Solving the Math Problem in Good Will Hunting

[@EllieSleightholm]

❗Minor correction❗: for the r = 5 condition, the (3, 3, 5) graph has an extra line attached to the middle node. This shouldn't be there! brilliant.org/EllieSleightholm - Try everything Brilliant has to offer for FREE for a full 30 days! The first 200 of you that click on the link will get 20% off Brilliant's Annual Premium subscription 👀

[@yanwo2359]

Yeah, I spotted that immediately. (I wish)

[@beancount811]

@@yanwo2359 I thought I drew I drew it wrong the first time around. Then I counted the edges and nodes again between what I had and the video, and halted. >.o

[@vivneb]

What a nice surprise! Thank you for this video. I put this question on a worksheet this year for a Discrete Mathematics course I tutor for fun. Only one student in the class had seen the film though.

[@EllieSleightholm]

@benvivian9280 Thank you! Aaah no way, I'd have loved to have you as a tutor 😂

[@leif1075]

​@EllieSleightholm Thanks for sharing Ellie. Thanks hope you cam follow.ypnonnmy.last comment..how can I be a great math whiz like Ramanujan? Happy Holidays and hope to hear from you!

[@MichaelWaisJr]

Robin Williams is not very funny in the film. 😕

[@RuthvenMurgatroyd]

This is my solution from before I watched your video (tbh your solution is so much neater): Every tree of order |V| has size |E| = |V| - 1; since Σdeg(v) = 2|E| = 2|V| - 2, for a tree of order |V|=10 we have Σdeg(v) = 2(10) - 2 = 18. Trees have at least two vertices of degree one. Let deg(v_1)=1. Then we know that Σdeg(v_{>1})=17. We get the desired trees by generating their degree sequences as follows: Initialize the degree sequence as 1111111111; accordingly, subtract 9 from 17 to account for the 9 vertices v_2,...,v_10 we have initialized as having degree 1. 8 represents the number of units we have left to "distribute" to each number in the sequence. The number of ways the units of 8 can be distributed is given by the partitions of 8-the terms of the each partition are added to distinct numbers in the sequence to get the degree sequence of one of the graphs (ex., 8 = 4+4 and the two terms of this partition are 4 and 4 which we add respectively to two numbers in the sequence to get 5511111111 as a degree sequence of one of the desired trees). Since we initialize all of the numbers in the degree sequence to be 1 we ignore all of the partitions which have a term equal to 1 as these will sum to give a degree in the sequence which is 2. The 7 partitions of 8 which do not have 1 (viz., 8, 6+2, 5+3, 4+4, 4+2+2, 3+3+2, 2+2+2+2) obtain the sequences 9111111111, 7311111111, 6411111111, 5511111111, 5331111111, 4431111111, 3333111111 respectively which are the graphs shewn in the video. I'm not completely sure if the degree sequence is unique for isomorphic trees but I guess a similar argument could be made for the adjaceny matrix. Merry Christmas! ♥️🎄✝️

[@RuthvenMurgatroyd]

Edited my comment and sadly lost the heart 😮‍💨 don't make the same mistake, my friends lol.

[@RuthvenMurgatroyd]

It's a Christmas miracle! Haha thanks a lot, Ellie.

[@HankC9174]

its also a problem of combinatorics : start with one node of n=9 , and move nodes to create extra nodes of n>2 . brute force also works rather simply.

[@carmensayago9354]

I love this series of solving math problems in movies so much ! Thank you so for always keeping math accesible and interesting !

[@EllieSleightholm]

@carmensayago9354 thank you so much! I'm having so much fun doing it!!

[@leif1075]

​@EllieSleightholm Thanks so much for sharing Ellie. I really hope you can respond to my other comment whenever you can. Thanks very much.

[@stevesanders6770]

I love your optimism that anyone could solve this 😂 - it was a bit like Shakespeare for me: understand most of the words individually but no idea what’s actually happening and switch off. Will look forward to further bafflement- new subs

[@jorgehidalgo4792]

Blasted lady! Now you got me curious about graph theory and force to learn more, and I am not a college graduate by I understand your explination. Thank you for taking the time to post it

[@brunomoura7719]

I just love your content here, on Math! Every now and then, when I fell myself unmotivated to study math(and I love it!), I often come here and watch some videos that, truly motivate me. Congratulations for your excelent job and thank you so much for helping me.

[@jamesjohn2537]

😂😂 i like the way you solve it with all the calm and logic behind it!! I love maths and am self studying mathematics heading towards applied maths but not let there!! I do like pure mathematics particularly prove writing and some of subjects related to applied maths. Am not yet into prove rather then studying calculus 3.

[@EllieSleightholm]

@jamesjohn2537 thank you so much! That sounds amazing - as much as I love applied maths, I also really love pure maths too!!

[@jamesjohn2537]

@@EllieSleightholm kindly, you welcome! I subscribed you know. keep up dear 💫💪.

[@jaymie-leacollingwood2753]

You explained this beautifully. It was precise, concise and clear 😍

[@yoursfithfully047]

hey can you explain the step where to added the edges with their respective number of edges to 18 . and nice effort by solving these kind of problems really helps a lot of people realize how beautiful math is

[@williamcaton3409]

18 is the sum of the degrees of the graphs. The degree of a node is the number of edges connected to it. There is a formula in graph theory which states that the sum of the degrees is equal to ×2 the number of edges and I'm guessing that's why she put the second expression equal to 18. I'm guessing the coefficients to the nodes is the amount of edges for each different amount of nodes not including the one "starting" node :)

[@Maraquitica]

Ahh, what a beautiful video, really 🤍 I am studying mathematics and your videos have been a game changer for me. What you are doing with your channel is beautiful :)

[@pikachupower5844]

Incredible! I’m so glad I found this. I had wondered about that calculation in that movie. Definitely subscribed.

[@healyrj2001]

Thanks Ellie. I always wondered what the problem was. You have explained it so well.

[@hisnameisdavid5082]

Watching you do maths is like going stargazing- it goes right over my head!

[@ChandrasegaranNarasimhan]

Thanks. It was a systematic way to think and do.

[@graemezimmerman109]

Really cool! I took a different approach; I tried to draw the most basic structures I could and then solve from there. It’s hard to explain without being able to draw it, but it’s interesting how there’s different ways to get to the end result

[@patricowen-meehan4210]

I am one of your amateur maths followers, didn't study it at uni, only to A level. I was confused lines and edges. But drew it out visually and now understand.

[@_DD_15]

You just got another subscriber! lovely video! Love the way you explain and the positive vibes!

[@Doomsquad99]

Just seeing the matrix behind ellie gives me nightmarish flashbacks to linear algebra.

[@jilow]

I had to pause to think a few times but it all made sense. Great job!

[@MinecraftMasterNo1]

Graph theory may not have as many applications as calculus but I would hardly call it "pure math", Ellie! Don't forget about us computer scientists! We love our graphs.

[@Zeitaluq]

Excellent diagram explanation

[@BreezeTalk]

Better than any mathematics teacher I have ever had.

[@davidgillies620]

It should perhaps be pointed out that the result that summing the degrees of the nodes of a graph yields twice the number of edges (7:03 - 7:14) is known as the Degree Sum Formula, which has as a consequence the Handshaking Lemma.

[@minyare]

Keep going! Ur videos are really motivate me xx

[@EllieSleightholm]

@minyare 🫶🫶🫶

[@ayandas8299]

This video is so much fun, thank you!!

[@debbiemartin3351]

I saw you in the Sidney newsletter and I've watched a few of your videos, they're great 😊 look forward to seeing more!

[@EllieSleightholm]

@debbiemartin3351 aaah no way!! Thank you so much for supporting my channel 🥰

[@dwittn9193]

Great video, will watch the movie soon now that I have learned about it. I really liked your calm manner of explaining! I think it would've been good though, if you would've explained a little bit more, how you got to the constraints in the beginning, especially the last one. I feel like you went over there pretty quickly.

[@juancampos1164]

Yup, i was just noticing. Very cool review

[@amdenis]

Wonderfully fun video! Question: In the sides equation =18, where does that expression come from, with the various n-1 * n’s ?

[@ehawkins730]

I have this same question

[@AkashGupta-th2nm]

It's because the equation double counts the edges. So we should have 9 edges, but since we're double counting, the equation sums to 18 edges.

[@williamreid6715]

I thought you eliminated a straight line through a node?

[@Senays_random_stuff]

This is great! I enjoyed it quite a bit. Thank you very much!! What note app did you use?

[@seventhunder777]

You are simply amazing.

[@tempest5343]

Hey Ellie, I've seen many people use the same app when doing anything on an Ipad and I'm not quite sure what it is. It would be really helpful if you could say what app you're using! Thanks for the interesting videos

[@wilsonjunior9122]

Yeah. She listed everything in her filming setup, but the software was used. I am curious, too.

[@jleroy33]

That penmanship! Nice!

[@Oath_Keeper1979]

I challenge you to solve the roswell equation from Big Bang Theory. Even Sheldon couldn’t solve it.

[@abbssaa]

I love your channel Ellie!!!🩷🩷

[@EllieSleightholm]

Aaah thank you so much!!!

[@David-hu2zx]

Is there a typo or did I miss something? In the r=5 solution, the second tree…. Its labeled 3.3.5, but isn’t it drawn 3.4.5?

[@florianbroeder2572]

12:51 isnt there more than 1 way to draw a 3-5-3 tree ? or is the explanation of homeomorphical incomplete ?

[@petern8576]

How would that one look like?

[@NOTycyborg]

Which app do you use on iPad to solve questions ?

[@iliaakhavin7528]

You are my role Model ❤🎉

[@Doz-13579]

This gives me a node on my left cheek!

[@eartphoze]

For it's no different that the special key char {, } and applying z= x^2 + y^2 to , values in conditional statement brackets, delimited from special key char , {,},, though another formula is audio for impedance, reactance, in ohms speaker voltage wiring, RLC, visual proof in the radius squared of a circle from magnetic direction drawing a cone with ellipse opened end for the calculation of magnetic , momentum, is like counting a syntax statements letters and squaring the length by each of its common parts a la carte, to get square root of each in separate segments for one line ending with ; and suming up after all have been square rooted, as a integer string sum.

[@jimmurphy7296]

Nice work.

[@ericerpelding2348]

Very good. Here is a movie with a math problem. Find the Fritz Lang movie "Cloak and Dagger" 1946. In it Gary Cooper solves a "line integral" problem.

[@StephenPheenixx]

What software are you using on your Apple device to annotate and more?

[@jacobwetherby]

I thought you couldn't continue the line with a node... so which is it?

[@basildaoust2821]

Excellent video, however, I would never remember the proof statements but I did enjoy the tutorial.

[@Igingerninja]

Oh I love these videos! New sub :)

[@EricPham-gr8pg]

All wave reflect and interfere and can have internal reflection in heavier materials so seed designing or supermarket and egg can be created by filed of geometric sound echo then light and heat kinetic water and ion can condense into seed or egg or sperm

[@JTL14789]

I don't understand the r=5 part where n1=8. Doesn't n5=2 satisfy the conditions without n1=8? Also maybe I'm just slow, but I didn't understand the part about edges = 18 until about an hour after watching the video and re-reviewing it a bunch of times. I thought it would have been helpful to spend a lot more time explaining the logic behind that ... = 18 equation with some demonstration of examples. It just seemed to appear very quickly out of nowhere, like some sort of mathematical sleight of hand.

[@UrvangJoshi]

Yes. For"r = 5", íf we want to stay consistent with other cases, "n5 = 2" is enough to satisfy the equation with 18. But "n1 = 8" is kind of additional information coming from first equation: as we have 10 total nodes, and 2 of them have 5 edges each (n5 = 2), the other 8 must have 1 edge each, aka n1 = 8. But then, value of n1 should have been written for other cases as well (when non zero).

[@petern8576]

Thanks for this nice video. For future videos, maybe some simple examples at 7:05 and 8:05 might be usuful, i.e. when you introduce new topics. Once you explained them in the comment section they were easy to follow.

[@user-cb3qr9dt2k]

WOW, Nice, I loved that movie, Have been fascinated to know more about the maths, and this one in particular from the movie. New Sub today.

[@rakibhassan-e9r]

can you please tell me what app you are using ?? It's really cool. I love your videos btw, very easy to understand.

[@paathimself]

What course would this math problem fall in? 300-400 or grad-level?

[@seannolan2120]

Where does the 18 in the second equation come from?

[@914Rocky]

Doesn’t n3 = 1 violate the first rule you mentioned that you can’t have a node with two lines passing through it. The only difference to the example you raised is that in your solution the segments are vertical. What am I missing?

[@UlricGrim]

I thought the same thing.

[@JTL14789]

Can't have a pass-through in isolation because that would be reducible. o-o-o reduces to o-o. Thinking about this as a route in a map helps too. Let's say you are trying to go from point A to point D. If point B is simply on the way from A to D, then you don't need to specify point B, and therefore, point B is reducible. Now let's say that you add point C as a new destination, but you decide to only get to it through point B. Point B is still reducible purely from the perspective of point A and point D, but it is no long reducible in the grand scheme of things because point C is now part of the mix, making point B relevant.

[@eartphoze]

I can tell you how this is possible to tie with RLC low, high pass filter in capacitance, the charge around an electric field from Maxwell's speed of light.

[@sidneybosike5270]

Im not a fan of math but this is strangely calming🙂

[@kendonblack6904]

So the last two are both 3,3,3,3 graphs? How do you prove that those are the only two? How do you prove that there aren't alternate versions of the other ones?

[@guidodenbroeder935]

By applying a search algorithm. Which is how the entire problem should be solved in the first place.

[@pedzsan]

You should explain the 2nd equation. Where did the 18 come from? I know where but the general audience probably does not.

[@Brandon-hd4vg]

yes please do, I am confused why we are randomly multiplying everything by 2.

[@pedzsan]

@@Brandon-hd4vg I *think* because each edge has two end points thus 9 * 2 = 18…

[@Brandon-hd4vg]

@@pedzsan oh that makes sense lol. thanks for the reply.

[@kjbunnyboiler]

Could you go through the “game show” problem from the movie “21”

[@running_cactus]

I'm wondering what sort of software do you use to capture your hand-written notes and hand-drawings?

[@MadduxJames]

What app do you use for writing??

[@Cedarsunrun]

Just curious, what would a problem like this be used for in real life?

[@zhugzug]

Welp, my brain fried, I suck with math.

[@fabiennem]

Can someone explain to me why N. = 1 in most cases but sometimes it can be 2?

[@robertoivanperezluna6467]

excellent video, and also, movies are lies, what a surprise jeje

[@mattdougherty6547]

I noticed that your 335 figure is actually a 345, is the middle node supposed to only have 3 lengths out from the node?

[@RedCoat232]

Saw this too, but I noticed in the description it's been corrected

[@LuisEmilioLopez24]

HI!,Could you solving the problem about Graphic Theory that appears in the film with matrices, please?

[@LuisEmilioLopez24]

solve*

[@HH-mw4sq]

Question: for r = 4, why isn't 3n4 + 5n6 = 8, not a solution? This would suggest a (4, 6) graph.

[@renatos.2033]

(4,6) is homeomorphic to (6,4) which appears at r=6.

[@HH-mw4sq]

@renatos.2033 - yes, that is correct. But her statement made it sound as though the (4, 6) graph was illegal.

[@UrvangJoshi]

By definition, "r = max k, where nk != 0". So when r = 4, you cannot have non-zero value for n6. That is why this case is impossible.

[@HH-mw4sq]

@@UrvangJoshi - thanks.

[@pikachupower5844]

I was thinking the same. Thanks for clarifying this. Now I can go to sleep!

[@colinmays8811]

The 3-3-5 graph has 11 nodes

[@EllieSleightholm]

@colinmays8811 read the pinned comment :)

[@colinmays8811]

yeah, you drew a 3-4-5 graph. you need to eliminate one of the edges.

[@prod.kashkari3075]

What is the reason for the coefficients in the second equation?

[@EllieSleightholm]

@prod.kashkari3075 The second equation looks at the number of lines passing through each of the nodes. We know that if we have n=10 nodes then there must be 9 lines connecting each of the nodes together (due to the condition that we can't have cycles). So we know that there must be 9 lines altogether. Thus, if we look at the number of nodes with lines k we have, n1 + 2n2 + 3n3 + 4n4 + ... + 9n9 = 18. The coefficients come from the number of lines that must pass through that node and we have 18 as we will count the lines twice for each node (remember a line is connected by two nodes). I hope that makes sense! Any other questions, let me know! :)

[@Slowe_]

I dont understand how in explaining nodes you say cant extend the line but then in the examples you have lines that appear to just extend like you said they could not

[@mathalysisworld]

nice!!

[@AthenaSaints]

N1 + 2N2 + 3N3 + 4N4.... = 18, what is the explanation of writing this equation? why is it equal to 18?

[@RuthvenMurgatroyd]

n_k is the number of nodes with degree k (i.e., the number of nodes connected by an edge to k other nodes). From this it follows that n_k nodes contribute k edges each hence the coefficient (except we count edges twice when we count both nodes that the edge connects so 18 is actually twice the number of edges or 9 as was noted before the equation your comment asked about).

[@UlricGrim]

What is the name of this type of math? It’s not calculus, for example. Is this some type of astrophysics? What would this be used for?

[@RuthvenMurgatroyd]

Graph Theory. It's a part of Discrete Mathematics. Applications are found all over computer science and elsewhere. One application of graph theory is in modeling networks. For example, let the nodes represent people and the connections represent friendships. Then the number of nodes a particular node is connected to (its degree) is the number of its friends and the nodes it's connected to are its friends. You can start asking questions like who is friends with everyone or no one or even who is friends with everyone within the least number of mutual friends. You get the idea.

[@travisneill704]

On the second solution I thought we couldn’t have continuous nodes but it looks continuous on the left in the graph. Am I mistaken?

[@christophermball]

By second solution, you are referring to the 2nd from top tree boxed in red correct? The left-side vertex shares 3 edges (hence doesn't violate irreducibility - exactly 2 edges).

[@angelamccarthy4731]

What is the purpose of drawing mathematical trees? Totally ignorant here. I didn't understand the purpose of calculus until it was broken down to why calculus is used (and not just to make high schoolers and college students cry). Thank you so much!

[@mishaerementchouk]

It is not as much about counting as about enumerating. Trees may appear in different contexts. As a toy example, say, you are making a chip and a lot of gates share the same input (this is called net in the biz). Hence, the connections will form a tree. You want to make sure that the total length of connections is the smallest but at the same time the connections are not overcrowded and so forth. So, while testing different arrangements of the gates on the board you need an efficient enumeration. Getting the total number of the trees is the first step in that direction.

[@mattikemppinen6750]

The study of these mathematical trees falls under the field of graph theory. Graphs are just sets of nodes with edges connecting them. They can be drawn to represent various kinds of situations/problems, like social networks, animals migrating, transport networks, travel planning, etc. Trees are just graphs without cycles, meaning there's no way to return to a node you already have visited without backtracking. A family tree would be a good example of a tree structure in everyday life, but I think the most widespread applications these days are in computer science and programming. I don't know if this 100% answers the original question, but I hope it helps!

[@angelamccarthy4731]

Thank you!!!

[@4tarsus]

Could you possibly boost your audio input? Would make it easier to hear your voice. Thanks!

[@EllieSleightholm]

Yes improved for my future videos!!

[@armanavagyan1876]

Please also the problem from beautiful mind Jhon Nash movie thanks very much for your videos)

[@James_Lindgaard]

On one math teacher's YT channel, people weren't interested in 2 x 2 = 6 but wondered about 8^0 = 8. There are other mistakes in science. What's been missed is that no one has gotten math and science to agree with function. This matters and by January you might be hearing me discuss these "basic mistakes" which lead to mistakes made by Newton and Einstein. I mean you update your smart devices, right? No one has updated math or science.

[@anglaismoyen]

what are you talking about?

[@James_Lindgaard]

@@anglaismoyen Math and science are based on function. 2 x 2 ≠ 4. Does 2 x 2 = -4? What is the difference between 2 x 2 and 2 x -2? They can't be graphed as an opposite function of 2. f(x) = y2 - y1/x2 - x1 give the answer. Try graphing it and you'll know I'm right.

[@NaotaUchiha-e8b]

I don't know why my dumbazz is watching this like I'll ever be good at math but it was a good watch.

[@pyro-e5918]

how did you come up with the 2nd equation? Not clear at all. Wish you had spent more time on the most important step.

[@davidc4408]

Do you have a networth target?

[@neilgerace355]

Is there an algorithm to calculate the answer without drawing?

[@EllieSleightholm]

@neilgerace355 I suppose you could form an algorithm that iterates the possible values for r which will give you the corresponding n_k values :)

[@schulem1409]

Hey, could you please make a video on math books?

[@EllieSleightholm]

@schulem1409 Yes, absolutely!!

[@schulem1409]

@@EllieSleightholm 😁👍

[@_GandalfTheGrey_]

For the 3 3 5 tree, i count 10 lines and 11 nodes…what am i missing?

[@mtb_sail_vids3645]

That is a mistake. The far right node should have degree of 3. (3,4,3) for the tree.

[@derekderek2570]

Maybe I didn’t understand but those final solutions were reducible

[@genghisgalahad8465]

So he wasn't making stuff up on the chalkboard? I feel like one of his clueless friend now!

[@TeslaElonSpaceXFan]

❤

[@HiQuantumSKY]

Sometimes I think... "I don't know what I have studied or where i will use it but I enjoyed it."😂 Sorry, I understood it well and know where to use it. As a Physicist I wonder if some 5th grade kid saw me doing that ... They will be like "what the hell, are you in KG"

[@georgejetson4378]

You lost me with that 2nd equation. Where did the number 18 come from and why is the left side of this equation set equal to it?

[@EllieSleightholm]

​ @georgejetson4378 The second equation looks at the number of lines passing through each of the nodes. We know that if we have n=10 nodes then there must be 9 lines connecting each of the nodes together (due to the condition that we can't have cycles). So we know that there must be 9 lines altogether. Thus, if we look at the number of nodes with lines k we have, n1 + 2n2 + 3n3 + 4n4 + ... + 9n9 = 18. The coefficients come from the number of lines that must pass through that node and we have 18 as we will count the lines twice for each node (remember a line is connected by two nodes). Does that help? If not, let me know!!

[@neonblack211]

didnt numberphile do this

[@Morgyborgyblob]

n(2) does exist. It's a straight line with a node at each end. n(3) doesn't exist. Or have I missed something obvious?

[@CK-ne9eh]

with a n(2)-knot it would be reducible, that is excluded in this task.

[@JackV-n3c]

as a software engineer, i would be willing to bet you £10k that I found a flaw in your logic

[@just-ask-why]

Fun one! Once you explained what the words meant I gave it a go and got 9 out of the 10! I didn’t get the last 3-vertices one, I think I just wanted to stop and see if I was right! Bit more thought would have completed it I figure.

[@lesalmin]

I understand that you solved the problem but I'm not clever enough to understand did you also prove that solution to be correct. 🤯

[@luisakehau1398]

Nice video 🎉 ... Really enjoyed, are we seeing a pattern in here ? Hahaha Greetings from Yucatán

[@EllieSleightholm]

@luisakehau1398 thank you so much! You are seeing a pattern indeed 👀 greetings!