Mathematical Prison Escape: Create challenge where prisoner has to use geo/trig to escape

[bob thompson]

Hello,
Recentely i started teaching Math at a local High-School, and noticed they really liked talking about crime-related topics, so in order to motivate them to study more i decided to make a challenge where a prisioner has to escape by using concepts of geometry and trigonometry, as well as analytic geometry, however i could not get anything out of my mind and would really appreciate if anyone has any ideas for a problem like this.
Thank you very much for the attention.

 

[stapel]

Hello,
Recentely i started teaching Math at a local High-School, and noticed they really liked talking about crime-related topics, so in order to motivate them to study more i decided to make a challenge where a prisioner has to escape by using concepts of geometry and trigonometry, as well as analytic geometry, however i could not get anything out of my mind and would really appreciate if anyone has any ideas for a problem like this.
Thank you very much for the attention.

Unfortunately, writing papers and completing projects for students is not a service which is offered here. ("Read Before Posting")

Please reply with whatever thoughts and efforts you have made thus far. Thank you.

 

[lev888]

Hello,
Recentely i started teaching Math at a local High-School, and noticed they really liked talking about crime-related topics, so in order to motivate them to study more i decided to make a challenge where a prisioner has to escape by using concepts of geometry and trigonometry, as well as analytic geometry, however i could not get anything out of my mind and would really appreciate if anyone has any ideas for a problem like this.
Thank you very much for the attention.

Prisoners can dig a tunnel from inside the prison building, bypassing the outside fence from below. Given various distances and the depth of the fence calculate the tunnel sections' angles and lengths.

 

[Dr.Peterson]

Hello,
Recentely i started teaching Math at a local High-School, and noticed they really liked talking about crime-related topics, so in order to motivate them to study more i decided to make a challenge where a prisioner has to escape by using concepts of geometry and trigonometry, as well as analytic geometry, however i could not get anything out of my mind and would really appreciate if anyone has any ideas for a problem like this.
Thank you very much for the attention.

What particular techniques do they know so far, that they could use?

Can you show us any idea you had, even if you decided it was not good enough? That might give us a place to start (and also show that you are serious about doing the work).

I'm wondering if you could make a diagram of the prison, fences, guard towers, etc. with dimensions labeled (but maybe not to scale, just a sketch), and they have to figure out how much space there is where an escapee could not be seen.

On the other hand, maybe teaching your students how to break out of prison isn't as good an idea as teaching them how to solve a crime ...

 

[lev888]

On the other hand, maybe teaching your students how to break out of prison isn't as good an idea as teaching them how to solve a crime ...

If you are imprisoned for a crime you did not commit, in order to solve the crime, you need to get out first. So, better learn trigonometry, or you'll end up digging a tunnel to some French dude's cell.

 

[bob thompson]

Unfortunately, writing papers and completing projects for students is not a service which is offered here. ("Read Before Posting")

Please reply with whatever thoughts and efforts you have made thus far. Thank you.

Hello, i'm very sorry about that.
As you can see, it's my first time using this website, and i am still not very much used to the rules.
Still, i had lots of ideas on how to make this work out, the first of them was to make the main building of the prison a circle, wich once you found the equation, it would also be possible to figure out a line that goes through both the prisioner's cell and an entrance to the sewers, and make him escape from there.
However i found this idea very lame, and wanted it to be more chalenging, that's why i started looking for help.

 

[bob thompson]

Prisoners can dig a tunnel from inside the prison building, bypassing the outside fence from below. Given various distances and the depth of the fence calculate the tunnel sections' angles

What particular techniques do they know so far, that they could use?

Can you show us any idea you had, even if you decided it was not good enough? That might give us a place to start (and also show that you are serious about doing the work).

I'm wondering if you could make a diagram of the prison, fences, guard towers, etc. with dimensions labeled (but maybe not to scale, just a sketch), and they have to figure out how much space there is where an escapee could not be seen.

On the other hand, maybe teaching your students how to break out of prison isn't as good an idea as teaching them how to solve a crime ...

Hello.
i had many ideas, the first of them was to make the main building of the prison a circle, wich once you found the equation that describes it, it would also be possible to figure out a line that goes through both the prisioner's cell and an entrance to the sewers, also having to make a pythagorean theorem in order to calculate the distance and the time it would take, than making him escape from there.
Yes, i left a sketch of what the prison looks like in my head attached in this message (As i said, i decided to leave the circle idea behind), but i insist, if you have any better design ideas or even conecpts that i could use, please share them with me, after all my drawing isn't the best and there's still lots of room for improvement (i thought it would be nice for the prisioner to be at cell 21, that´s why i marked it).
About the techniques they know, we have just finished calc II and analytic geometry, and i am currently preparing their final exam.
I decided to make this in order to cheer them up for a little bit, they had a rough year and i wanted to make a special challenge before they leave school for good.

 

[blamocur]

I remember a nice puzzle (but don't remember the solution), which can be modified to fit some prison scenario: you are lost in a forest of infinite length and constant width of, say, 1 mile. Figure out a walking path to get you out of the forest so that the maximum possible (or, maybe, the average) length would be minimal. In the prison case that could be a tunnel dug to the outside of the fence where the prisoner knows the shape and the dimensions of the fence but doesn't know the location of their cell.

 

[bob thompson]

Hi!
That sounds really interesting, i'll research a little bit better.
But still you have a link or something of the full original problem?

Edit: i was able to find the problem you mentioned(or at least a similar one) and there was no solution on how to get out of the forest.

 

[lev888]

Hi!
That sounds really interesting, i'll research a little bit better.
But still you have a link or something of the full original problem?

Edit: i was able to find the problem you mentioned(or at least a similar one) and there was no solution on how to get out of the forest.

Can you come up with a solution?

 

[blamocur]

Hi!
That sounds really interesting, i'll research a little bit better.
But still you have a link or something of the full original problem?

Edit: i was able to find the problem you mentioned(or at least a similar one) and there was no solution on how to get out of the forest.

I don't even remember if I saw it online or elsewhere. But a similar problem is posted here.

 

[blamocur]

I don't even remember if I saw it online or elsewhere. But a similar problem is posted here.

And another one here.

 

[bob thompson]

Can you come up with a solution?

Hey!
No, i can't...
I don't know if my skills as a mathematician eventually faded away as i began getting used to teaching more basics and more introductory subjects rather than more sophisticated ideas or if i simply can't see the solution...
After all, if i'm in a forest with infinite lenght to all directions, no matter where i go, or what i do, i'll always be in the forest, in a 2 dimentional point of view, of course, Hahahahahahahahaha!

 

[bob thompson]

And another one here.

Hello!
I found the idea of the problem very interesting, and i must admit, i had lots of fun trying to figure it out by myself, hahahahaha!
(I'm getting a bit rusty with this type of problem)
I really want to use something similar in my problem, still need to figure out how, but i really wanna make it so that he needs to reach the sewers or calculate a tangent line.
I Think i will try to design it so that he doesn't know exactly at wich point the sewers intercects his cell, an thus must make a spiral, as your original problem with the ant...
I really don't know what to do just yet, however, your problems, both the forest one and the ant one gave me some pretty wild ideas wich i'm going to give a more elaborate thought tomorrow.
Thank You very much!

 

[lev888]

Hey!
No, i can't...
I don't know if my skills as a mathematician eventually faded away as i began getting used to teaching more basics and more introductory subjects rather than more sophisticated ideas or if i simply can't see the solution...
After all, if i'm in a forest with infinite lenght to all directions, no matter where i go, or what i do, i'll always be in the forest, in a 2 dimentional point of view, of course, Hahahahahahahahaha!

Well, fortunately, it's not infinite in all directions. Given that it's 1 mile wide, there are several methods to get out. Let's say you walk straight for 1 mile and you are still in the forest. What do you do next?

 

[bob thompson]

Well, fortunately, it's not infinite in all directions. Given that it's 1 mile wide, there are several methods to get out. Let's say you walk straight for 1 mile and you are still in the forest. What do you do next?

Hey!
I see you are using the problem from reddit, the problem i thought about was the one presented in the original post, right here in the forum...
But given the fact that it's 1 mile wide, and i walked 1 mile in a straight line but i am still inside the forest, than i suppose that if i start making a trajectory in the shape of a circle, i should be able to find my way out.
Another way to view this is by considering my position as the origin of the forest and then walking a mile to either points (1,0); (0,1); (-1,0) or (0,-1), than following the path described by x² + y² = 1.
As i mentioned, i am getting very rusty with time, and this solution may be wrong, but still, i guess it's an idea...
I noticed you're very active here, thank you very much for your ideas!

 

[blamocur]

As I mentioned in post #8, because you don't know where you are inside the forest you either want to find the best path for the worst case scenario, or the best path on average. I don't remember solutions for either problem, but I am sure some can be found online.

 

[bob thompson]

The reddit post you sent in post #12 has some solutions in the comment, still i found it more fun try and solve it by myself.
You seem to be very creative so i am going to share with you my ideas so far, and would really appreciate if you could give me any type of advice:
I thought of making the prison also have a laundry room, where he could steal some clothes from, and use those to make a rope of any lengh, than by mesuring the angle formed by the sunlight reaching the floor from the window of his cell, and by having a rough idea of the distance between his cell and the prison wall, he could calculate the wall's height and use that to know what size of rope he must make in order to escape.
I also thoght of using the idea of making the prison have some sort of plumbing system, and having him calculate the distance between the cell and the wall by knowing there's a pipe wich goes under all restrooms, thus by knowing the coordinate of every bathroom he coud than find out the equation to that line, wich could then be used to calculate the distance between his cell and the wall.
I also had the idea of making he escape by using the reaction between CaO3 and sulfuric acid wich expands in volume after reacting with water making plaster of paris than gypsium, with the math revolving around him calculating the necessary amount of CaCO3 and Sulfuric acid in order to generate enough strenght to make a hole in the wall (i chose this idea, because the chemical reaction wouldn't produce lots of noise and also would definetely be much quicker and practic than trying to dig a hole by hand).
Anyways, any type of idea and feedback is very welcome, please do not hesitate in sharing your thoughts with me.

 

[blamocur]

Seems you have enough ideas for a whole game, not just one puzzle

I also thoght of using the idea of making the prison have some sort of plumbing system, and having him calculate the distance between the cell and the wall by knowing there's a pipe wich goes under all restrooms, thus by knowing the coordinate of every bathroom he coud than find out the equation to that line, wich could then be used to calculate the distance between his cell and the wall.

Aren't there some unstated assumptions, like all the pipes form a straight line and it is the shortest distance from the cell to the wall? Also, why is this distance important? If the prisoner is digging a tunnel than it must go beyond the prison fence, not just its wall. Or is there no fence in this prison (urban type)?

I wish I were as creative as you think I am, but whatever creativity I have is mostly in solving math problems, not in inventing them.

Good luck with your project!

 

[bob thompson]

Seems you have enough ideas for a whole game, not just one puzzle


Aren't there some unstated assumptions, like all the pipes form a straight line and it is the shortest distance from the cell to the wall? Also, why is this distance important? If the prisoner is digging a tunnel than it must go beyond the prison fence, not just its wall. Or is there no fence in this prison (urban type)?

I wish I were as creative as you think I am, but whatever creativity I have is mostly in solving math problems, not in inventing them.

Good luck with your project!

Hey!
Yes, there are some assumption i created in order to make the problem more practical to be solved, such as the one in the plumbing system...
About the line, i was thinking of using the line's equation to determinate the distance between the cell and the fence wich would then be used to calculate the height of the fence, since (fence's distance to cell)/(fence's height)=( Window Shadow's lengh)/(Window's height), (i'm using the proportions found in pythagorean triangles with equal angles) then allowing us to finally know exactly how many clothes he must steal from the laundry in order to make the perfectly sized rope to escape...
You could make the whole problem even fancier by asking how much time he would need to stay in the prison in order to properly execute his plan, if given the avarage time he takes to make each task, such as saying the laundry is only retrieved every tuesday and thursday and that he can only steal on one of those 2 days.
But still i found you very creative, many ideas i only had because you talked to me about concepts that although very distant from my initial thoughts, helped me develop a series of other wanderings that i wouldn't have accessed otherwise.
My Students' finals are next week, i hope they'll at least like the final version of the problem, i'll try to post here as soon as i think it's good enough, but still i really want to know if you people got any other resourse i could use to make the problem more elaborate and fun.