Sorry not sorry about the vertical video, I love the multi-platform lifestyle! Feel free to support me on Patreon and lobby for your preferred video ratio ;) www.patreon.co...
The joy of avoiding counting should never be underestimated.
@metallsnubben
Жыл бұрын
A hard-to-translate quote by my old maths teacher: "i matte ska man tänka mycket och räkna lite" - in maths you should think a lot and count* a little (but "räkna" in swedish is both "count" and "calculate", same origin as "reckon" in english)
@reidflemingworldstoughestm1394
Жыл бұрын
Counting. The only thing worse than addition.
@vaxjoaberg
Жыл бұрын
I remember being exposed to this problem way, way back in junior high school. It seemed an insurmountable problem at the time but when the solution was revealed I began to understand what math was for.
@Wolf_Avatar
Жыл бұрын
My thought process went like this: There are 5 "up" and 5 "across". So when you mix up the 10 moves, you get 10!, but 5 of them are indistinguishable "up" and 5 are indistinguishable "across", so you divide 10! by 5! twice. Which as it turns out is exactly how to calculate 10 choose 5.
@oli_k
Жыл бұрын
restoring my faith in maths here
@madeline6951
Жыл бұрын
ok I know this video has only just been posted, but it's criminally underrated nonetheless
@Rojo9149
Жыл бұрын
Wake up yall, new Ayliean video 🎉
@gtziavelis
Жыл бұрын
Oh Pascal, you rascal! Oh Blaise, what a malaise!
@nxpnsv
Жыл бұрын
Very enjoyable!
@Ayliean
Жыл бұрын
Thank you, I’ll take any excuse to draw straight lines :)
@roxanne3590
Жыл бұрын
I love watching these, you remind me of younger Vi Hart when they were just starting out on KZitem. Don't be surprised if one of your videos totally blows up (even more than the Mono-Tile video) one of these days, this stuff is gold
@Exception-mk3xh
Жыл бұрын
Woahh, this is a brand new perspective on a problem that I've heard and tackled ages ago. That is a surprisingly genius, simple, AND beautiful way of looking at it. If you want to see another cool solution to this problem, just read further below. When picking three random paths, there is an interesting observation to be made: Path 1: → → → → → ↑ ↑ ↑ ↑ ↑ Path 2: → → → ↑ ↑ → → ↑ ↑ ↑ Path 3: → ↑ → ↑ → ↑ → ↑ → ↑ All of these paths have the exact same number of arrows, 5 ↑ arrows and 5 → arrows. So each path is simply a different arrangement of the same 10 arrows. The question now is, how many arrangements of ↑'s and →'s can we make? We can think of each arrangement as "10 slots" for all the 10 arrows to go in. And if you think about it, we only need to focus on **choosing 5 slots out of the 10** for one type of arrow (let's say ↑), since all the other unfilled slots can be interpreted as being filled by the other type (→). The **bolded statement** from the last paragraph can be determined via combination formula: 10C5 = 10!/5!(10-5!) = 10!/5!5! = 252 And to tie this very well with the video, the combination formula is VERY related to Pascal's Trangle. To solve nCr, you just need to go the the (n + 1)th row of the triangle and find the (r + 1)th number. For example: n = 6 and r = 5 7th row of pascal's triangle: 1, 6, 15, 20, 15, 6, 1 6th number = 6 and it's true, 6C5 = 6
@Ayliean
Жыл бұрын
Don’t you dare ask me to draw all 3432 paths for an 8x8 board…
@RaglansElectricBaboon
Жыл бұрын
It seems almost as though you're daring us to ask you..
@kennynolan736
Жыл бұрын
There is something wildly satisfying about seeing all of the possible paths hand drawn out like that. I'm curious how you chose the order to put them in, it's not obvious to me!
@pbenikovszky1
6 ай бұрын
Oh I used to ponder on this topic so much when I was in uni, I found it so interesting that it is the same to choose 5 out of 10 and count the number of permutation of 10 elements, where 2 elements repeat 5 times each. Then you realize that counting the permutation of the elements A,A,A,A,A,A,A,A,B,B which is 10!/(8!*2!) is the same as 10 choose 2 or 10 choose 8, and now you see whey 10 choose 8 is equal to 10 choose 2 :)
@homomorphichomosexual
10 ай бұрын
woah! i remember i first learnt about this when i started reading about dynamic programming since you can model the number by a recurrence relation, but i never actually noticed it's just pascals triangle :D love your videos, definitely very vihart-y
@reidflemingworldstoughestm1394
Жыл бұрын
That was awesome. I can't wait to see Pascal's square.
@nobodi6448
Жыл бұрын
i love your videos! so much enthusiasm! and math! and presented with pretty visuals! 🤩
@aner_bda
Жыл бұрын
Pascal's Tri... Square?
@TranscendentBen
Жыл бұрын
There's a Project Euler problem, maybe more than one, that uses this.
@jakethomas6123
11 ай бұрын
Beautiful!
@mint530
Жыл бұрын
!!! very good video
@vishal1278
Жыл бұрын
Lovely! I am a bit confused about the combinatorial approach. You’d have to choose 10 steps (out of some total possible number of steps) in total right? Why, in your formula, we choose only *five* steps (out of ten)? Five steps wouldn’t take us to the top right corner…
@gx8fif
Жыл бұрын
I think the audio has been swamped out by the music. I can hear you, but not well, on PC speakers
@Ayliean
Жыл бұрын
Thanks for letting me know, upping the sound setup is on the to do list :) I’ll try to get you that sweet audible chocolate soon!
@xxnotmuchxx
Жыл бұрын
1:50 weee
@klgamit
Жыл бұрын
Does this generalize to n dimensions?? i.e. will any cell in such n dimensional lattice will have the sum of "previous cells" possible paths as its own number of possible paths? (I think the number of "previous cells" is equal to the number of unit vectors in the space, which is just the dimension itself, because for example in 3D we have 6 directions, of which we allow only the 3 positive ones for progressing "forward" in the lattice... but I'm not sure :))
@metallsnubben
Жыл бұрын
I think this checks out, in particular each "outer face" of a cube would be the "pascal's triangle square" here since it's only using 2 of the n directions
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