Is it possible? Simple questions, not so simple solutions
Get free access to over 2500 documentaries on CuriosityStream: curiositystream.com/majorprep (use promo code "majorprep" at sign up)
STEMerch Store: stemerch.com/
Support the Channel: / zachstar
PayPal(one time donation): www.paypal.me/ZachStarYT
Versión en español de este video: • ¿Es posible? Preguntas...
Instagram: / zachstar
Twitter: / imzachstar
Join Facebook Group: / majorprep
Mathematics for Computer Science (MIT Course)
►Lectures: • Lec 1 | MIT 6.042J Mat...
►Course Page (assignments and textbook found here): ocw.mit.edu/courses/electrica...
Animations: Brainup Studios (email: brainup.in@gmail.com)
►My Setup:
Space Pictures: amzn.to/2CC4Kqj
Magnetic Floating Globe: amzn.to/2VgPdn0
Camera: amzn.to/2RivYu5
Mic: amzn.to/2BLBkEj
Tripod: amzn.to/2RgMTNL
Equilibrium Tube: amzn.to/2SowDrh
►Check out the MajorPrep Amazon Store: www.amazon.com/shop/zachstar
If you get a pen thicc enough, you can connect all of them with 1 line
If the dots are big enough, you can use only three really long lines at slight angles.
@@ryancw714 Or if the piece of paper is large enough
Or if all dots are overlapped
Or if you can fold the paper
A line has no width.
3:20 imagine pausing the video to try to solve it for like an hour jsut to come back, unpause it, and see its impossible
but if you're smart enough you will understand the impossibility and start trying to prove it
Who would do that
That is comendable engagement, and you likely understand why better than those who just shake their heads in agreement. Stay a skeptic in all matters!
if you couldn't figure out that it's impossible then you couldn't solve it
Mohammed jawahri that doesn’t mean u can’t solve it
"Is it possible to start with 4 students and infect the whole class?" 2020: Say no more.
worse ror what happened to this guy?
worse ror Patient Zer0 lives while thousands of other people died... Quite ironic
666 likes, seems fitting
@@Zack_Zander whos Patient 0
@@DanksterPaws I'm pretty sure patient zero just refers to the first human recorded to have a particular disease
The infected question is actually similar to multiplying water in minecraft, the best way is to place a water source across the diagonal of the cube, so every block becomes a water source
That's what I immediately thought! And i figured the most efficient way to infect students would be put the infected kids diagonally across the room, just like the way you would fill an area with water in minecraft
I love that that’s what I thought of as well
Just wanted to comment that!
Ikr!!
life imitates art..
I have discrete math course flashbacks. All of this is fun and games, until you are graded for it.
ikr
My discrete final is in 10 hours, very much so not looking forward to it :|
It shouldn't be that way... School conditions us to be scared of math but once you actually look at it from a fun and games angle it's actually really interesting and useful in certain applications.
Yea
@@melaniemedina8037 Must depend on the school/teachers/kids. Nobody was "scared" of math at my school. Maybe some who were less than average but that's about it
1:53 the third problem is actually how water works in Minecraft
C o p i e d
qsaification I was going to write the exact same thing!!!
**looks at number of likes** nice
@@user-go5he1fe1g It is actually pretty obvious to those who have played Minecraft, that's the first thought that came into my mind. It doesn't surprise me if you saw comments similar to mine :)
@@qsaification oh k
3:16 that moment when you've placed enough water sources in Minecraft to know the answer immediately.
I was thinking same
The police thief problem at 8:26 is similar to a concept in chess called triangulation where you lose a tempo with your peices and force your opponent to play an unfavorable move due to him having no good move(like the Red Red city of police and the robber in this example)
I was thinking about that as well haha, came to the timed comment section to see if anyone else spotted it.
Do you play chess?
kzhead.info/sun/lalvdbKjm6B9Y5s/bejne.html Puzzle 3 in this video is the most insane version of this ever.
@@null3007me too haha
In germany we call this Zugzwang
3:06 and that's why you sit in the corner of the class room.
Best chance to survive a zombie apocalypse
@@mayocult2025 taking classes online is an even better strategy
@@HPD1171 true
@@HPD1171 now we all have to take online classess
HPD1171 Welp,
there is something unique about this channel
It has 1 odd node.
Yeah. It's like a hybrid of 3b1b n socratica but something more
Everything about this channel is perfect but what makes it unique is HIM ⬆️
Yeah - confusion
As with Arvin Ash
Theory: whenever someone says "no pun intended" in a video/presentation... The pun was intended.
2:27 *MINECRAFT WATER PHYSICS WANTS TO KNOW YOUR LOCATION*
The classroom problem is the infinite water source problem in minecraft. Just sayin’...
Yooo thats what I was thinking.. spooky
Wie alle Mainguaftler das selbe denken xd
I thought that too :D
_uses Minecraft to simulate this problem_
Damn all you, I thought I was original.
Can't believe you didn't mention triangulation in chess in the thief/cop problem.
This problem reminded me exactly the same thing
Magnus carlsen has joined the chat.
Yes, I thought of the same.
I thought exacly the same too !
Bobby Fischer has entered the chat
2:16 Fun Fact: This puzzle can be visualized in Minecraft using Water since they share the same mechanics. Goal would be to make the hole filled with all source blocks.
Thank you so much! I've tried with at least a dozen classrooms by know and I couldn't figure out how there were always survivors! I've used this info four times now and it has worked without fail!
I found a different way to think about the infection problem: Suppose all in infected students are inside a rectangle smaller then the 5 * 5 grid. Since all healty students outside the rectangle share at most a single border with an infected person, the infection can never spread outside the rectangle. With 4 or fewer infected persons you can allways draw rectangles around them, so that there are remaining students outside the rectangles. Only with 5 infected students, that are placed along the diagonal, the surrounding rectangle is equal to the complete grid, so everyone can be infected (and also will be).
I might have interpreted this wrong but what about 4 people along the diagonal being infected (ignoring the center)? Now you cant draw a rectangle around everyone and have students outside of it.
MajorPrep For the thief problem, what prevents the thief from doing the same thing as the police? That is, go to the neutral city and switch states so once again the police can’t land on the same color city as that of the thief. Seems to me that then there is no guarantee that the thief can be caught.
@karagi101 If the thief has the opportunity to go through that city on their own before the cop then they will change the state of the game to one they'd lose. the cop could just follow them and it'd result in that same pin. But either way from the beginning the cop has the ability to go through the city first and from there the thief couldn't get to it themselves (at least in the layout from the video).
I think this solution works but nerds expanding a little: draw bounding boxes around each subset of students who don't neighbour each other, including diagonal neighbours. Each subset won't be able to infect beyond their bounding box, and with n infected students the biggest bounding box you can construct has an area of n² - so 4 students can't create a 5x5 bounding box, but 5 can.
@@zachstar I think the OP means 'rectangle' not in terms of an outer hull of all the points, but a connected sub-matrix with no 0 rows or columns. In your example of 4 along the diagonal, there will always be 1 row/column which is all 0s, either making a single 4x4 'rectangle', or two disconnected 'rectangles' either a 1x1 & a 3x3 or two 2x2 ones. Each of those separated connected-sub-matrices cannot expand beyond their own borders. The 0 row/column can never become populated with any infected students. Thus, the problem can be thought of in terms of matrices, matrix degrees, and linear independence (at least for figuring out the minimal n; the reasoning doesn't quite work when the # of initial infected students is allowed to be larger than the minimal n, since 0 rows/columns can then sometimes become infected).
The police-theif game at 8:26 can win you a bet
How exactly will you propose the bet? Be this thief so I can prove to you that I can capture you as a cop for $x
@@con_pi_tour_dice7147 Or I bet the cop will/will not catch the thief
Can it really? how dumb are people? you can solve it with very minimal bruteforcing/pattern recognition...
@@con_pi_tour_dice7147 Yes, what elde would the bet be? lmao
1. Thou Shant Lewd Kaori Any wise person could increase the complexity of the problem while preserving that basic odd edge which is the key to solve this problem. In this way your brute force approach shall be in force forever.
love this video - good work Zach
If we bring the question at 3:13 to Minecraft "Can you use only 4 water bucket to fill the 5x5 grid?" it will be a lot easier to think
Only be a robber in states with an even number of cities! Life hack...
What?
But what if the robber followed the police into that void?
@@Mr48two if that happened he would still be caught because if he touches the red then the police is on either neutral or and it will be your move or he will have to turn back and get cornered
Imagine someone getting caught and not being able to understand why they caught him... "But the math checks out!" Would be another case to study the Dunning-Kruger effect
gotcha i.imgur.com/RYDDp16.png
Ah, math that i dont understand. Perfect.
h
it was explained perfectly thou
U actually reminded me of me math hw for tomorrow, god bless you
Why, it's actually easy
Yes marvelous
The problem of the infected students could be viewed as Minecraft water sources spreading. It's really the same concept.
Actually yes bro it's same concept 👍
LMAO didn't think of that
I love you, your content and your account, thank you so much for everything, hug from Portugal ❤
Never thought this would become useful, but the infection problem is just minecraft water spread mechanics. At least 2 adjacent water sources are needed to form a new one, and the least amount you can fill a square with is a diagonal of them, which would be the same as any given side. This just multiplies it by 4 and uses the perimeter. That's hilarious
@@qbwkp because Minecraft is an accurate model of the universe.
Just VACCINATE YOUR CHILDREN
Diagonal is only one possible solution. It can be done without putting any sources on the diagonal. Here's an N=5 solution that doesn't use the diagonals. 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0
@@erinelizabeth9545 Someone mentions Minecraft water physics You:
@@sbunny8 Or a more systematic way of doing it is to cover two connected sides of the square like an "L" as in: 00001 00001 00001 00001 11111 But then that would be wasting students since there are infected students that already have two neighbours at the start. So the most efficient way is to draw the "L" without infected students that already have two neighbours: 00001 00000 00001 00000 10101
This was a surprisingly insightful video, connecting the simple idea of whether something is possible, to how to identify and understand invariants, to the final revelation of Euler's Characteristic, which I've seen before but never really understood. Thanks!! 😊 You're on par with 3Blue1Brown, Mathologer, et al., but a little briefer, more collected, and more to-the-point. It makes your videos a little easier to digest, not requiring quite as much time spent. Truly great work, MajorPrep! Thanks again!
The way I looked at the disease question is that when you initially select the infected desks, you are immediately choosing the parameters of how far the infection can spread. If you have a 5x5 square, and you put a student on (1,2) a student on (1,4) a student on (2,3) and a student on (2,5) then your parameters are going to be where the farthest x and y points meet. Since you're given a maximum of 4 people to infect, your parameters can never exceed 4x4, or 16 squares.
Like water source blocks in Minecraft
and the furthest you can place your edges are at the corners. But you can’t have students at just (1,1) (2,2) (4,4) and (5,5), because they would just create 2 x 2 squares. You have to connect them which would mean you would have to have five students at least
Idk if it was assumed that the classroom was a square, but my solution was: square root of area - one
For the students infected one, I'm surprised he didn't mention that the students have to be in a continuous diagonal line for maximum infection. And that the students form the edge of the square/rectangle, so it's impossible for the infection to move outward
Labelling rows with letter and columns with numbers, place infected students at a1 c1 e1 e3 and e5. This would create an L shape and the rest of the students would be infected spreading from the corner
@@BryanLu0 Good find!
It don't have to be diagonal for maximum efficiency
@@BryanLu0even other "shapes" work 00100 00000 10101 00000 00100 Works too But I think the l shape, is the on, that Takes the longes time to reach every cell
The infected students thing is basically the same as filling up a hole in Minecraft such that the entire hole is filled with source blocks
still_lava115 ya you need water source on diagonal, iirc
functionally identical if you assume water cannot be picked back up and you only have 4 water buckets
I CAN DO THE CONNECT THE 9 DOTS IN 1 LINE!!! *Uses a crayon without the wrapper and uses the side*
A line has no width in geometry.
@@windows_sky r/ihavereddit
Y’all eat your crayons with or without the wrapper?
@@avamatthews1459 you monster
@@avamatthews1459 imagine eating the crayons 1 and 1
No, no, they just close the schools at that point. Believe me, we know. -The Future
Let's get 20 dislikes too, so it's also signed 2020.
Anyone who plays chess can solve this puzzle 8:56 easily, we have the same concept in our game. Sometimes enemy king (let's call it black) is in the worse position, but you can't pushem away or change the situation on a board with your king, because black repeats your moves and don't let your king walk in. This triangle in the right bottom is the key, in chess we call that triangulation, when black can repeatedly do only 2 moves to repeat the position, but you can do 3 and so make the black go away. Surprised that knowledge of chess helped me with some math problem, chess actually is a great game to sharpen your mind because quite often you've got to solve such puzzles on a board to win. Great video btw, it took me several hours to watch it and solve everything
For anyone wondering how the V-E+F=2 works, here's a short explanation: When there's only 1 vertex, there is obviously no edges and the only face is the outer section, so V-E+F=1-0+1=2. If you add another vertex a new edge is also created, so the sum doesn't change. You can keep adding new vertices like this, and as long as you don't connect any old ones V-E is always going to stay the same (and since you're not creating any new faces F is not going to change either). If you connect two existing vertices, though, *only* an edge is going to be created - thus V-E is going to decrease by one. Meanwhile, a face is inevitably going to be created, and the total sum is going to remain two.
As always, awesome video, man! I usually save your videos to watch them right after I wake up. Helps me wake up with the perspective I want for the day.
This channel is awesome! I'm considering a career in statistics, and this channel is just feeding that idea. I love this channel!!!
13:31 ah yes, I didn't see the answer until you made that simple shape! Thank you
My favourite KZhead channel, you completely change the way I view mathematics and it’s applications
At 14:56 you could also just teleport right away and start from the outside point and weave your way around. Both ways work because you're starting at an odd numbered node.
Yep! I actually saw that after having the animations done but didn’t want to change it lol
@@zachstar Lmao. It's all good. Always interesting to see new ways to view and approach a problem. Even ones where it's not complicated, but ideas can vary. Also, I would've done the same thing too lol. Love your videos. Keep up the good work, but dont overwork yourself.
The reasoning behind why it's possible with the teleport hole, is that it just makes the two "nodes" the same (since you can travel between them without restrictions), so you can just add their degree and get 14, which is even. Then you just start in one of the odd nodes :).
So at 14:38 is the puzzle solvable? Or do the odd nodes need to be adjacent to either each other or the worm hole?
For the first problem, you can connect all dots in 3 lines. Just connect all the columns then tilt the lines and expand. The line will then intercept at some point
What do you mean?
I came up with a slightly different method for the infected students. I noted that in order to completely span the length of a side, you need at least n/2 + 1/2 students, arranged in a X O X O X straight line. (I think for Even n, it's n/2 + 1, X O X O X X). You need to do this for horizontal, and vertical, otherwise the infection won't be able to reach every row/column. While you can share an infected student between the two lines (e.x. lines are top row and leftmost column, sharing the top left corner infected student), that still leaves n/2 + 1/2 + n/2 - 1/2 = n. For evens, it works out to n+1. So, it's not possible to infect all students with fewer than n infections.
Superb! Enjoyed that.
This topic turned out to be very interesting, although it did not seem to me at first. Thank!
I watch a lot of math videos, but this was the best I've seen so far!
I have could like to say that you have improved my way of my thinking
I subbed three years ago when I decided to go into engineering and didn't know what to major in (ended up choosing electrical). Glad I stayed subbed because your videos are becoming more and more interesting. Happy to see that you have gained so many subscribers because you seriously have such high content quality.
Thank you for the comment and glad you've stuck around!
Is it possible? apparently sometimes it is and sometimes it isnt... Who could have known Great video btw
Thank u so much of how to learn so much more and i was looking for a kind of website where it can teach me how to learn other equations and other lessons u inspire me to learn and solve problems the right way so thank u
It has probably already been said, but something like the problem about the thief actually happens a lot in chess. When you reach the latest stages of the game with very few pieces (kings and pawns), you can find yourself in positions that would benefit one side or another depending on who has to move next. So the strategy involved is to "lose a tempo/move" by triangling with your king (using 3 moves to come back to the same spot, instead of 2) so you get the same position handing the turn to the other player and therefore forcing him to let you some space on the board. It's kind of hard to portrait really, but it is not that of an advanced tip you learn in chess, but still it made me instantly find the solution for this problem
Ima big fan of u und ur videos. Keep making great videos. 👍👍👍
11:00 we litterally learned this in my a level further maths class today
2:42 See also: Minecraft water source generation
I came up with a different (less elegant) solution to the nine dots problem. You can do it by drawing an m-shape with four line segments, with the only catch being that you would have to go back over already drawn lines.
lol the chasing the thief problem is so obvious to anyone who's played roguelikes
Exactly what I was thinking lol
This channel surprisingly gives me feels like the Vsauce used to do.
all hail vsauce
Watch Ted Ed
I'm pretty sure that I saw an video with these exactly same problems in one of the Michael's channels.
@@KenKelvinN he has spread his contents over so many channels and its so hard to pinpoint this in his contents.
I LOVE problems like these. I hope to come up with my own someday
The problem at 8:43 reminded me of the opposition in chess and how often you need to triangulate the king in alot of pawn endgames
This came up in my recommended and the infected problem felt too real, very informative though good work!
when you intuitively know the proof around 4:00 because its the same as minecraft water physics
Wow..I loved the way you explained the topic which I find difficult to understand 👏👏
This was a nice video overall. We need more of graph theory problems.
2:22 Ah the good old minecraft pool problem There's a more intuitive way to resolve it though
The same goes with the thief problem
My biggest problem is my inability to think outside the box, but almost all the ones that required logical thinking I was able to rationalise an answer. Nice video, btw.
Funny thing is in my mother tongue no one explained it to me as clear as you did :) I might even start to like maths now - thank you so much!
I love how you changed the music when Euler's characteristic was revealed.
"simple question, not so simple question" Seems like exam questions
When watching the part at 4:10 i just kept thinking about infinite water sources in minecraft
Thank you. This really helped me fall asleep.
The infected students' problem reminds me of the segmentation problem in Image Processing (region growing and region splitting)
These videos are so high quality it's amazing
Do Sprouts theory! If it exists, I don’t know. This video just reminded me of Sprouts, maybe it’s the dots and lines...
4:40 only 3*3 is infected but only with one recursion and the other is increase in log
these type of videos are usually frustrating but i actually learned something
Well, I gotta say, in the light of the first puzzle, where we have to use the space outside the square, I figured the answer to the second puzzle would for sure be to just let some of the rectangular tiles partly hanging outside the grid. That's perfectly doable.
I THOUGHT THE SAME THING
The first one was a challenge our math teacher once gave us. Me, not knowing the point of the riddle, just connected three points downwards, and then FOLDED the paper, so that the line actually connected all nine. I just thought of controlling space to get what I want, because I didn't got it right. Big Brain or dumb failure of the rules? He said its good. Sorry for the bad sentences, I usually don't speak english very often.
I guess you were allowed to lift the pencil off the paper then
You could also solve it by backtracking the pencil along existing line segments. You're not taking the pencil off of the paper and are not creating additional line segments, merely re-tracing existing segments.
@@XeNoX_off Thank you for your help!
When you add the hole in the house problem you are merging the 2 nodes, basically adding up the numbers on them. Thus you remove 2 odd vertices adding an even one and the graph is traversable.
For the infection example: the perimeter solution is a nice mathematical proof for it, but you could also word it differently. Given any n×n grid, the number of infected students must be equal to or greater than the number of diagonals in the grid. The diagonal configuration means that you'll start of with all sides of an infected square adjacent to a non-infected one, meaning no edges are shared between infected squares. It also means that at least 6 non-infected squares will be infected, then 4, then 2. Funnily enough, the only reason I know of the solution I outlined above was due to a slightly different problem for someone I was tutoring in a math class geared towards teachers teaching math for K-5.
9:48 is like triangulation of the king in chess
3:42 ... one of the instructors last name is " Van Dijk " !
1:53 This problem actually has a pretty interesting application in Minecraft: Water blocks technically have two different states: Source water blocks that can be collected, and rushing water blocks that cannot. Rushing water can be converted to source water using the same exact rules shown, and it is also proven there that the absolute minimum source water blocks needed to fill a pool of nxn size has to be n blocks.
Very well done.
WHAAATTT THEEE *BEEP* It's my first video on this channel and how come I not realise till 7:49, its Zach
0:20 There can be thicc line wich connect all dots xD
8:30 - I solved this as soon as I saw it, because of a well-known triangulation concept in chess that goes the same way in king-pawn endings, with the same effect of "switching" the move order.
For the cop and robber one, I saw that bottom corner and realized I could catch him there, but gave up when i couldn’t think of a way to force him there, I didnt quite think of that
6:57 rip that non infected guy
the problem at 5:18 (infected students) felt realy easy to me. I could just tell how it would work for some reason. Then I realized, it is the same as water spreading mechanics in minecraft.
I knew solving it felt familiar. Making aquariums in Minecraft is useful.
3:20 the infection can never grow outward towards the edge without there being an initial infection, so you must have infected nodes on the edges, then work from there
The catch the theif solution is similar to a manouvre in the game of chess. The extra town (neither blue or red) allows the manouevre of 'triangulation' which is a bit like skipping a turn.
I thought this video would be about an unlock pattern, lol
1:07 this is some weird minesweeper
one thing I was thinking about during the 6 rooms problem as I was trying to solve it was that the graph of the nodes wasn't planar, so when he connected that to the problem being solvable on a taurus, i had this HUGE moment of ecstasy. like i used to just look at those problems and think, "oof unsolvable, big sad" but it's just impossible in 2D, it's doable in 3D, the problem just isn't presented that way so I never thought about that!
You mentioned that for all the dots to be connected either all dots must have even edges or 2 dots can have odd edges, but I drew a figure in which both these conditions are failing!
0:34 Intend your puns, coward!
My mother once had me try to solve the 1st puzzle on this list and while it did take me like 20 minutes (I refused to give up!), the phrase "thinking outside the box" occurred to me as I was taking a bathroom break (lol), so I asked my mother if the dots were restrictive. As in, was I ONLY allowed to draw directly from dot to dot. She said no, I didn't have to... after which it pretty much solved itself xP. But yeah, that even occurring to me did take a long time...
we need a part 2!!!
0:44 Any grid XY can be covered by rectangulars where XY/xy=natural number
3:35 beaver flu: a rare variant of bird flu that lasts forever *uh oh*
Oh Yes
I believe the "9 dot game" is where the term "think outside the box" actually comes from, so you were actually right on point.
the cop and thief puzzle is like the opposition of kings in chess, you have to gain a tempo to get the opposition and once you do your opponent cant defend/ attack and thats what the diagonal does
It's past midnight. I should be sleeping. But first, let's solve problems I will never encounter.