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MIT PRIMES/Art of Problem Solving

CROWDMATH 2017: The Broken Stick Problem

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Broken Stick J
Question about publishing the solution
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myfavouritenumberis73
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This topic is linked to Problem 6.
Y by Adventure10, Mango247
I have a little question about putting up a solution, does I have to put it up here or can I put it up somewhere else?
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Question about publishing the solution
myfavouritenumberis73   0
Nov 27, 2018
I have a little question about putting up a solution, does I have to put it up here or can I put it up somewhere else?

0 replies
myfavouritenumberis73
Nov 27, 2018
0 replies
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Strategy on computing the probability to (c)
myfavouritenumberis73   1
N Nov 24, 2018 by Nirmay
(C)

A simpler case should be used, where n=1.

Since the equation of a line is y=mx+c, all the possible values of m and c using an inequality should be plotted on a graph, where the x-axis is c and y-axis is m. Then another region should be drawn which points inside, (c,m), that would give the equation of a line that would cut the square into to unequal regions. Then the ratio of these two regions should be calculated. This would probably require one to look at individual cases of what some of the lines would look like.

1 reply
myfavouritenumberis73
Oct 16, 2018
Nirmay
Nov 24, 2018
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Expected Area of Largest Perimeter
adamodoherty   2
N Nov 8, 2017 by GeronimoStilton
This is a problem which I think was discussed before but never made it all the way to a thread. In the random lines problem, what is the expected area of the longest piece of perimeter which has not been interrupted by random lines? What is the probability that no piece is longer than $x$?

We're playing the game on the unit square, which obviously has area $4$. A lower bound then is to define an uninterrupted piece of perimeter which we can call $p$ (make it the longest piece), we then say that for $n$ random lines the probability that $p$ is interrupted is approximately $2p/4$ and that the expected value of $p$ when intersected is obviously $p/2$.

This is obviously not a super great lower bound but it should suffice as a start.
2 replies
adamodoherty
Oct 10, 2017
GeronimoStilton
Nov 8, 2017
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Problem 6a
RadioActive   21
N Oct 16, 2017 by v_Enhance
[quote]The square $[0,1]^2$ is cut into several regions by picking two random points on the border of the square at a time and connecting them to form $n$ "random" lines. What is the expected number of regions? [/quote]

Solution
21 replies
RadioActive
Mar 19, 2017
v_Enhance
Oct 16, 2017
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Dope Lower Bounds on Area Being less than 1/2 with random li
adamodoherty   17
N Oct 9, 2017 by v_Enhance
The best way to approach this probem is finding some $1/2$ area subset and then finding the probability that this area is not inteferred with after $n$ lines. We can do the area is equal to $1/2$ idea with a number of different shapes. These shapes which we can use will provide a lower bound which I have constanty improved upon.

If we start with a square which is tilted and placed in a corner (of course the square with area $1/2$), without loss of generality we place the first point of a random line on the bottom half of the right side across from our tilted square, the only way our square is not interferred with is if the next line crosses to the left side and in the opposite half as our square. For this the probability is equal to $1/6$ therefore a lower bound on the lines not interferring with the square is $1/6^n$.

A slightly better lower bound is created using different shapes, for example the circle which shares a center with the square in question and has radius $1/ \sqrt(2\pi)$. There is an analytical way to solve the problem with a square but it's not pretty so I just made a similation. Using a simulation with the probability that a line misses the circle is around $p=0.345$. So an estimate now is that the lower bound of the area being less than $1/2$ is $0.345^n$.

This is a pretty crude way to calculate a lower bound but on the bright side it's very easy and I think it yields another question:

What shape does the best at avoiding random lines and why?

Some more questions to consider:

What is the average area largest area after break number $n$?

What is the expected value of the largest piece of perimeter between random lines after break number $n$?

I also tried equilateral triangle which is somewhere in between a square and a circle. I'm struggling to see why this is, perhaps it's something obvious which I'm missing...

UPDATE: I got around $0.32^n$ with an octagon, kind of seems like circle might be the ideal shape though at this point I have absolutely no idea how to prove it.
17 replies
adamodoherty
Aug 3, 2017
v_Enhance
Oct 9, 2017
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Problem 6 employing Option (d) to choose lines
goodbear   13
N Aug 5, 2017 by v_Enhance
We use option (d), Draw a line through the center in a random direction, to attempt problem 6.

[quote](a) What is the expected number of regions?[/quote]
This is obviously $2n,$ as there are $2n$ angles at the center, and one region for each angle. The probability that two lines coincide is zero, as there is an infinite number of lines to choose from.

[quote](b) What is the expected number of triangles?[/quote]
This is just $2n - 4,$ as each of the $2n$ regions are triangles between two of the lines and one edge of the square, except for the four regions containing the corners. The probability of a line going through one of the corners is zero, because there are an uncountable number of options for the line.

[quote](c) What is the probability that some part has area at least 1/2?[/quote]
If $n=1,$ since all lines passing through the center of the square split the square in half, this probability is 1.
If $n>1,$ since the square was already split in half by the first line, each piece will be guaranteed to be less than $\frac 1 2,$ as each piece is further split by the additional lines.
13 replies
goodbear
Feb 9, 2017
v_Enhance
Aug 5, 2017
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Quick question
adamodoherty   2
N Jun 21, 2017 by adamodoherty
Sorry to bring this up as I'm sure it has already been addressed but how are we defining "random lines" for this problem?
2 replies
adamodoherty
Jun 20, 2017
adamodoherty
Jun 21, 2017
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Question on 6d
RadioActive   2
N Apr 8, 2017 by v_Enhance
I'm not sure how fitting in is defined. For example, are we allowed to rotate the regions?
2 replies
RadioActive
Apr 6, 2017
v_Enhance
Apr 8, 2017
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Random lines
LaTeX_turtle   4
N Mar 1, 2017 by v_Enhance
Option d probably cannot be considered a random line. Also, options a-c all reference the square in some way. How about choosing a line with a random slope and intercept, or (equivalently) a random point and angle? And then, from those, picking the ones that intersect the square? Would that be equal to any of the options given? It wouldn't be equal to option b, because for option b, the lines nearer the corners of the square (smaller intersect with the square) would be less likely. I'm saying we just define every line to have equal probability to every other.
4 replies
LaTeX_turtle
Feb 6, 2017
v_Enhance
Mar 1, 2017
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Probability of a Line
chocopuff   2
N Feb 7, 2017 by shinichiman
The inspiration behind this is from goodbear mentioning here at the very bottom that random does not mean uniformly random.

So I propose that one important question we should first tackle is given a line through the square, what is the probability that if we choose a line through the square that it will be that given line? (Should we count identical lines off by a translation?)

Also, we should develop a well defined notation that will be able to describe any line.
2 replies
chocopuff
Feb 7, 2017
shinichiman
Feb 7, 2017
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Question about publishing the solution
myfavouritenumberis73   0
Nov 27, 2018
I have a little question about putting up a solution, does I have to put it up here or can I put it up somewhere else?

0 replies
myfavouritenumberis73
Nov 27, 2018
0 replies
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