 2 times in total.

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Re: hard probability problem, lines in the x-y plane [#permalink]

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21 Nov 2012, 23:42
mikemcgarry wrote:
In the x-y plane, the square region bound by (0,0), (10, 0), (10, 10) and (0, 10) is isolated. A boundary point is any of the 36 points on the edge of this region for which both coordinates are integers; boundary points are indicated as purple in the diagram. Square J, bound by the points (4, 4), (4, 6), (6, 6), and (6, 4), are shown in green. If two boundary points are selected at random, and the line segment connecting these two is drawn, what is the probability that this line segment touches or passes through Square J?
Attachment:
10 x 10 region with Square J.JPG

This is a very challenging question. Answer will follow after some discussion ....

talking about flawed questions.. isnt this question flawed? There are 40 boundry points as per the picture, but question mentions 36.
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Re: In the x-y plane, the square region bound by (0,0), (10, 0) [#permalink]

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24 Nov 2012, 15:21
With the 36 points, I'm getting 1/35. Not sure though!

I will explain after the OA is posted. I dont want to confuse anyone if it is incorrect.

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Re: In the x-y plane, the square region bound by (0,0), (10, 0) [#permalink]

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25 Nov 2012, 10:30
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My ans

2/35..

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Magoosh GMAT Instructor
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Re: hard probability problem, lines in the x-y plane [#permalink]

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26 Nov 2012, 11:53
Vips0000 wrote:
talking about flawed questions.. isn't this question flawed? There are 40 boundary points as per the picture, but question mentions 36.

Yes, I corrected that in the question. I would submit that there's a difference between a simple oversight that can be noticed by anyone and rectified in seconds, and a question that fails to specify the fundamental parameters of the very situation about which it asks.

The OA I get for this question is $$\frac{32}{195}$$. Any confirmations? I will soon post a complete solution.

Mike
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Mike McGarry
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Re: In the x-y plane, the square region bound by (0,0), (10, 0) [#permalink]

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26 Dec 2012, 05:32
Mike,

My mind's boggled. Need the OE....XD
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Magoosh GMAT Instructor
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Re: In the x-y plane, the square region bound by (0,0), (10, 0) [#permalink]

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26 Dec 2012, 11:40
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eaakbari wrote:
Mike,
My mind's boggled. Need the OE....XD

Dear eaakbari,

I was waiting for someone to ask for the OE before posting it. Here it is.

Mike
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Mike McGarry
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Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

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Re: In the x-y plane, the square region bound by (0,0), (10, 0) [#permalink]

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11 Jan 2014, 10:20
Hi Mike,

Nice question. Do we know whether any such question has appeared in the GMAT in past or not? And what is the probablity of getting one such question

Sorry for my ignorance

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Magoosh GMAT Instructor
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Re: In the x-y plane, the square region bound by (0,0), (10, 0) [#permalink]

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12 Jan 2014, 20:13
Rohan_Kanungo wrote:
Hi Mike,

Nice question. Do we know whether any such question has appeared in the GMAT in past or not? And what is the probablity of getting one such question

Sorry for my ignorance

Dear Rohan_Kanungo,
This is probably a notch harder than what would appear on the GMAT, although if you are acing everything else on the math section and the CAT is throwing the hardest possible questions at you, then who knows? Perhaps you could see something of this sort.
I hope that helps.
Mike
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Re: In the x-y plane, the square region bound by (0,0), (10, 0) [#permalink]

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28 Apr 2014, 23:35
I dont agree..Even if u are acing everything else on the GMAT..this is pure labor work..counting..This question took me a good 7-8 mins..And I was not sure where I am gonna end up..
No smart tricks to be applied here..I guess this is not a GMAT Q..Not even at 800
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Re: In the x-y plane, the square region bound by (0,0), (10, 0) [#permalink]

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29 Apr 2014, 13:41
JusTLucK04 wrote:
I dont agree..Even if u are acing everything else on the GMAT..this is pure labor work..counting..This question took me a good 7-8 mins..And I was not sure where I am gonna end up..
No smart tricks to be applied here..I guess this is not a GMAT Q..Not even at 800

Hmmm. I could see this question taking a bit more than two minutes, but 7-8 minutes seems extraordinarily excessive. It sounds as if you did not choose the most efficient route. I point out that many of the most difficult GMAT math questions have this quality --- they are inordinately time-consuming, unless you have the insights that shorten the calculations. This calculation is still time-consuming, but symmetries reduce the calculations considerably. Understand: just because one person, even a person very good at math, winds up spending a lot of time to solve a problem, that doesn't necessarily mean it's too hard to be on the GMAT: indeed, that is precisely what characterizes the hardest math problems on the GMAT.
Does this make sense?
Mike
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Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

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Re: In the x-y plane, the square region bound by (0,0), (10, 0) [#permalink]

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30 Apr 2014, 00:56
mikemcgarry wrote:
JusTLucK04 wrote:
I dont agree..Even if u are acing everything else on the GMAT..this is pure labor work..counting..This question took me a good 7-8 mins..And I was not sure where I am gonna end up..
No smart tricks to be applied here..I guess this is not a GMAT Q..Not even at 800

Hmmm. I could see this question taking a bit more than two minutes, but 7-8 minutes seems extraordinarily excessive. It sounds as if you did not choose the most efficient route. I point out that many of the most difficult GMAT math questions have this quality --- they are inordinately time-consuming, unless you have the insights that shorten the calculations. This calculation is still time-consuming, but symmetries reduce the calculations considerably. Understand: just because one person, even a person very good at math, winds up spending a lot of time to solve a problem, that doesn't necessarily mean it's too hard to be on the GMAT: indeed, that is precisely what characterizes the hardest math problems on the GMAT.
Does this make sense?
Mike

It took me a complete 1 minute to get hold of the problem & decide an approach..tried some counting with formulas and PnC..messed up...3 minutes already...Started counting again..as there as so many cases and you have to take care of repetitions and close outliers..It took me a good 7 mins & all the way along I am constantly doubting my approach..coz I feel GMAT never tests such counting problems...
Think of it is how can some one come up with 7 lines and 6 lines in each case without an exactly symmetrical figure at hand..Some cases are too close..
If there is some approach to beat the time taken in counting the numbers of lines from each point..it is possible that this be done in 3 mins..Else I dont think this is a 2-3 mins question
I think options with highly varying values can also help here
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Re: In the x-y plane, the square region bound by (0,0), (10, 0) [#permalink]

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30 Apr 2014, 10:48
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JusTLucK04 wrote:
It took me a complete 1 minute to get hold of the problem & decide an approach..tried some counting with formulas and PnC..messed up...3 minutes already...Started counting again..as there as so many cases and you have to take care of repetitions and close outliers..It took me a good 7 mins & all the way along I am constantly doubting my approach..coz I feel GMAT never tests such counting problems...
Think of it is how can some one come up with 7 lines and 6 lines in each case without an exactly symmetrical figure at hand..Some cases are too close..
If there is some approach to beat the time taken in counting the numbers of lines from each point..it is possible that this be done in 3 mins..Else I dont think this is a 2-3 mins question
I think options with highly varying values can also help here

Dear JusTLucK04,
To some extent, this conversation might be irrelevant. This problem probably is much harder than anything that would appear in the GMAT: I didn't create it as an example of "you will need to know this on the GMAT" --- instead, I created it as a challenge problem in respond to another problem that, I felt, was quite ill-defined. I think we can agree that pretty much no one can solve this in 90 seconds, and therefore, it is outside the scope of GMAT math.

The entire issue of the "close" lines, the segments that are either just inside or just outside the square --- that's very subtle. If one had to perform a separate y = mx + b calculation for each "close" line, then of course, the problem would take forever. Classifying those "close" lines can be very quick with subtle visual reasoning skills, even without a scaled diagram drawn. That's always very tricky, because what is a laborious thing for one person to discover, another person can see immediately. Much of Geometry and Coordinate Geometry has that quality, and in much simpler problems, this kind of "seeing" can still be an enormous time-saver. I would say: do not underestimate Geometry and Coordinate Geometry. Do not underestimate the power of deep visual reasoning.

See this blog for not-as-hard problems in Coordinate Geometry:
http://magoosh.com/gmat/2014/challengin ... questions/
The last three are "counting" problems involving Coordinate Geometry.

Mike
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Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

Kudos [?]: 8459 [1], given: 102

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Re: In the x-y plane, the square region bound by (0,0), (10, 0) [#permalink]

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30 Apr 2014, 12:34
mikemcgarry wrote:
JusTLucK04 wrote:
It took me a complete 1 minute to get hold of the problem & decide an approach..tried some counting with formulas and PnC..messed up...3 minutes already...Started counting again..as there as so many cases and you have to take care of repetitions and close outliers..It took me a good 7 mins & all the way along I am constantly doubting my approach..coz I feel GMAT never tests such counting problems...
Think of it is how can some one come up with 7 lines and 6 lines in each case without an exactly symmetrical figure at hand..Some cases are too close..
If there is some approach to beat the time taken in counting the numbers of lines from each point..it is possible that this be done in 3 mins..Else I dont think this is a 2-3 mins question
I think options with highly varying values can also help here

Dear JusTLucK04,
To some extent, this conversation might be irrelevant. This problem probably is much harder than anything that would appear in the GMAT: I didn't create it as an example of "you will need to know this on the GMAT" --- instead, I created it as a challenge problem in respond to another problem that, I felt, was quite ill-defined. I think we can agree that pretty much no one can solve this in 90 seconds, and therefore, it is outside the scope of GMAT math.

The entire issue of the "close" lines, the segments that are either just inside or just outside the square --- that's very subtle. If one had to perform a separate y = mx + b calculation for each "close" line, then of course, the problem would take forever. Classifying those "close" lines can be very quick with subtle visual reasoning skills, even without a scaled diagram drawn. That's always very tricky, because what is a laborious thing for one person to discover, another person can see immediately. Much of Geometry and Coordinate Geometry has that quality, and in much simpler problems, this kind of "seeing" can still be an enormous time-saver. I would say: do not underestimate Geometry and Coordinate Geometry. Do not underestimate the power of deep visual reasoning.

See this blog for not-as-hard problems in Coordinate Geometry:
http://magoosh.com/gmat/2014/challengin ... questions/
The last three are "counting" problems involving Coordinate Geometry.

Mike

Agreed..I guess it was just me being pissed of at myself for not being able to solve the question....I am targeting a 700 Plus and you mentioned somewhere that this is something one can expect at that level..
Thank You for the link..I attempted all questions..Awesome stuff..Got them right except the last one
Thanks a lot.. again..
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Re: In the x-y plane, the square region bound by (0,0), (10, 0) [#permalink]

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08 Sep 2014, 15:24
mikemcgarry wrote:
eaakbari wrote:
Mike,
My mind's boggled. Need the OE....XD

Dear eaakbari,

I was waiting for someone to ask for the OE before posting it. Here it is.

Mike

Hi Mike,
Is there another way of determining how many lines run through the square?
If it's purely by looking at the image, isn't there a chance we'll count some false-positive or miss a few false-negative?

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Re: In the x-y plane, the square region bound by (0,0), (10, 0) [#permalink]

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08 Sep 2014, 17:17
ronr34 wrote:
Hi Mike,
Is there another way of determining how many lines run through the square?
If it's purely by looking at the image, isn't there a chance we'll count some false-positive or miss a few false-negative?

Dear ronr34,
Keep in mind, this problem is much much harder than a typical GMAT Quant problem.

Go back and look at the explanation carefully. It was NOT done purely by looking, purely by visual estimation. Instead, I used sophisticated reasoning about proportions in each case to determine exactly which lines did or didn't go through the square. Because this is a super-hard problem, I just didn't spell out every last detail of my reasoning in the OE. Go back to the OE and see if you can do all the proportional reasoning on your own. That would be good practice for any proportional reasoning you may need on the GMAT.

Algebra is an incredibly poor choice for this problem: that would take 10x longer. Proportional reasoning is quick and efficient.

Mike
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Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

Kudos [?]: 8459 [0], given: 102

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Re: In the x-y plane, the square region bound by (0,0), (10, 0) [#permalink]

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10 Sep 2014, 23:45
mikemcgarry wrote:
ronr34 wrote:
Hi Mike,
Is there another way of determining how many lines run through the square?
If it's purely by looking at the image, isn't there a chance we'll count some false-positive or miss a few false-negative?

Dear ronr34,
Keep in mind, this problem is much much harder than a typical GMAT Quant problem.

Go back and look at the explanation carefully. It was NOT done purely by looking, purely by visual estimation. Instead, I used sophisticated reasoning about proportions in each case to determine exactly which lines did or didn't go through the square. Because this is a super-hard problem, I just didn't spell out every last detail of my reasoning in the OE. Go back to the OE and see if you can do all the proportional reasoning on your own. That would be good practice for any proportional reasoning you may need on the GMAT.

Algebra is an incredibly poor choice for this problem: that would take 10x longer. Proportional reasoning is quick and efficient.

Mike

Hi Mike,
I've already read your detailed OE. But it basically starts off by counting how many lines run through the square from the bottom corner. My problem is, during the test, on a maybe smaller screen than I have at home, I might estimate that instead of 7 lines running through it, maybe 9 lines run through it.
This would of course change the whole answer, as well as maybe affect further calculations...

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Re: In the x-y plane, the square region bound by (0,0), (10, 0) [#permalink]

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11 Sep 2014, 10:37
ronr34 wrote:
Hi Mike,
I've already read your detailed OE. But it basically starts off by counting how many lines run through the square from the bottom corner. My problem is, during the test, on a maybe smaller screen than I have at home, I might estimate that instead of 7 lines running through it, maybe 9 lines run through it.
This would of course change the whole answer, as well as maybe affect further calculations...

Dear ronr34,
My friend, first of all, this question is NOT representative of what you see on the GMAT. Repeat: NOT representative of what you will on the GMAT. I was inspired by another post to create a challenge problem. Among other things, sometimes very strong math students like the practice of a challenge question. You should appreciate this problem for the skills you might learn from it. You categorically should NOT be thinking about: how would I do this exact problem on the GMAT, because it is well beyond what the GMAT would give you.

Furthermore, I think you are still falling in the trap of relying too much on the diagram. The diagram is purely a convenience here: it may help a user organize her or his thoughts & approach, but it is absolutely not necessary at all to solve the problem. There is absolutely no visual estimation in the solution to this problem. There is absolutely no number in the problem that one needs to get from the diagram. I strongly suspect that you still don't fully grasp the power of proportional reasoning and symmetry to produce exact answers in this situation.

For instance, from the midpoint of the left side, (0, 5), extend a line right through the upper left corner of the green square, (4, 6). This line has a slope of 1/4, so it goes through (8,7) and (10, 7.5). If it intersects the left side at (10, 7.5), a line can start at (0, 5) and go through below this line and intersect the square: (10, 7) & (10, 6) are possible endpoints of segments that work. Obviously, a horizontal line would intersect, so (10, 5) could be the destination, and finally there would be the two symmetrical lines below the horizontal: five lines total from this midpoint.
Again, from a corner, say, (0, 0), consider the line through the upper left corner of the square (4, 6). This is a line that passes through (2, 3), then (4, 6), then (6, 9), and passes through the top at x = 6 2/3, between 6 & 7. Therefore, endpoints could be the three points (7, 10), (8, 10), and (9, 10) at the top, obviously the corner (10, 10), and by symmetry, three more points down the top of the right side, for a total of seven.
Notice
(a) absolutely nothing is approximated; absolutely nothing is estimated. It is all 100% rigorously precise.
(b) the diagram is purely a convenience, but one can and should visualize all this in one's head.
(c) I took time to spell all this out, so that you could could follow my thinking, but these are the sorts of things that I "saw" in a single flash. When put all this into left-brain words, it appears lengthy and laborious, but in fact it's an immediate right-brain apprehension. That's the kind of facility with proportional reasoning for which you should strive.
From any of the 40 boundary points, you can compute the exact line that would go through a corner, and using proportional reasoning with that slope, you can see exactly where it will hit the opposite side. Those lines will form the "boundaries" and between those lines are segments that work. I cannot emphasize enough: this "seeing" is not necessarily related to the diagram at all: all this "seeing" should happen in your head. This is the kind of visual reasoning & proportional thinking that will really help you on the GMAT.

Does all this make sense?
Mike
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Re: In the x-y plane, the square region bound by (0,0), (10, 0)   [#permalink] 11 Sep 2014, 10:37

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