@andrew_ray
13:10 I love that in this animation, when two numbers are combining, each turns into some portion of the sum proportional to the value it contributes to that sum (so that when 1 + 1 turns into 2, each 1 turns into half of the two, but when 3 + 1 turns into 4, the 3 turns into three quarters of the four and the one into only one quarter). You absolutely didn't have to do that, but I find it strangely pleasing that you did.
@pakkiufung883
The fact that 3blue1brown wrote a poem about this problem when he was a kid is mind-blowing Edit: Ok, I know there are some angry replies stating that ‘when he was younger’ doesn’t mean kid, but to an *11-year old Asian kid whose main language isn’t English*, it’s mind-blowing all the same.
@danishazhar86
I'm just impressed that you were able to make a poem AND a FREAKING SONG about this problem as a CHILD.
@thebighg
Found this channel an hour ago from the short presenting this problem. Immediately hooked to see why. I finally found a place that can explain math so well and visually beautiful that people are drawn to it and can love math again. And this is the power of right education. Thank you.
@InactiveAccount325
A more intuitive way to understand the formula for number of areas from n chords, is that for each additional chord, it creates a new area, and another new area for each chord it intersects. So the total number of areas is the original circle + the number of chords + the number of chord intersections.
@wolfelkan8183
I don't know much about diophantine equations, but if I've learned anything from Matt Parker, the first step is to write some dodgy Python code to brute force check if there's another power of 2. Obviously, we can't go to infinity, but this will at least identify if the conjecture is obviously false. I can confirm that other than the cases already discussed, the number of regions will never be a power of 2 if the number of points is less than 15 billion.
@sapemi08
I'm 63 and once I read somewhere the phrase: "The secret of life is to make it look easy". Sir, you are the person that makes maths look easy. Thank you so much for your work. Greetings from CDMX!
@marcusscience23
The reason Euler’s formula works for both polyhedra and planar graphs is because the former can be transformed into the latter by stereographic projection, preserving the relations of vertices, edges, and faces. Additionally, the reason Pascal’s triangle shows combinations can be understood this way. With n items, there are a certain number of ways to choose k of them. With (n+1), you can either include the new item and end up with (k+1), or not include it and have k. Both of these are carried onto the next row.
@sideswipe3996
I love how you wrote a poem over this problem it really shows how invested you are in it and I read the poem, I have tried many times in school to write a poem, and my brain just can’t do it
@jucom756
This is a fun problem to give to high schoolers right before they get binomial, and then revisit after, because it uses a lot of the concepts you learn with binomials and polynomial approximations.
@warasilawombat
Euler really was such a gift to humanity
@Luke-me9qe
I love that he does not just get satisfied to answer the issue but explains also why it is the way it is. Plus in a nice and calm way.❤
@jdmhexagon2584
I have an MMath, and the last few minutes of this video made my jaw drop in a way that none of my lectures ever did. Look up the word "elucidate" in the dictionary and you'll find this video. Wonderful work.
@Blaisem
13:27 "circling" back to our original question. I see what you did there. But my favorite part of the video is how you hunt for the underlying "why." That's exactly how I like to learn, and I appreciate that you delve into that for just about every component of the explanation.
@larsprins3200
To solve Moser's circle problem you don't need to use Euler's theorem for graphs (but the detours to Euler and the Pascal triangle certainly make the video a lot more educational). Once you know that there are (n choose 2) chords and (n choose 4) intersections, you continue as follows: The original circle without chords is 1 region. Now start drawing chords, one after the other. When you start drawing a chord from a point on the circle, the chord starts cutting an existing region up into two new regions right away, thus adding one region. Whenever a chord crosses a chord drawn earlier at an intersection, it finishes cutting up a region and starts cutting up a further region into two new regions, thus adding one further region. The total number of regions is thus 1 + "the number of chords" + "the number of intersections". The answer to Moser's circle problem is thus 1 + (n choose 2) + (n choose 4). Yet another great video by 3Blue1Brown, thanks!
@billyjoethethird8436
The explanation of Euler's characteristic is simply mindblowing. It's much easier to understand than the inductive proof!
@tobybartels8426
You can tell that something must be up, because if you place zero points on the boundary of the circle, then the question still makes sense, and the number of regions is again 1, while the 2ⁿ formula would (nonsensically) suggest that the number of regions should be ½. If you realize that the sequence begins 1, 1, 2, 4, 8, 16, rather than thinking that it begins only 1, 2, 4, 8, 16, then it becomes less surprising that the next term isn't really 32. This approach to math problems, thinking about what comes before the sequence that you're trying to extend, is sometimes humorously called ‘negative thinking’. It can be tricky to do, but sometimes it helps.
@tonitete
well, there is not another power of 2 in at least the first 10,000,000 numbers, maybe later on there is another, but my CPU is crying so i better stop here.
@romanceano7519
It is so nice to feel again being a kid in math class, marveling on the the magic of how everything connects and wondering what's next. Thank you Grant for all this hapiness.
@Nathan-wp1ir