This pattern breaks, but for a good reason | Moser's circle problem

There's another great sequence that goes, 1,2,4,8,16,30,60,96... The number of divisors of n!

“Simple yet difficult” problems are always lots of fun. You go through pages and pages of papers, and suddenly you’re like.. wait a minute.. OH COME ON.. that shoulda been easier 😂

I'm just impressed that you were able to make a poem AND a FREAKING SONG about this problem as a CHILD.

Being able to understand the math behind this problem is a true human blessing, let alone the fact that a stranger on the internet shared it with me

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.

Well done Grant. My 11 year old son was able to follow this and his curiosity was satisfied. For context, his formal maths education has not included functions, nor algebraic manipulation, nor factorials, nor the multiplicative principle for counting permutations. He had met Pascal's triangle but knew very few of its patterns. And a huge amount of the terminology was brand new: graph, edge, permutation, function etc. In other words, you have done a superlative job of communicating. The fact that he was able to follow and understand the reasoning when so much was new to him is testament to a job well done.

The resulting integer series are also known as the "Parker powers of 2"

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.

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, it’s mind-blowing all the same.

13:18 Love how in the merging animation size of a number ends up proportional to its value. I.e. when 1 and 4 merge to 5, then 1 takes 20% and 4 takes 80%

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.

31… it better not be computing pi

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.

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.

The thought and care that you put into the logical sequencing of the concepts, the animations, the selection of how to illustrate a spoken statement, are beyond fantastic. The improved intuitive understanding that results is thus extremely solid and greatly satisfying. That's exactly how I would wish to perfectly explain anything to anyone. Thank for doing this! (that's an exclamation point, not a factorial 😂)

3:55 music and the chords lighting up line up so perfectly I'm glad you redid this video. Original was great but this one has just a little something on top that makes it fantastic

Euler really was such a gift to humanity

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.

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.

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.