2048 Induction Extra singingbanana English (auto-generated) English

2048 Induction ExtraYouTube↗ IA↗(long shot)
Education 66.5K views 859 likes 12 dislikes 10 Apr 2014 Show Contributors 1

singingbanana 228K subscribers United Kingdom YouTube↗

Contributed subtitles

Dutch Angela Flierman

Click to Play Video

as so this is Steve mold and we've just

>>James: So this is Steve Mould... >>Steve: Hello.

made a video about internet sensation 2048 the game and that's on Steve's

>>James: and we've just made a video about internet sensation 2048, the game. >>Steve: Yes.

channel Steve the brave is equal Steve

>>James: That's on Steve's channel-- Steve the Brave? Is that...?

most even the brave with a be because I

>>Steve: It is called Steven the Brave, with a 'v,'

set it up ages ago and you know you must

because I set it up ages ago, and you know....

you be paying for a cherry should have a

>>James: What a stupid name for a channel! You should have a sensible name, like I've got! [laughing]

sensible name like I've got I've seen

>>Steve: Like Singingbanana.

yeah so go see Steve an eyes a video out

>>James: Yeah! So go see Steve and I's video,

we'll put annotations so yeah but we

we'll put annotations somewhere.

thought we'd do a little extra bit yeah

But we thought we'd do a little extra bit. >>Steve: Yeah.

so make sure they're match two things

>>James: Some extra bit of mathsy things,

which will be on my channel because

which will be on my channel,

we're gonna do some proof by induction

because we're going to do some proof by induction! Hoorah!!

ah but a proof by induction my favorite

>>Steve: A bit of proof by induction. My favorite. >>James: Excellent!

excellent so in the full video we were

So, in the full video, we were talking about

talking about the score that each tower in 2048 can make maybe I'll do this

the score that each tile in 2048 can make.

quickly if you haven't seen this if I

Maybe I'll do this quickly, if you haven't seen this

joined look I've got all these props now

If I joined--look, I've got all these props now... [Steve chuckles]

I'm join to two tiles together they

If I join two 2 tiles together,

merge they create a four tiles which I

they merge--they create a 4 tile--

haven't got some where they like a for

which I haven't got somewhere nearby--

tile you get four points for that and we

a 4 tile--you get four points for that.

were talking about you know how many

And we were talking about, you know, how many points would you get

points would you get for something like a sixteen tile to make a sixteen tower

for something like a 16 tile?

you need a pair of eights to make up to

To make a 16 tile, you need a pair of 8's.

make an 82 pair for sore point for a new

To make an 8, you need a pair of 4's,

pair tubes right yeah

and for a 4, you need a pair of 2's--right. >>Steve: Yeah.

and so in the full video though we

>>James: And so, in the full video, though,

described as a big pyramid yes

we described it as a big pyramid. >>Steve: Yes.

so the sixteen will be at the top next

>>James: So the 16 would be at the top,

big pyramid all the twos then eight two

it makes a big pyramid, all the 2's--there are eight 2 tiles at the bottom.

tiles at the bottom and it all

And it all accumulates up,

accumulates up and all they score adds

and all their score adds up

up and makes the sixteen tile right and

and makes the 16 tile--right. [Steve chuckles]

the pyramid if I didn't in general the

And the pyramid, if I did it in general,

pyramid would have key levels clear

the pyramid would have K levels, >>Steve: K levels--yeah.

level so the apex would be 2 to the K

the apex would be 2 to the K [2^K}.

yeah that's my general tone clay levels

>>Steve: Yeah. >>James: That's my general tile.

and each leveled add up to 2 to the K to

K levels, and each level would add up to 2 to the K [2^K].

the K all right so they form you perform

>>Steve: 2 to the K--all right.

you know right at me I worked out is if

>>James: All right. So the formula--the formula, right-- that I worked out,

you want to make two to the K and you score all together okay

is--if you want to make 2 to the K [2^K], you score all together,

there's one lots of two to the K so this

(K-1) lots of 2^K.

pyramid you make each level scores to

So this pyramid you make--

decay and this K 2-1 levels the bottom

each level scores 2 to the K. [2^K] And there's K to the minus 1 [K-1] levels--

level wouldn't count you get those for

the bottom level wouldn't count,

free right now proof by induction

you get those for free. >>Steve: Right. >>James: Now, proof by induction--

because there's a nice is that formula

because there's a nice--

really true is it true in general so to

is that formula really true? Is it true in general?

do this in general if I have great a 2 to the K tile yeah and another 2 to the

So, to do this in general, if I have--great!--a 2 to the K tile [2^K], Steve: Yeah.

cave tile but they would then join

>>James: and another 2 to the K [2^K] tile,

emerge and measure K plus 1 T 2 k plus 1

they would then join and merge, and make-- >>Steve: 2 to the K plus 1! [2^K+1]

>>James: 2 to the K plus 1! [2^K+1] Exactly!

so this one like I said scores K minus 1 times 2k this one scores the same I'll

So, this one, like I said, scores (K-1) times 2^K. This one scores the same,

try and write in but oh this is really difficult to do we said no it no second

I'll try and draw that in-- this is really difficult to do! >>Steve: We said no second takes allowed!

takes allows no second takes of us is raw raw man alive do you know it is more

>>James: No second takes allowed! >>Steve: This is raw--raw maths! >>James: Man alive! Do you know what?

difficult to write this at a 37 degree annual than it is upside up right 2 to

It's more difficult to write this at a 37 degree angle than it is upside down!

the K then we would merge them and when

Right. 2 to the K. [2^K] Then we would merge them,

you merge them you make this one which

and when you merge them, you make this one.

means you put this one you get 2 to the

Which means for this one,

k plus 1 all right right right

you get 2 to the K plus one [2^K+1] points. >>Steve: Right, right, right.

so I'll get that as again oh yeah so

>>James: So I'm going to add again--

you've got these two that you've already

>>Steve: Oh, yeah! So you've got these two that you've already got

got yeah you've got the scores for those

>>James: Yep! >>Steve: You've got scores for those.

so that I'm going to get a school for

>>James: So that's how much-- >>Steve: AND you're going to get a score for this one.

this one then join them every times on that so k plus 1

>>James: --then join them, which adds on that. >>Steve: 2 to the K plus one. [2^K+1]

here's your formula so how much does

>>James: Here's your formula.

this score all together

So how much does this score all together?

well yes I want to be expect to get yeah

>>Steve: Yeah--so what do we expect to get?

so apparently well if I add all this

>>James: Yeah! So apparently--well, if I add all this together,

together why I would expect with my formula yeah

what I would expect with my formula-- >>Steve: Yeah?

my formula what was it was the K minus 1

>>James: My formula--what was it? It was the (K-1)2^K.

to decay but now I'm using K plus 1 so

But now I'm using K+1 there

substituting k plus 1 K we've never had it where I've ever ever had it before

>>Steve: Substitute in K+1for K-- >>James: Wherever I had it--wherever I had it before--yeah.

yeah so it's the formula I expect to get

So the formula I expect to get

it is is and if I'm going to write it in

>>Steve: is... >>James: is--and if I'm going to write it in--

yes well I expect to get is K times choose 2 k plus 1 right so these two

>>Steve: Yeah? >>James: what I expect to get is K times 2 to the K plus 1. [K2^K+1]

things are equal

>>Steve: Right. So if these two things are equal... >>James: Yeah....

proof by induction proof by induction

>>Steve: Proof by induction! >>James: Proof by induction! Inductive step--

inductive step and well let's cut a long

and, well, let's cut a long story short--

story short and you do they're the sons

you do the sums there-- >>Steve: Right--

they're just not try this

>>James: which is not too hard-- >>Steve: you multiply this--

you've got to lots of this so you can

>>James: You've got two lots of this

this Sonny goes up a step to k plus one

So you can--this suddenly goes up a step to K+1,

and and and add on another team to the k

and then add on another 2^K+1,

plus one and we'll get you came opps of

two to the k plus one and so it does

K lots of 2^K+1.

work the inductive set burst work the

So it does work-- the inductive step does work,

base case does work we can check it out

the base case does work,

with the force yeah boom great proof by

we can check it out with the 4's. Boom!

induction with my induction laser a

>>Steve: Great! >>James: Proof by induction! >>Steve: Proof by induction, ladies and gentlemen.

lovely job useless was another proof I'm

Lovely job. But there was another proof by induction.

yeah there's another book my section if

>>Steve: Yeah, there is another proof by induction, if you want to do it.

one day this is like what is the highest

This is, like, what is the highest score you can get

score you can get on a board oh here we

go

Oh, here we go! OK, so--

okay so I've gotten a two pops I'm going

>>James: Look--I made two props. [Steve laughs]

to use them but whether I made two props

I'm going to use them both! >>Steve: Might as well.

right so there's the same yeah so in the

>>James: I made two props! >>Steve: Right, they're-- >>James: They're both the same.

main video we talked about the highest achievable tile yeah in 2048 the aim of

Yeah, so, in the main video, we talked about the highest achievable tile in 2048. >>Steve: Yeah.

the game is to get 2048 but you can get further yeah then that and to get the

>>James: The aim of the game is to get 2048, but you can get further than that. >>Steve: Yeah.

height achievable tile and well

>>James: And to get the highest achievable tile--

basically if you want to get 2 to the K

well, basically, if you want to get 2 to the K [2^K]--

all these titles are power isn't to yeah

all these tiles are powers of two-- >>Steve: Yeah.

you want to get 2 to the K you want you need K tiles

>>James: So you want to get 2 to the K [2^K], you need K tiles--

okay free cells I'm sorry in a free

or K free cells--I'm sorry--you need K free cells-- >>Steve: Aaahh!

cells to do that just to make that point

>>James: to do that.

there no it hammer this home but let's

Just to make that point, then-- I won't hammer this home--

do that let's play a game

but let's do that.

this one tells me some peaceful go so I

Let's play a game! This one curls. >>Steve: Let me hold onto it. >>James: Make yourself useful.

knew I was here for a real an okay so

>>Steve: There you go. I knew I was here for a reason. [James chuckles]

first like I want to make a for tile I need two of these two times like that

>>James: OK. So, first of all, if I want to make a 4 tile, I need two of these 2 tiles, like that.

and so I need to free sales so that I can put these tubes in and then they

And so I need two free cells so I can put these 2's in,

would merge to make a for tile like that

and then they would merge, to make a 4 tile, like that.

so the next thing I want to do is to

So the next thing I want to do is

make an eight tile so to do that you're

to make an 8 tile.

out swipe yeah you do generate a thing

So to do that, I would swipe--you know, you'd generate a thing,

like that you swipe again you generate

like that--you'd swipe again,

another one but if you can see here to

you generate another one.

make 8 which is 2 to the power 3 I do

But you can see here, to make 8, which is 2 to the power 3,

I do need three free cells.

pre-sell yeah so then I would merge them

>>Steve: Yeah, OK.

that makes another four and get one

>>James: So then I would merge them--that makes another 4--

there you go and then I would merge

if I can get one--there you go!

these fours to get an eight okay

And then I would merge these 4's, to get an 8.

bubbly right so the next pile up is to to power four sixteen to make that well

There you go. Lovely! >>Steve: Right. >>James: So the next tile up is 2 to the power 4 [2^4], is 16.

now I need to start yet on the road I

To make that--well, now I need to start-- >>Steve: You need another 8.

need another eight which means let's cut

>>James: I need another 8. Which means,

this short which means I need to make

let's cut this short--

another eight I know I need three cells

which means if I need to make another 8, I know I need 3 cells to make another 8. >>Steve: Oh, yeah!

together eight so I can use these I've

>>James: So I can use these.

got three three here so I used those

I've got three free here,

three and eventually I've created my

so I use those three,

number eight but I had my eight they're

and eventually I've created my--another 8.

occupying the fourth just sitting there

But I had my 8 there, occupying the fourth--just sitting there,

taking up that cell so altogether I

taking up that cell,

didn't need to have four cells to

so all together, I did need to have four cells, >>Steve:Yeah.

finally join these aids together and get

>>James: to have finally joined these 8's together,

my sixteen tile now that's going to

and get my 16 tile.

happen as you keep going up yeah you're

Now, that's going to happen as you keep going up.

going to need the K cells to make 2 to

>>Steve: Yeah. >>James: You're going to need K cells to make 2 to the power K [2^K].

the power ok but so to be clear you needed force-free

>>Steve: But--so, to be clear,

cells to make this 16 so now if you

you needed four free cells to make a 16,

wanted to make another 16 you couldn't

so now, if you wanted to make another 16,

do it because you've only got three free

you couldn't do it,

because you've only got three free cells.

yeah I need four to make another sixteen

>>James: Exactly. Yeah. I need four to make another 16--I've got three left.

I've got three left okay exactly so in

>>Steve: OK. >>James: Exactly. So, in the full video,

the full video we've said if I'll just

we've said it, but I'll just say that point again--

say that point again in the full game

in the full game,

before by four grid is 16 cells so in

the 4x4 grid is 16 cells,

theory if everything goes your way

so in theory, if everything goes your way,

everything lands where you need it to be

and everything lands where you need it to be,

you could hypothetically get 2 to power

you could, hypothetically-- [Steve chuckles]

16 okay which is 65,000 something something like 60,000 and I was pretty

get 2 to the power 16 [2^16]-- >>Steve: OK. which is 65,000-something. >>Steve: Something like that. 65 and some change.

much an inductive argument again

>>James: And that's pretty much an inductive argument again. >>Steve: Yeah.

yeah they said the number 3 cells you

>>James: The number of free cells you need

need to do you need to be in the right sort of positions they can't just be in

do need to be in the right sort of position--

a row that doesn't work not the way the

they can't just be in a row--

game works when you swipe oh yeah of

that doesn't work--not the way the game works

course but say I think works perfectly well in you can prove it probably going

when you swipe. >>Steve: Oh, yeah, of course. Yeah, yeah. >>James: But, I think it works perfectly well in a square--

from like a 2x2 or 3x3 to for this clear

>>Steve: You could prove it, probably, going from a 2x2 to a 3x3 to a 4x4.

I think there's clearly an inductive

>>James: There's clearly--I think there's clearly an inductive argument

argument and a hand crossed the t's and dot the aisles here but I think that I

and I haven't crossed the t's and dotted the i's, here.

mean Fermat said about some of the thing

But I think there is. >>Steve: I mean, Fermat said a very similar thing

yeah let me have a perfectly marvelous

[Steve laughs] >>James: Yeah, we have a perfectly marvelous

demonstration of indicative arguments

demonstration of the inductive argument,

which video Steve Shortell painter and if you

which this video is too short to contain.

have been thanks for watching

And if you have been, thanks for watching!

Similar videos:

Subtraction 2 | Addition and subtraction | Arithmetic | Khan Academy
Khan Academy
708.6K views 887 likes 142 dislikes 19 Jan 201027% match

New Wikipedia sized proof explained with a puzzle
singingbanana
720.8K views 12K likes 234 dislikes 24 Feb 201427% match

ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12
Numberphile
8.3M views 102.9K likes 0 dislikes 09 Jan 201427% match

Maths Puzzle: The self descriptive number solution
singingbanana
175.1K views 4.8K likes 42 dislikes 09 Jan 201626% match

BOLD and BEAUTIFUL in PEACOCK Colors! ~ Fluid Acrylic Painting ~ Acrylic Pouring
MOLLY'S ARTISTRY
239.7K views 9.2K likes 210 dislikes 22 Nov 202026% match

Basic addition | Addition and subtraction | Arithmetic | Khan Academy
Khan Academy
3.4M views 4.7K likes 482 dislikes 10 Mar 200726% match

Graphing logarithmic functions | Exponential and logarithmic functions | Algebra II | Khan Academy
Khan Academy
965.1K views 2.4K likes 97 dislikes 08 Nov 201126% match

MY NEW CAR
Markiplier
2.4M views 143K likes 2.1K dislikes 23 Oct 201726% match

Probability with playing cards and Venn diagrams | Probability and Statistics | Khan Academy
Khan Academy
987.7K views 2.6K likes 85 dislikes 03 Aug 201126% match

Add 500 To Each Number In an Excel Cell - Duel 192
MrExcel.com
1.5K views 94 likes 0 dislikes 21 Aug 202026% match

3 Simple PVC Creative Caches ANYONE Can Make! (GCNW)
Behind the Cache
848 views 58 likes 0 dislikes 20 Aug 202026% match

MINECRAFT: How To build A Modern House / Best modern House 2013 - 2014 ( hd ) Tutorial mansion
Typhoon Cinema
19.1M views 109.3K likes 0 dislikes 09 Sep 201426% match

Solving a more complicated equation | Linear equations | Algebra I | Khan Academy
Khan Academy
1.9M views 2.7K likes 222 dislikes 15 Sep 201026% match

How to Solo over ANY CHORD Using the Pentatonic Scale - Steve Stine Guitar Lesson
Steve Stine Guitar Lessons
950.2K views 30.1K likes 301 dislikes 13 Jul 201826% match

Brawlhalla Diamond Gimp & Rec Tutorial With Real Examples
Ramkatt
1.4K views 75 likes 3 dislikes 16 Jun 202026% match

VLOG #1 - My new workshop!
The Practical Engineer
12K views 527 likes 15 dislikes 19 Oct 202025% match

Basic subtraction | Addition and subtraction | Arithmetic | Khan Academy
Khan Academy
1.2M views 1.1K likes 213 dislikes 10 Jul 200825% match

McDonald's® DOUBLE HOT 'N SPICY McCHICKEN® Review ✌️🌶️🐔 | Peep THIS Out! 🕵️‍♂️
Peep THIS Out!
5.7K views 277 likes 19 dislikes 17 Sep 202025% match

HOW TO MAKE THE BEST CUP OF COFFEE AT HOME Using YourSuper Mellow Yellow Superfood Powder
Dad Reviews with @LaneVids
644 views 16 likes 0 dislikes 15 Dec 201925% match

Crochet Little Girls Handbag | Bag-O-Day Crochet TUTORIAL #91 Subtitles Available 21 Languages
Bag-O-Day Crochet
1.1M views 7.2K likes 428 dislikes 05 Jul 201425% match

This is Swedish FIKA 🇸🇪 [S3 - Eps. 18]
Itchy Boots
329.6K views 29.7K likes 93 dislikes 23 Sep 202025% match

Basic multiplication | Multiplication and division | Arithmetic | Khan Academy
Khan Academy
2M views 5.6K likes 771 dislikes 28 Jul 200925% match

BUYING ALL FORTNITE CHRISTMAS UPDATE DLC | Fortnite Battle Royale
SXVXN
749.3K views 8.3K likes 246 dislikes 14 Dec 201725% match

Fiat Barchetta Timing belt | Timing belt cover removal | Check for worn cam belt | Part 10
PAULIECABS
1.1K views 35 likes 0 dislikes 27 Sep 202025% match

DIY Lace Up Top | Make Thrift Buy #40
Annika Victoria
677K views 20.5K likes 176 dislikes 09 Jul 201625% match

Settings Privacy About Contact Donate