Hope you guys enjoy! Two things to mention here. 1) The video that explains the last method for the 'lost fisherman problem' is already up on patreon and will be out in just a few days from this one's release. 2) Most of these examples/stories actually came from 2 books I recently read and those are linked in the description if you want to learn more math/optimization.
@@naswinger It's on patreon now and will be out on the channel in 3 days. And for everyone asking about how this comment was posted 5 days ago, videos are posted early on patreon (plus I need to make them unlisted in order to get approval from the sponsor).
@@abhijanwasti7991 hmm i think you could start doing the 1km circle and then move to the 1.04km towards the end i did some drawing on paint and came to the conclusion that you do 1km circle normally until you completed 3/4 of it and then stop turning and move in a straight line -> 1 + 3pi/2 + 1
Let's frame the question better: you travelled 100 km away from the shore. Suddenly, it becomes foggy, and a violent gust tilted your boat for an unknown angle. Your fuel tank is leaking, and the engineer cannot stop it completely. All communication stopped. Survive the day.
Very very fun boat problem, thank you! The best I managed to get was ~6.4 The strategy I came up with is: - go in a straight line for some distance r - draw the two tangents to the unit circle that pass through your current location, label them A and B - follow A until you're on the unit circle - follow the arc of the circle until your direction of motion is perpendicular to B - go in a straight line towards B The total distance in the worst case scenario is given by r + sqrt(r²-1) - 2arccos(1/r)+ 3π/2 + 1 The optimal solution is for r ≈ 1.16
This was really clever! Only thing I'm not seeing is where you got the 3arccos(1/r), the way I drew this out I'm getting that it would be 2arccos(1/r), assuming the worst case is where you barely miss the shore and drive in an arc until you reach basically where 3pi/2 is on the unit circle and then drive perpendicular to the shore until you hit it (which would be 1km). In that case you'd drive the entire 3pi/2 arc - 2arccos(1/r). Or maybe I just didn't interpret your method correctly.
How do you actually come up with these amazingly cool solutions?...wow ...is it easier to come up with such knowing optimizations ? P.S. I have not yet learnt calculus ... probably will be learning them this year... ~ A sophomore (10th grade)
@@sudheerthunga2155 for me it didn't involve much calculus, the key was to reimagine the problem as stretching a rope around a circle with a couple of extra conditions. It was a really fun process and not tedious at all. The calculus only came at the very end for finding the optimal angle, but there you might as well just plot the graph.
Just recently got interested in math. Wish I had before. Like in school, everyone asked "why do we need to learn this?" and no one could ever answer. "Because you should". It's as if they tried their best to make it as uninteresting and boring as possible. Had someone talked about humanity going to Mars or like the golden ratio... what a difference it would have made.
Yep. One of the failures about education is that teachers often can't communicate why is the thing important, or how it may be fun or interesting. I was never interested in biology for example, and yet I caught myself watch full series about evolution of life by Aron Ra right from the single cell organisms up to humans, and thats 40 something episodes 5-15 minutes each. Which is a far too much for a guy who is not interested in biology =)
I was the same way in high school but learned to actually think math is pretty cool. You can almost tell the future with it. For example, everybody places an x where they guess a ball is going to land where you would calculate it with math and predict it perfectly every time. The look on their faces when you accurately predict where EXACTLY that ball will land 10 times 50 times in a row.
@@gabrielbarrantes6946 I have done a ton of research into the current US education system and thought a lot about how to fix it and I can 100% definitively tell you that lack of curiosity is not at all an issue when first learning about these topics. School has a way of taking things you like and forcing you into the most uninteresting and uninspiring part of those subjects especially in the beginning. Talk to any child in grades 1st to 6th and you will generally find they like things like math. There are even alternate education systems that you can put children in right now that are completely driven by the child's curiosity as in they have no set curriculum and those systems have proven to even give better results than regular school on tests.
Hey man. I just wanted to thank you for the resource this channel is. At my high school we’re required to right a 4000 word essay on anything. I chose math and just had a bunch of trouble finding a topic within that. Thanks to videos like this I settled on optimizing baysean search theory. I honestly could not of done it without this channel and would of failed a class. Just wanted to say thank you
For anybody curious about the answer to the minimum spanning tree question, the algorithm they use is called Kruskal's Algorithm. Essentially, they sort the distances and, going over the sorted list, connect any two currently unconnected nodes.
move in a spiral pattern, so that the distance between the lines of that pattern (measuring at an angle that goes through the centre) never exceeds the view distance in the fog. you'll also find survivors that fell out of the boat, if they're lucky and you don't reach shore first
also what i thought first, till he mentioned the "what if we go the 1km at an angle approaching 0" then i realise, ok we gotta think in maths, not practical reality
1:08 *solves by logic* Stop. Since it's day, the wind breeze should go to the land, I feel the air hitting my face, so it should be the other way right?
This is pure genius. I read a book with similar content, and it was pure genius. "The Art of Saying No: Mastering Boundaries for a Fulfilling Life" by Samuel Dawn
One of my friends figured out how to find the Fermat point for math club in high school. One of the rest of us in the room (might have been me, but I think it was another kid) realized that you could take weights on strings and hang them through holes, and the stable point was the Fermat point. Bell should have hired a 10th grader to help them.
@5:00 Also, if you had turned right like in the figure, don't start your circle turning left, turn right again. Much more efficient. You will reach in (half the distance-1.04) or whatever distance.
6:41 Fermat point is shortest distance task, if you have extra variable let say bandwidth, minimum length is dominant, but bandwidth can change location point.
12:40 fun fact there is a much much simple method to solve such kind of problem...and even a bigger fun fact it does not use computers of any sort....ENTER THE SOAP BUBBLE. How?? u may ask...to be brief about it... You may take two sheets of acrylic sheets saperated by a small distance,all the points of interest are simulated by a screw joint of a sort holding the two sheets together. now dip the whole thing in a soap solution, the path traced by the soap bubble is the optimised one... you found your solution in no time...😊😊😊
So why are people spending potential millions and tons of manpower to make lasers that basically simulate what you can do with acrylic? Perhaps it's so that the information can be quickly analyzed by some computer and sent to the person who requested it. It still doesn't make quite as much sense but whatever.
@@ոakedsquirtle It depends on how many times you need to solve the problem. If you need to solve it once, no big deal. If you need to solve it 10 times per second do you still think building acrylic sheets and soaking in soap would be feasible?
Before watching a solution. I ended up with this equation: S(x) = x(1+ 2pi - 2acos(1/x)). Solving for the minimum gives roughly 6.995... at x = 1.044... I think that's a definite improvement compared to 7.28... at x=1. *EDIT:* ok, 30 seconds passed, and he repeated my solution word by word. And it's not even the best one D: gotta try more. *EDIT2:* Before I watch further. I got another equation: S(x) = x + 2sqrt(x^2-1) + 2pi - 4acos(1/x), this one gives the minimum of 6.459.. km at x = 1.242... that's my second guess.
A better solution is to go in a distance of "d", then go through an OUTWARD SPIRAL. Suppose the distance from the spiral from the original to the point that the spiral goes one complete circle is "f", i.e. f>d, r = (theta/2pi) * (f-d) + d. The final solution is in terms of both f and d, while under the constraint f>d. Let's say L is the worst-case scenario distance of travel, the resultant f and d are found by performing partial derivatives on L with respect to f and d, set the partial derivatives to zero. Oh well, I give up on the calculation..... and I assume the rate of radius growth of that spiral is linear to the angular distance of travel. If that is not the case, the ultimate solution perhaps requires variation calculus...... and I don't know what I am talking about.
Voronoi diagrams could also help to solve minimum distance problems like the triangle one. Given 3 points as the example, the point were the three Voronoi regions meets is the point that has the minimum distance between all three. There is a similar problem were a company has N werehouses that serve a certain region of a city and you have to find the best way to split these regions. The solution is given by the Voronoi diagram. Anyway, thanks for the great video Zach, have a good one!
To find the Fermat Point isn’t that the same as drawing a line from each vertex so it is perpendicular to the opposite line for two vertexs then where the lines intersect that would be the Fermat point? Please answer
I learned optimization and other than calculate the size of the structure built with certain prices of certain materials, I used it for calculating optimum stadium ticket price in a football manager game when I knew the current audience numbers, the current price and the size of the stadium. For maximum profit that is. But like for what wouldn't use optimization! But so we just assume that we can't see the shore until we're on the beach with the boat?
in the boat problem how do you know that you circle is big enough to reach the shore at all? wouldn't the worst case scenario be that you end up going directly away from shore and your circle ends up being too small to reach it?
You can slightly decrease the velocity for the basketball problem. This is only because a basket ball can hit the front of the rim and still go in. I don’t know how much but you could decrease it by a bur
Optimiztion is in a way common sense if you have a high IQ . That's why most of the times people with higher IQ solve problems faster by finding a pattern/Algorithm in their brain . I don't know if i have high IQ or not but believe me i also solve my in general life problems by finding patterns and just using my common sense , i solved most of the problems in the video by just using common sense .
you said you can't remember where the shore is. What if you go relative further out into the ocean, instead of towards the shore. Your area you will ancher around, and do a full circle will all be within the water
You travel *at least* the same distance away from your anchor, as your anchor is from the shore. This will guarantee that you always hit the shore before completing a full circle. (e.g. your anchor is 5km away from shore, then travel *at least* 5km in any direction).
A good current real life explanation of functions and calculus: a pandemic. All the infection rate models are based on functions, and add, subtract and multiply hundreds of functions (factors) that influence infection rate. These growth curves then are analysed using calculus to measure rate of changes : too high, we have a problem so increase safety measures. Low : open up some activities to find the balance point. And yes, they probably factor in extra measures to counter the 20% of people (or states) who won't follow measures. The other interesting problem was asymptomatic disease: we are used to seeing runny noses when people are sick. This disease you can show nothing for 2 weeks and spread it. That means cases you measure today are actually indicating spread 2 weeks ago, and you have to factor that in (you're actually measuring data that is 2 weeks behind what is actually happening today), another reason for calculus to determine what IS happening today.
Better solution: Go in the opposite direction of the boat's compass and travel that direction for 1 km. or Go in the opposite direction in which the waves are hitting any given side of the boat. Travel in that direction for 1 km. Both methods save well over 6km's in travel.
Regarding the boat problem, what if you drove into the wrong direction, so not towards the shore, but away from it. Thereby ending up, say 2 kilometers from shore, thus not reaching the shore using the 1km circle. Is it just an assumption that the rough direction of the shore is known or am I missing something?
7:00 wouldnt it be easier to just take the angle bisectors and let them intercept? This method also gives you the abstract solution for any given triangle...
We learn stuff in math cause people use math words in documentation and explanations of the worklife challenges, and we need to understand what we are reading so we don't spend time in our job being interrupted all the time by learning foundational advanced math concepts on top of the domain skills of the job. A few times a project in a job that isn't teaching or pure mathematics would actually require to do the math too. Mostly it's reapplication. Like say a comp sci person might reuse math already applied by someone ells.
In the boat example, if you are entirely unsure of where shore is and the fog hinders vision, then why wouldnt the most efficient way back be to turn the boat exactly 180 degrees and travel that path 1.05 km then do the circle thing if you dont hit shore?
Using test taking strategy, I would say nautilus spiral? Depending on the domain the answer is always choice C or the golden ratio (or e if it's larger than 2) or "they're the same" or Fibonacci or nautilus or there exists no such foo.
I understand you are what the point of the video is. I simply don't agree with the boat scenario at all. If you go on a boat trip for whatever reason, you should 1) Know the weather. 2) Know where you came from or at the very least have a sense of direction. and 3) Have someone that's experienced in boating and/or have someone onshore that can call for help in a situation like this.
It was my favorite yet most frustrating part of Calc 1 in high school. I loved it's direct applicability. The ones I hated the most was like filling up a cone, determining the rate at a certain height
I'm shit at math. I pretty much gave up on them when I was like 13 because well, I always had decent grades everywhere by doing the bare minimum I didn't feel like working to get better at it. That and I had a terrible teacher. I don't know why tho, but seeing how powerful they are at optimising stuff makes me regret giving up on them. I know I particularly enjoyed graph theory, finding the quickest routes and stuff... Oh well.
So you're saying you'd putt he anchor down where you were at 1km out, and then go 1km out and in a circle around that anchor and you're guaranteed. But that if you made any small error ( the 1 plus 2 pi thing) then you'd miss the shore by a smidge?
Yes you are. We left an anchor with rope at our "Lost Point" and drive in any direction as far or further than our distance to shore. Now drive in circle. You will always hit the shore because a circle with radius equal or greater than the distance from shore will intersect with the shore, no matter what. You were thinking of taking our guess drive as center, but we take it as our start around the perimeter. The only thing to think about then is to optimize *how far* we drive in any direction. He always showed the worst case, which is bad for understanding the problem, granted.
Surely for the first puzzle your given solution only works if you go in the rough direction of the shore - if the random angle you choose is further out to sea, your circle no longer hits the shore?
Ok, my answer for the boat. You want to turn 180° in place, which should be a maneuver all sailors can execute using a compass in the fog and then sail 1k back to the shore.
In the boat example you are in a situation where you do not know the direction of the shore, am I missing something in thinking that guessing the exact opposite direction of the shore would be the basis for a worst case scenario. If you know generally (within 180 degrees) of which direction the shore is, which seems to be implied given the worst case scenario doesn't consider you actually getting further away from the shore, then couldn't you travel 1km in your guessed direction. Then because you know the angle from the original point from which you traveled (say 1 degree in a bad guess, ending up just less than 1km from shore and just short of 1km North/Parallel to shore from the point at which you became disoriented) - you could then do a quarter rotation in the same direction of the original angle (ending up at 91 degrees rotation from where you started) hitting the shore before making the entire quarter rotation. Let me know what I am missing.
Yup, but use a triangle instead of a square. The square inside the circle has a perimeter of 6.93 km and the triangle has a perimeter of 5.2 km, both are which are lower than the circle minimum distance of 6.995.
sqrt(2)+1+pi+1= 6,5559 km Drive sqrt(2)=1,4143 km straight. Then 1 km in an angel of 45° straight back to the 1km inner circle. Sail the half-circle (pi=3,1416 km) and take the shortcut of 1 km to the worst-case shore.
@@JordanMatrix Yeah. Like the better method in the Video could be easily "optimized" after going to the 1,04km radius going back to the 1km radius on a nice angle.
I'm lost. The most efficient route is to rotate an angle greater than 90 degrees from your current direction and proceed forward. For all initial angles from shore, having traveling 1km, this would yield a < 2km return distance.
I’m confused doesn’t the ship have radar and staff locating where the ship is at what time or whatever? I know math is great and all but I’m pretty sure sailor have like water maps and stuff.
the problem with the boat thing is that the proposed methods only work if you move the boat in a direction that actually brings you closer to the shore. it completely falls flat if you move in a direction that leaves you further away from the shore.
For the first problem, isnt it inaccurate to compare different "worst-case scenarios", given that the diff distances require the boat to approach the shore at different angles? Wouldnt optimisation require all scenarios to be taken into consideration? Such that for each distance that the boat should travel before making the circle (be it 1km, 1.04km, 1.183km), each possible scenario for all 360 degrees should be taken into consideration, the distance for each scenario calculated, and the average distance travelled derived to truly optimise the solve. This does sound like a brute-force approach to finding out which is the most 'optimal', but still more optimised than the soln you proposed.
Heres a question from a non-maths person that ive been pondering. Im sure this has a name and is studied but i havent managed to find it. Say you have some number of people. Each person will derive different amount of happiness from x dollars. The more or less you give them, the more or less happiness they will derive (like a function). Now, say you have x dollars to distribute amongst these people. What formula would you use to maximise the total happiness (assuming you could sum the happiness of distinct people) of the people? You should find the optimum distribution wherein the sum of the outputs is the greatest.
With that first problem... If you're disoriented enough, you could end up say turned 360 degrees but think it was only 180, so you boat til you're 2.04 km from shore, and due to how lost you are, you could then arc _away_ from the shore. The circle will never hit it. It's why I'd always go for a spiral - It accounts for that Unless there is something I've missed which is possible
Hi Zach Star, awesome video. I had a question about text books because I'm going to highschool soon and wanted to get ahead in math and science, so I was wondering if you could tell me how to find the right textbooks to learn a subject
When I'm looking for a good textbook for math/science class I usually search quora and reddit for that same question and I always find some kind of response. Often there are good amazon reviews but they can be misleading as well depending on how technical or rigorous you are looking for a book to be.
So you are just going to ignore going parallel to the shore, or better yet any angle beyond that? You said disorientated, not vaugly aware of which side the shore is of you which with a few moments to think would lead you to think "oh I'll just average this feeling" which leads you to the shore
okay so i would say. go 1 km in the direction you're facing, turn right, go 1 km, turn right again you're now 180 from your first position so go 2 km, then go right, you're now 180 from ur 2nd position so go 2 km worst case scenario the Shore is East, you start facing south 1 km, then west 1 km, then north 2km, then east 2km, 6km total traveled.
Hi Zach. On optimization problems. How about variable situations like a factory making something out of a variety of parts, materials, etc. Does the optimization solution for something change under a different viewpoint.
For the first problem: What i dint rlly get. What if ur going 1 km out on the sea? Like if i lost orientation it msy happen so how does it help me to make a circle
Hi Zach, I'm researching on teaching and real-life examples. For the boat: mention the boat is lacking a functioning compass to catch a more pragmatic audience. Is this a classical greek example? From which books did you get it?
I remember James Grimes on his old channel did a similar problem to the Fermat point with four locations instead. His solution was to use a soap film to simulate how it should look like.
just move to your right 4 times... 1) move 1km to right 2)move 1km to right 3) move 2km to right 4) move 2km to right... max 6km wich is already better than example.
Using that approach I think you'd need another 2km move to the right to be completely sure of getting back to shore if after the first move you only just missed the shore on your left hand side
This might be the dumbest idea....but why not just switch off the motor? Wouldn't the waves eventually bring the boat back to shore since it's just 1 km away? (Unless ofcourse if the winds are too harsh which would drift me sideways and away.)
For the airport communication probem is what you did with the equilateral triangles just finding the middle point of two edges? Or is it more then that? (Sorry for mistakes I didn't learn maths in english)
You can measure exactly how far your boat drives, you can remember any point in the ocean to use as the center of a circle, and you can drive a more or less perfect circle around this center point? Nothing of this is possible without a GPS and if you have a GPS, you can also located the direction of the shore. Sorry but I hate those dumb school questions that make no sense in reality. Reminds of me of the question "A tower is built at a complete flat country. This tower is x meters high. Assuming the earth is a perfect circle with a radius of y and there are optimal conditions, how far could you look from the top of that tower?" Of course that question was related to earth curvature. But some student wrote "If you look upwards, there's no limit how far you can look other than maybe the size of the universe". He got full points on this one as technically he's correct and the fault was that the teacher didn't specify the question accurately as he didn't say "what's the widest point *on earth* you could still see from the top of this tower".
In the first problem, If you drive in any arbitrary direction you have 50% chances that you drive further away from the shore, and making a circle there would never get you to the shore. The worst-case scenario is that you drive along the path you came and drive let's say another 1 km away( 2km in total ). Your approach would, at best, get you to the starting point ( 1 km away from shore ).
Hope you guys enjoy! Two things to mention here.
1) The video that explains the last method for the 'lost fisherman problem' is already up on patreon and will be out in just a few days from this one's release.
2) Most of these examples/stories actually came from 2 books I recently read and those are linked in the description if you want to learn more math/optimization.
5 days ago?
wait how is it 5 days ago
where can i find that video about the most optimal solution to the boat problem?
@@naswinger It's on patreon now and will be out on the channel in 3 days. And for everyone asking about how this comment was posted 5 days ago, videos are posted early on patreon (plus I need to make them unlisted in order to get approval from the sponsor).
@@Peter-q1p7t definitely does, thought that was a given
Ahhh eliminating your enemies while using the least amount of cannon resources
Then use the rest of the powder as fireworks to celebrate victory.
th-cam.com/video/XPCgGT9BlrQ/w-d-xo.html 💐
And they say war is bad..
@@noop9k It is, because now a new generation hates you.
When another would destroy you, the soul must be the warrior.
zach star: we can do better but I wont explain it
me: aww
zach: just kidding
me: yay
zack: but not in this video
me: aww
I think the solution is a spiral. Not sure tho.
@@abhijanwasti7991 hmm i think you could start doing the 1km circle and then move to the 1.04km towards the end
i did some drawing on paint and came to the conclusion that you do 1km circle normally until you completed 3/4 of it and then stop turning and move in a straight line
-> 1 + 3pi/2 + 1
Gotta tease a little bit
turned off notifications, disliked, unsubscribed, unfunded, demonetized, and reported for child abuse
Oof
Put the boat in reverse dummy.lmao
Let's frame the question better: you travelled 100 km away from the shore. Suddenly, it becomes foggy, and a violent gust tilted your boat for an unknown angle. Your fuel tank is leaking, and the engineer cannot stop it completely. All communication stopped. Survive the day.
Y. Z. Survive the day =/= Go back to shore
@@joelmiller2601 go back to shore == survive the day
ninja lame Sorry; but no.
@@y.z.6517 Then there is a high chance that we go far and far away from the shore
Optimization was one of my favorite parts of Calculus. I really appreciated how applicable it was to the real world
tedskins2000 same!
same!
same
!emaS
@FBI ! I'm not under arrest am I? 🤔
Very very fun boat problem, thank you! The best I managed to get was ~6.4
The strategy I came up with is:
- go in a straight line for some distance r
- draw the two tangents to the unit circle that pass through your current location, label them A and B
- follow A until you're on the unit circle
- follow the arc of the circle until your direction of motion is perpendicular to B
- go in a straight line towards B
The total distance in the worst case scenario is given by
r + sqrt(r²-1) - 2arccos(1/r)+ 3π/2 + 1
The optimal solution is for r ≈ 1.16
This was really clever! Only thing I'm not seeing is where you got the 3arccos(1/r), the way I drew this out I'm getting that it would be 2arccos(1/r), assuming the worst case is where you barely miss the shore and drive in an arc until you reach basically where 3pi/2 is on the unit circle and then drive perpendicular to the shore until you hit it (which would be 1km). In that case you'd drive the entire 3pi/2 arc - 2arccos(1/r). Or maybe I just didn't interpret your method correctly.
@@zachstar yep! You're right, shouldn't be doing maths at 12am I guess! The actual value becomes 6.4 then, at r≈1.16 oops
This is super cool!!
How do you actually come up with these amazingly cool solutions?...wow ...is it easier to come up with such knowing optimizations ?
P.S.
I have not yet learnt calculus ... probably will be learning them this year...
~ A sophomore (10th grade)
@@sudheerthunga2155 for me it didn't involve much calculus, the key was to reimagine the problem as stretching a rope around a circle with a couple of extra conditions. It was a really fun process and not tedious at all. The calculus only came at the very end for finding the optimal angle, but there you might as well just plot the graph.
6:35 Delta fixed a triangle problem, how ironic Δ
I wouldn't say that's ironic. It's just coincidental and funny.
th-cam.com/video/XPCgGT9BlrQ/w-d-xo.html 💐
Dear calculus students, enjoy it while you can! Calculus was such a cool course and in my opinion the math only gets messier from here! Thanks Zach!!
Math always get messier as you progress.
@@pouzivateljutube2995 It's called entropy 😁
Differential equations is a relief 🥵
Damn things get bad after Fourier sequences...
As for the Taylor's theorem and what not, it still doesn't penetrate my head.
Everything after 1+1 is messier.
Just recently got interested in math. Wish I had before. Like in school, everyone asked "why do we need to learn this?" and no one could ever answer. "Because you should". It's as if they tried their best to make it as uninteresting and boring as possible. Had someone talked about humanity going to Mars or like the golden ratio... what a difference it would have made.
Yep. One of the failures about education is that teachers often can't communicate why is the thing important, or how it may be fun or interesting. I was never interested in biology for example, and yet I caught myself watch full series about evolution of life by Aron Ra right from the single cell organisms up to humans, and thats 40 something episodes 5-15 minutes each. Which is a far too much for a guy who is not interested in biology =)
I was the same way in high school but learned to actually think math is pretty cool. You can almost tell the future with it. For example, everybody places an x where they guess a ball is going to land where you would calculate it with math and predict it perfectly every time. The look on their faces when you accurately predict where EXACTLY that ball will land 10 times 50 times in a row.
To be fair, you never asked yourself how a plain can fly, how a computer works, how wireless communications works, lack of curiosity is the main issue
th-cam.com/video/XPCgGT9BlrQ/w-d-xo.html 💐
@@gabrielbarrantes6946 I have done a ton of research into the current US education system and thought a lot about how to fix it and I can 100% definitively tell you that lack of curiosity is not at all an issue when first learning about these topics. School has a way of taking things you like and forcing you into the most uninteresting and uninspiring part of those subjects especially in the beginning. Talk to any child in grades 1st to 6th and you will generally find they like things like math. There are even alternate education systems that you can put children in right now that are completely driven by the child's curiosity as in they have no set curriculum and those systems have proven to even give better results than regular school on tests.
Hey man. I just wanted to thank you for the resource this channel is. At my high school we’re required to right a 4000 word essay on anything. I chose math and just had a bunch of trouble finding a topic within that. Thanks to videos like this I settled on optimizing baysean search theory. I honestly could not of done it without this channel and would of failed a class. Just wanted to say thank you
ib xdd ?
For anybody curious about the answer to the minimum spanning tree question, the algorithm they use is called Kruskal's Algorithm. Essentially, they sort the distances and, going over the sorted list, connect any two currently unconnected nodes.
this is the first time I can say that graph theory helped me solve a problem in any circumstance other than school
This was definitely and exactly what I was suggesting. This absolutely makes for an awesome and informative series man! Keep it up
th-cam.com/video/XPCgGT9BlrQ/w-d-xo.html 💐
I love these videos where you show the application of different kinds of math, I will be studying this this year and I'm so exited for it
th-cam.com/video/XPCgGT9BlrQ/w-d-xo.html 💐
move in a spiral pattern, so that the distance between the lines of that pattern (measuring at an angle that goes through the centre) never exceeds the view distance in the fog. you'll also find survivors that fell out of the boat, if they're lucky and you don't reach shore first
also what i thought first, till he mentioned the "what if we go the 1km at an angle approaching 0" then i realise, ok we gotta think in maths, not practical reality
1:08 *solves by logic*
Stop. Since it's day, the wind breeze should go to the land, I feel the air hitting my face, so it should be the other way right?
I never understood the use of calculus until i started machine learning, Everything makes sense now
Love this one. Sharing with my class (I'm a prof teaching calculus and physics). Best wishes Zach
I started this channel to help my class. Thanks for the inspiration Zach.
This is pure genius. I read a book with similar content, and it was pure genius. "The Art of Saying No: Mastering Boundaries for a Fulfilling Life" by Samuel Dawn
3:28 Angry pacman
Yes
One of my friends figured out how to find the Fermat point for math club in high school. One of the rest of us in the room (might have been me, but I think it was another kid) realized that you could take weights on strings and hang them through holes, and the stable point was the Fermat point. Bell should have hired a 10th grader to help them.
9:18 Basketball was invented in 1891, so this problem from 1686 probably did not involve a basketball.
EDIT: 11:10 nvm you got me there
Watch the whole video before commenting.. Why don’t people do this more often?
@@joelmiller2601 That's put me in my place...
Now I'm off to watch the rest of the video...
Joel Miller Thats like the teacher telling you to ask your questions after the lecture
Like I’d even remember
@@ThePenisMan You probably have a note. Write your question and read it out if it's not answered by the end of the lecture.
Bastion Barrick that is an incredible waste of paper
i´ve benn watching this channel for the last 2 months and love it. Great job man. I really like your content. Keep the good work. Cheers.
Get out of my head! My Dad was an Engineer. I was thinking about this concept the other day. Thanks!!!!
gotta love the idea to "drive a perfect circle" around your starting point before you went for about 1km trough thick fog .. xD
@5:00 Also, if you had turned right like in the figure, don't start your circle turning left, turn right again. Much more efficient. You will reach in (half the distance-1.04) or whatever distance.
Le joie de vivre is watching your fascinating videos!! Thanks!
6:41 Fermat point is shortest distance task, if you have extra variable let say bandwidth, minimum length is dominant, but bandwidth can change location point.
12:40
fun fact there is a much much simple method to solve such kind of problem...and even a bigger fun fact it does not use computers of any sort....ENTER THE SOAP BUBBLE.
How?? u may ask...to be brief about it...
You may take two sheets of acrylic sheets saperated by a small distance,all the points of interest are simulated by a screw joint of a sort holding the two sheets together.
now dip the whole thing in a soap solution, the path traced by the soap bubble is the optimised one...
you found your solution in no time...😊😊😊
So why are people spending potential millions and tons of manpower to make lasers that basically simulate what you can do with acrylic? Perhaps it's so that the information can be quickly analyzed by some computer and sent to the person who requested it. It still doesn't make quite as much sense but whatever.
yea... when I came accross it... I too was shocked...
@@ոakedsquirtle It depends on how many times you need to solve the problem. If you need to solve it once, no big deal. If you need to solve it 10 times per second do you still think building acrylic sheets and soaking in soap would be feasible?
Before watching a solution. I ended up with this equation:
S(x) = x(1+ 2pi - 2acos(1/x)). Solving for the minimum gives roughly 6.995... at x = 1.044... I think that's a definite improvement compared to 7.28... at x=1.
*EDIT:* ok, 30 seconds passed, and he repeated my solution word by word. And it's not even the best one D: gotta try more.
*EDIT2:* Before I watch further. I got another equation:
S(x) = x + 2sqrt(x^2-1) + 2pi - 4acos(1/x), this one gives the minimum of 6.459.. km at x = 1.242... that's my second guess.
A better solution is to go in a distance of "d", then go through an OUTWARD SPIRAL. Suppose the distance from the spiral from the original to the point that the spiral goes one complete circle is "f", i.e. f>d, r = (theta/2pi) * (f-d) + d. The final solution is in terms of both f and d, while under the constraint f>d. Let's say L is the worst-case scenario distance of travel, the resultant f and d are found by performing partial derivatives on L with respect to f and d, set the partial derivatives to zero.
Oh well, I give up on the calculation..... and I assume the rate of radius growth of that spiral is linear to the angular distance of travel. If that is not the case, the ultimate solution perhaps requires variation calculus...... and I don't know what I am talking about.
I was just thinking about spiral
Voronoi diagrams could also help to solve minimum distance problems like the triangle one. Given 3 points as the example, the point were the three Voronoi regions meets is the point that has the minimum distance between all three. There is a similar problem were a company has N werehouses that serve a certain region of a city and you have to find the best way to split these regions. The solution is given by the Voronoi diagram. Anyway, thanks for the great video Zach, have a good one!
To find the Fermat Point isn’t that the same as drawing a line from each vertex so it is perpendicular to the opposite line for two vertexs then where the lines intersect that would be the Fermat point?
Please answer
What you described is the orthocenter of the triangle and, sadly, it's not the point with the desired property.
12:56 Certain triangles (BCD, BCF, EFJ, GHI and HIJ) just fill my heart with so much pain...
I learned optimization and other than calculate the size of the structure built with certain prices of certain materials, I used it for calculating optimum stadium ticket price in a football manager game when I knew the current audience numbers, the current price and the size of the stadium. For maximum profit that is.
But like for what wouldn't use optimization!
But so we just assume that we can't see the shore until we're on the beach with the boat?
in the boat problem how do you know that you circle is big enough to reach the shore at all? wouldn't the worst case scenario be that you end up going directly away from shore and your circle ends up being too small to reach it?
You can slightly decrease the velocity for the basketball problem. This is only because a basket ball can hit the front of the rim and still go in. I don’t know how much but you could decrease it by a bur
Optimiztion is in a way common sense if you have a high IQ . That's why most of the times people with higher IQ solve problems faster by finding a pattern/Algorithm in their brain . I don't know if i have high IQ or not but believe me i also solve my in general life problems by finding patterns and just using my common sense , i solved most of the problems in the video by just using common sense .
I just wrote my optimization exam this morning. Why didn’t I see this before then 😫
you said you can't remember where the shore is. What if you go relative further out into the ocean, instead of towards the shore. Your area you will ancher around, and do a full circle will all be within the water
You travel *at least* the same distance away from your anchor, as your anchor is from the shore. This will guarantee that you always hit the shore before completing a full circle. (e.g. your anchor is 5km away from shore, then travel *at least* 5km in any direction).
A good current real life explanation of functions and calculus: a pandemic. All the infection rate models are based on functions, and add, subtract and multiply hundreds of functions (factors) that influence infection rate. These growth curves then are analysed using calculus to measure rate of changes : too high, we have a problem so increase safety measures. Low : open up some activities to find the balance point. And yes, they probably factor in extra measures to counter the 20% of people (or states) who won't follow measures. The other interesting problem was asymptomatic disease: we are used to seeing runny noses when people are sick. This disease you can show nothing for 2 weeks and spread it. That means cases you measure today are actually indicating spread 2 weeks ago, and you have to factor that in (you're actually measuring data that is 2 weeks behind what is actually happening today), another reason for calculus to determine what IS happening today.
for optimization problems, LP can be put to use since that's what it's for.
Better solution:
Go in the opposite direction of the boat's compass and travel that direction for 1 km.
or
Go in the opposite direction in which the waves are hitting any given side of the boat. Travel in that direction for 1 km.
Both methods save well over 6km's in travel.
Imagine if you used this ancient technology called a compass.
Or better yet, just look for waves, they likely lead back to the shore
Polar inversion.
@@otheraccount5252 Just remember which way is north before going out into the water. If, somehow, this happens, just go "south".
@@thetimelords911 You clearly don't understand the concept of being lost.
*compbutt
Regarding the boat problem, what if you drove into the wrong direction, so not towards the shore, but away from it. Thereby ending up, say 2 kilometers from shore, thus not reaching the shore using the 1km circle. Is it just an assumption that the rough direction of the shore is known or am I missing something?
7:00 wouldnt it be easier to just take the angle bisectors and let them intercept? This method also gives you the abstract solution for any given triangle...
We learn stuff in math cause people use math words in documentation and explanations of the worklife challenges, and we need to understand what we are reading so we don't spend time in our job being interrupted all the time by learning foundational advanced math concepts on top of the domain skills of the job.
A few times a project in a job that isn't teaching or pure mathematics would actually require to do the math too. Mostly it's reapplication.
Like say a comp sci person might reuse math already applied by someone ells.
In the boat example, if you are entirely unsure of where shore is and the fog hinders vision, then why wouldnt the most efficient way back be to turn the boat exactly 180 degrees and travel that path 1.05 km then do the circle thing if you dont hit shore?
Using test taking strategy, I would say nautilus spiral? Depending on the domain the answer is always choice C or the golden ratio (or e if it's larger than 2) or "they're the same" or Fibonacci or nautilus or there exists no such foo.
Brings me so many memories. Why can't professors use these type of examples?
Okay video! Thanks for uploading!
I understand you are what the point of the video is. I simply don't agree with the boat scenario at all. If you go on a boat trip for whatever reason, you should 1) Know the weather. 2) Know where you came from or at the very least have a sense of direction. and 3) Have someone that's experienced in boating and/or have someone onshore that can call for help in a situation like this.
It was my favorite yet most frustrating part of Calc 1 in high school. I loved it's direct applicability. The ones I hated the most was like filling up a cone, determining the rate at a certain height
This is what I hope to learn in an msc in supply chain management
How is it going?
I'm shit at math. I pretty much gave up on them when I was like 13 because well, I always had decent grades everywhere by doing the bare minimum I didn't feel like working to get better at it. That and I had a terrible teacher.
I don't know why tho, but seeing how powerful they are at optimising stuff makes me regret giving up on them. I know I particularly enjoyed graph theory, finding the quickest routes and stuff... Oh well.
So you're saying you'd putt he anchor down where you were at 1km out, and then go 1km out and in a circle around that anchor and you're guaranteed. But that if you made any small error ( the 1 plus 2 pi thing) then you'd miss the shore by a smidge?
About the lost fisherman, in nature similar problems often cause spiral shaped trajectories.
I think better d is a d such that the integral of the path value (sum) is minimum rather than just the worst case path to be minimum
What if we decide to go further into the sea? Then we wouldn`t hit the shore, right? Am I missing something?
Yes you are. We left an anchor with rope at our "Lost Point" and drive in any direction as far or further than our distance to shore. Now drive in circle. You will always hit the shore because a circle with radius equal or greater than the distance from shore will intersect with the shore, no matter what.
You were thinking of taking our guess drive as center, but we take it as our start around the perimeter.
The only thing to think about then is to optimize *how far* we drive in any direction.
He always showed the worst case, which is bad for understanding the problem, granted.
Surely for the first puzzle your given solution only works if you go in the rough direction of the shore - if the random angle you choose is further out to sea, your circle no longer hits the shore?
Very ‘Presh Talwalker’-like, enjoyed it!
Ok, my answer for the boat. You want to turn 180° in place, which should be a maneuver all sailors can execute using a compass in the fog and then sail 1k back to the shore.
In the boat example you are in a situation where you do not know the direction of the shore, am I missing something in thinking that guessing the exact opposite direction of the shore would be the basis for a worst case scenario. If you know generally (within 180 degrees) of which direction the shore is, which seems to be implied given the worst case scenario doesn't consider you actually getting further away from the shore, then couldn't you travel 1km in your guessed direction. Then because you know the angle from the original point from which you traveled (say 1 degree in a bad guess, ending up just less than 1km from shore and just short of 1km North/Parallel to shore from the point at which you became disoriented) - you could then do a quarter rotation in the same direction of the original angle (ending up at 91 degrees rotation from where you started) hitting the shore before making the entire quarter rotation.
Let me know what I am missing.
I think the lost boat problem can be optimized further by moving with a square path inside a (one kilometer circle plus some small distance.)
Yup, but use a triangle instead of a square. The square inside the circle has a perimeter of 6.93 km and the triangle has a perimeter of 5.2 km, both are which are lower than the circle minimum distance of 6.995.
sqrt(2)+1+pi+1= 6,5559 km
Drive sqrt(2)=1,4143 km straight. Then 1 km in an angel of 45° straight back to the 1km inner circle.
Sail the half-circle (pi=3,1416 km) and take the shortcut of 1 km to the worst-case shore.
ツTsusday
It seems good, perhaps it could be optimized a little bit further if you play with the initial interval length and angles.
@@JordanMatrix Yeah. Like the better method in the Video could be easily "optimized" after going to the 1,04km radius going back to the 1km radius on a nice angle.
Can’t determine what direction it came from, can draw a perfect circle. Nice!
I'm lost. The most efficient route is to rotate an angle greater than 90 degrees from your current direction and proceed forward.
For all initial angles from shore, having traveling 1km, this would yield a < 2km return distance.
i literally coded Dijkstra's algorithm just two days ago and it is fun to look at the result
As always, a job well done!
I’m confused doesn’t the ship have radar and staff locating where the ship is at what time or whatever?
I know math is great and all but I’m pretty sure sailor have like water maps and stuff.
the problem with the boat thing is that the proposed methods only work if you move the boat in a direction that actually brings you closer to the shore. it completely falls flat if you move in a direction that leaves you further away from the shore.
Where is the video link for the solution to the first puzzle ?
please do reply when you get the link
I too, was looking for the link..
It's not available yet! Will be out in 3 days and is available on patreon now.
@@zachstar Thanks
booksc.xyz/dl/78506901/ac16a2
Very cool concept. Please keep making such videos?
For the first problem, isnt it inaccurate to compare different "worst-case scenarios", given that the diff distances require the boat to approach the shore at different angles?
Wouldnt optimisation require all scenarios to be taken into consideration? Such that for each distance that the boat should travel before making the circle (be it 1km, 1.04km, 1.183km), each possible scenario for all 360 degrees should be taken into consideration, the distance for each scenario calculated, and the average distance travelled derived to truly optimise the solve. This does sound like a brute-force approach to finding out which is the most 'optimal', but still more optimised than the soln you proposed.
Heres a question from a non-maths person that ive been pondering. Im sure this has a name and is studied but i havent managed to find it. Say you have some number of people. Each person will derive different amount of happiness from x dollars. The more or less you give them, the more or less happiness they will derive (like a function). Now, say you have x dollars to distribute amongst these people. What formula would you use to maximise the total happiness (assuming you could sum the happiness of distinct people) of the people? You should find the optimum distribution wherein the sum of the outputs is the greatest.
Calculus was my favorit math course after Linear Algebra . i really enjoyed Optimization problems .❤
With that first problem... If you're disoriented enough, you could end up say turned 360 degrees but think it was only 180, so you boat til you're 2.04 km from shore, and due to how lost you are, you could then arc _away_ from the shore. The circle will never hit it. It's why I'd always go for a spiral - It accounts for that
Unless there is something I've missed which is possible
Hi Zach Star, awesome video. I had a question about text books because I'm going to highschool soon and wanted to get ahead in math and science, so I was wondering if you could tell me how to find the right textbooks to learn a subject
When I'm looking for a good textbook for math/science class I usually search quora and reddit for that same question and I always find some kind of response. Often there are good amazon reviews but they can be misleading as well depending on how technical or rigorous you are looking for a book to be.
I love this channel! Thank you.
So you are just going to ignore going parallel to the shore, or better yet any angle beyond that? You said disorientated, not vaugly aware of which side the shore is of you which with a few moments to think would lead you to think "oh I'll just average this feeling" which leads you to the shore
I love videos like this. Now to learn calculus... As soon as I can afford it (both time and money.)
How does Ant Colony Optimization fit into the idea of calculus optimization? How would one attempt to compare different optimization algorithms?
why don't you just hit reverse and go backwards toward the shore on the boat?
okay so i would say. go 1 km in the direction you're facing, turn right, go 1 km, turn right again you're now 180 from your first position so go 2 km, then go right, you're now 180 from ur 2nd position so go 2 km
worst case scenario the Shore is East, you start facing south 1 km, then west 1 km, then north 2km, then east 2km, 6km total traveled.
Hi Zach. On optimization problems. How about variable situations like a factory making something out of a variety of parts, materials, etc. Does the optimization solution for something change under a different viewpoint.
For the first problem:
What i dint rlly get. What if ur going 1 km out on the sea? Like if i lost orientation it msy happen so how does it help me to make a circle
Hi Zach, I'm researching on teaching and real-life examples. For the boat: mention the boat is lacking a functioning compass to catch a more pragmatic audience. Is this a classical greek example? From which books did you get it?
In the description, it's called 'when least is best'
Worst case scenario in the first example is continuing to drive away from the shore? Where the circle created does not hit the shore
I remember James Grimes on his old channel did a similar problem to the Fermat point with four locations instead. His solution was to use a soap film to simulate how it should look like.
Just half way into the video..
And I say awesome..
Keep it up , man
just move to your right 4 times...
1) move 1km to right
2)move 1km to right
3) move 2km to right
4) move 2km to right...
max 6km wich is already better than example.
Using that approach I think you'd need another 2km move to the right to be completely sure of getting back to shore if after the first move you only just missed the shore on your left hand side
Thank you for such an informative video on why maths is actually useful! Now I can convince my friends too... 😁
This might be the dumbest idea....but why not just switch off the motor? Wouldn't the waves eventually bring the boat back to shore since it's just 1 km away? (Unless ofcourse if the winds are too harsh which would drift me sideways and away.)
For the airport communication probem is what you did with the equilateral triangles just finding the middle point of two edges?
Or is it more then that? (Sorry for mistakes I didn't learn maths in english)
You can measure exactly how far your boat drives, you can remember any point in the ocean to use as the center of a circle, and you can drive a more or less perfect circle around this center point? Nothing of this is possible without a GPS and if you have a GPS, you can also located the direction of the shore.
Sorry but I hate those dumb school questions that make no sense in reality. Reminds of me of the question "A tower is built at a complete flat country. This tower is x meters high. Assuming the earth is a perfect circle with a radius of y and there are optimal conditions, how far could you look from the top of that tower?" Of course that question was related to earth curvature. But some student wrote "If you look upwards, there's no limit how far you can look other than maybe the size of the universe". He got full points on this one as technically he's correct and the fault was that the teacher didn't specify the question accurately as he didn't say "what's the widest point *on earth* you could still see from the top of this tower".
4:04 whoa this graph looks like the angle of deviation vs angle of incidence graph for a thin prism. Even the equation is the same
So in the boat one... why cant you just go in reverse 1km?
What happens if you drive 1KM perpendicular away from the shore? how do you "guarantee" that you will hit the shore?
Travel more as 1 km maybe 1,5 or so and then going a triangle with a inner circle of 1 km is better i think.
In the first problem, If you drive in any arbitrary direction you have 50% chances that you drive further away from the shore, and making a circle there would never get you to the shore. The worst-case scenario is that you drive along the path you came and drive let's say another 1 km away( 2km in total ). Your approach would, at best, get you to the starting point ( 1 km away from shore ).
What if you drive 1 km away from the shore?