Hey! Heads up. This video contains disturbing amounts of betrayal, deception, and abandonment.
So if you're the faint of heart be warned. Also, there's this. With that out of the way...
Hey. Remember me? I'm Cary KeyHole, and I'm so smart I made it into elementary school. Jealous?
It's okay. A lot of people are. One day in fourth grade, I was pretending to be sick,
so I spent the whole day watching TV in bed.
(Future Cary here. Don't worry. I was a super obedient kid.
I stayed home because I really did have a cold. In this video,
I just wanted to seem edgy to impress y'all. I guess
I just pretended to pretend being sick.) There weren't many interesting shows on TV,
especially because it was mid-day when everyone over the age of five was either in school or work,
so I was forced to watch baby cartoons on the Disney Channel. Now don't get me wrong,
I was a typical 10 year old boy, so I loved shows like Foster's Home for Imaginary Friends,
Ben 10, and Avatar: The Last Airbender. But this baby stuff?
There's just a little too much awkward staring at the camera for me to enjoy. No Dora,
I don't see Swiper even though he's the only thing that's moved on my screen for the last 10 seconds!
Anyway, the TV schedule was giving me a marathon
of Mickey Mouse Clubhouse, which teaches counting numbers and shapes,
but I think the only thing I learned is that Mickey Mouse's ears always point to the camera,
even when his head doesn't, leading to potential infinite twisting which makes me nervous. To prevent this discomfort,
I will hereafter draw Mickey Mouse with one ear since
infinite twisting is not possible this way. In one episode of the marathon, Mickey Mouse and friends were climbing some
beanstalk to meet a giant, like that Jack and the Beanstalk
fairy tale. The beanstalk itself had branches that the crew could climb like a ladder,
but they had to pay attention to whether the branches were little or big.
This beanstalk happened to follow the pattern, 'Little Little Big, Little Little Big' as you went up.
So Mickey and the gang would shout that pattern as they climbed.
I guess if they got a branch wrong, they'd fall to their deaths. I don't know that part wasn't made clear.
Well, after the clan got to the top and fluffed around with a giant for a bit, it was time to return!
What!? I don't remember what Mickey wanted to do with the Giant!
It's irrelevant. Edit: They were trying to get Donald's pet chicken,
Booboo back from the giant. That IS relevant. On the way down from the beanstalk,
Mickey said something like, "Now we have to say our pattern backward! Big little little, big little little!" Now this is crucial!
Based on pictures of the beanstalk,
there is no separator between repetitions of this sequence,
which means there is no beginning and no end, only a cycle.
The exceptions are the top and bottom, but I'll deal with them later. What I want you to notice
is that the cyclical pattern upward of 'Little Little Big' is
actually the same as the pattern downward of 'Big Little Little', just offset by one.
The truth is; The Mickey Mouse mob could have actually used the same
mantra to both ascend and descend the beanstalk.
Are you concerned that the repetitions no longer align with the end points of the stalk anymore?
You shouldn't be. Sequences can legitimately start in the middle without issue. Need proof? I was born on a Tuesday!
All this is to prove that when Mickey Mouse said that the original sequence wouldn't work, and we'd be forced against our will to
reverse it before we could descend to freedom, he was LYING.
Lying to friends is already a shameful atrocity,
but Mickey made the emotional wound cut extra deep by first building trust with his listeners as a teacher.
He taught us skills that were indeed truthful like the fact that the path to the blue stars leads to the tallest tower,
but it was all a ploy to make us feel comfortable
so we'd lower our guards. And then when we least expected it,
he plunged the knife of deception deep into our
unsuspecting hearts, and for that I have no forgiveness.
As an innocent child sitting in bed watching this crime unfold on TV, I
was devastated.
Outside, I felt utterly debilitated,
as if with every passing second my muscles were turning colder and colder into lifeless stone.
But inside, it felt like fiery ants were crawling and chewing up my insides! The torment was relentless.
Oh! I also wondered to myself,
how could the giant have grown this beanstalk such that Mickey WOULD require a different mantra when descending?
Specifically, what is the minimum-length sequence the giant could grow such that it forwards is different from it backwards
unlike the beanstalk shown in the show? Seems like an interesting question, huh? Let's call this an
apalindromic beanstalk. Since palindrome means something that's the same backwards,
and what we want is the opposite of that. Assuming our only types of branches are little and big, (which I'll abbreviate into L
and B,) then there are only 8 possible sequences of 3 letters, all of which are identical to their inverses.
Obviously, any sequence shorter than 3 letters, is also identical to its inverse
so we'll need to go longer than 3 to find it. A brute force solution would just be to check all sequences of longer and
longer lengths until we found one that is apalindromic, but that feels lazy and wouldn't give us an
understanding for why the solution is what it is. So let's think a little harder. My first key observation
was that a cluster of consecutive letters that are all the same is always the same backwards.
Let's call this a chunk. If we want to reverse a sequence consisting of three chunks,
we don't actually have to reverse the whole thing,
we just have to reverse the order of the chunks. Chunks are actually a big deal because here's point two:
Every sequence is made entirely of chunks. Even a single letter is a chunk of length 1.
The number of chunks a sequence has is just the number of similarly lettered consecutive regions within it, but hold on!
There's only two types of letters,
so when one region ends we know what the next region must consist of,
meaning every chunk of B's must be followed with a chunk of L's and every chunk of L's must be followed with a chunk of B's.
If two adjacent chunks contain the same letter then they just be the same chunk,
so now we know the number of chunks is just the number of times the letter alternates plus one.
But hold on! So far,
we've been considering these sequences as if they have a beginning and an end, but as we proved with our "Detective Work," earlier,
they don't. They're cycles.
That means that if the chunk at the beginning contains the same letter as the chunk at the end,
then they're actually in the same chunk. Cool, right? Now, imagine you start at any point along the cycle and you go around once.
You will witness the letters alternating a certain number of times,
but since you end up on the same letter you started on, that number of times must be even.
Since the beginning of each chunk is defined when the letter changes,
we have reached our next breakthrough. Every sequence has an even number of chunks.
For example,
these have two, while these have four, while these have six. Notice that there's one exception:
When the sequence consists of entirely one letter and that has one chunk.
Although, I'd argue it has zero chunks because if you cycle around it infinitely many times,
there's still one chunk spread across infinitely many iterations and one over infinity is zero
but we can see that this type of sequence will always be palindromic so we can ignore them. Recap!
We're trying to find the shortest cyclic sequence such that when it's reversed, it's different, and now we know two things:
Reversing a sequence is the same as just reversing the order of the chunks and
every sequence has an even number of chunks. Instead of sorting sequences by a letter length,
let's sort them by chunk length. As I said before, 0 and 1 are out of the picture
so the lowest number of chunks we should check is 2. We must ask ourselves, "Can sequences of two chunks be
apalindromic?" Lets check!
Now, the length of these chunks is arbitrary, and as you'll see soon, it's actually irrelevant.
But to simplify things, let's substitute the first chunk with an X, and the second chunk with a Y.
Don't forget this equivalence. It's also nice that all these symbols are the same reversed,
so we don't need any more notation than this.
So, every single two chunked sequence can be written as, "X, Y, X, Y, X, Y, X" and so on, right?
Well, what is, "X, Y, X, Y, X" reversed? Still "X, Y, X, Y, X", which means this sequence is unchanged and
palindromic. That means, every two chunker fails. [stock baby crying sound'
Next, we'll move up two sequences with four chunks. For this one, we can replace the four chunks with the labels, "X1,
Y1, X2, and Y2" but remember that if this chunk and this chunk are the same length
then they're identical and in that case we should represent them both with the same symbol that divides the four chunker's into three
categories those where both pairs of chunks are identical
Those where one pair of chunks is identical, but the other is different and those where both chunks are different analyzing the first category
it's pretty obvious that this is just a two-chunk sequence doubled up
so every beanstalk in this category fails. (Crying) The second category isn't as clear cut.
But if we define X1 and X2 to mark the two chunks with the same letter
but different length and Y to mark the ones that are the same letter and the same length
then every four-chunker of this category can be written as X1 Y X2 Y and so on.
Reversing it and using the X1's as anchor points to realign the sequence
we see that everything still matches up and this sequence is palindromic as well.
(Crying)
The issue here is that the Y's are just acting as delimiters that don't give us any more
information. Like glue. So even though the x's are differentiable when they flip around
we don't know which one belongs with which other one so we can easily shift the chunks around to get a 100% match.
Rest in peace. As you probably guessed the third category is where it gets interesting.
Now we have to mark all four chunks with different symbols. When we reverse this boy, and align the x ones as anchor points
sure,
the X2's also line up by virtue of being two away from the X1's. But the Y's have
flipped positions! Meaning that this sequence is different from its reverse.
So we can in theory say that this is a success!
Yaaay!
But I think it's time to show you an example of this. So like we said the third category
consists of four chunker's where both pairs of chunks are different
and the only way to differentiate two chunks is with a different length
so to achieve minimum total length
we'll make one of the X
chunks have one B and another x chunk have two B's (as two and one aren't the same)
and we need one Y chunk to have one L
and the other Y chunk to have two L's it actually doesn't matter which way we put it in
So let's plop them in. Our final beanstalk is big little big big little little.
When we reverse it
we get little little big big little big
and there is no way to make the reverse line up with the original.
Looking at the sequence we can kind of see how it works: having different lengths allows us to say,
"Hey only pay attention to the chunks with one letter because those are our
flags when reading it forward the flag with the big branch is in front,"
Since these two consecutive one chunker's are surrounded by stuff
We don't care about when the whole thing is reversed and the two consecutive one chunker's are still surrounded by stuff
we don't care about we can then see that the flag what the little branch is now in front and something has changed.
So there's our answer
the giant has to grow a beanstalk with a cycle length of at least six if he wants Mickey Mouse and his
Posse to be forced to recite a different mantra when going up versus down.
But what if it wasn't a beanstalk?
(DRAMATIC STING)
But a bean TRELLI S?
(EXTREMELY INTENSE DRAMATIC STING, followed by darkness...)
Yeah, I gotta stop it with the Inception sounds.
Anyway, hello again, and how dare you watch such a stupid video!
But first of all, can I just say that this upcoming December 23rd is scaring me?
That's because, once we hit that date, I will have been on YouTube for half my life, and that makes me feel old.
Hmm... actually,
I take that back.
Because, I've been telling IRL friends that I do YouTube for so long,
that it's hard for me to imagine that fact NOT being true.
But, I wasn't a YouTuber for the majority of my life? What did I do with all my time? Whatever. Oh second thing,
thanks for 300k subs (that's one step closer to impressing my parents
And I could make a dumb Sparta joke, but I won't.)
Okay, I hit 200,000 last September. And 300,000 this September. Oh!
Okay. It happened!
It happened! Oh my god, I didn't even finish my sentence! What the heck guys? You were interrupting me! How rude.
Lastly, while I was gone, I got a Instagram and no, I was not born in 1994,
I was just the 94th Cary Huang to sign up for Facebook, ok?
Also, you just know I'm active on that platform because the number of my Instagram posts is a power of ten!
*inhale*
WOW Ok, I gotta go finish some take-home midterm exam now.
Actually though, that's what I have to do right now. So BYE.