2048 strategy and maths

okay I've been playing this game 2048 like everyone else in the world and so has my friend Jana crying hi so in fact we were talking backstage before a show yeah because that's what we do that's the kind of world we live in we closed a personal showbiz friends that's right ah and we started talking about this game so we were both acknowledged it's it and I said swiping away are you ignoring me for the same reason to ignore yeah and we said you oh no madam I use many different reasons to ignore me and but we started talking around this loads of massive balls so we held we make a video on the internet by storm exactly yeah absolutely 2048 it but we thought well yeah it lends itself well to a touchscreen graphical interface right so I've made this you've made a sort of low fibers yes so we can in lieu of you actually playing the game will show you how it works on this sort of cardboard cuts and abrasions yeah this is the Home version of 2048 and so we'll just quickly go through the books yet gone and so you've got four by four or grips you start by having two tiles which have the number two written on them so two two tiles and they're in random cells yeah and then you swipe up/down let alright so some I'm gonna play this so I'm going to swipe downwards like this so we go swipe okay like that so everything falls down to the extreme edge okay great and then a new tile appears a new tile appears so that's generating new tile then Iran themself okay now I'm going to swipe towards me go right why and these two tiles have joined together so these two twos will join together their scores will add together they'll create a new tile so two twos make you have to score four points for making the four so already the maths is very complex and you add ungenerous and you play another - all right so I'm going to swipe down again - why now so these didn't combine so those then combined setting values okay so if I wanted to open up now a new tile appears a new tab appears quite right so put that one in so I'm going to swipe to me again so swipe this way jump add a new tile I think to make it a bit different I'm going to add a for tile oh because it does turn how about ten percent of the time and they do generate a for tile for you just throw that into it okay so I'm gonna swipe up you look like oh yeah why so those two twos combine combine we've got another four here this is swiped off as well this is smart top okay now what's that what's the aim of the game and the aim of the game is to create the 2048 tile so one of the questions we get all the time is what's the best strategy for playing this game you've completed this game yeah I've completed this calc up to 2048 yes have you completed the game yes way great so we've antastic will all work I'm glad you got the 4096 the 4096 tile or less to some power because you get something you get to tears before taking them either you could be game well some people might give up when they reach the 2048 tile but not people like me so you've you've got you carry on playing at it I carried on playing so you do get the option you can carry on playing ok listen let me explain if you haven't played the game how it feels right you start playing the game it feels very good because you know you're like this is one of I've just made 128 now I'm going to make tuna busy you know you get to 512 now this is easy you get to 1024 you like okay right one step to go on close now right well yeah one step well you can't heart rate there yeah so it slowly dawns on you actually you've got it when you get to 1,000 got to play that whole game again to make another 1024 and you can combine those finally together yes and then because what happens is you play it it's fun and then it becomes addictive and then you realize it's hard and you just play it for the rest of your life you lose your job your wife leaves you you lose access to your kids but then eventually you get 2048 and you're it's like being set free like this wonderful moment where you can carry on with your life okay but now I am told that Jake is grime let's go 4098 so I've got to start playing again I don't feel too bad because I people have tweet to me their scores and I'm now aiming to get to be next times the 8000 title so you've had a screen row of someone yeah and whether that's true or not whether it was a fake hmm we'll get to later okay so my strategy is to get the largest tile the largest numbered tile into the bottom right corner get a corner and to wedge it in there so I'll care of my biggest tile there in the corner and then from there I want some numbers kind of decreasing outwards for oh they're so out of five 16's like this right and I kind of make an out sort of like a triangle like that lords on the fringes I want the smaller numbers things for of twos well I get that wedged in there though so if I can complete this row in this column and that doesn't get moved it gets wedged in there and everything gets concentrated and focused on making the biggest tile possible in that corner right so you wouldn't if there was something for missing you wouldn't swipe that way because you might end up with a - suddenly in there all of this over here so you keep that you everything you do - you can to keep that yeah this is interesting because I was looking at some a eye that someone a program guide called Matt Overland he built some AI to play this game right and he encoded various strategies into the game and this strategy is one of the ones that he encoded it he called it a mana mana chromis now he didn't know what he call it they call it monotonic monotonic I'm on a 2ct when it is the okay but it's a nasty but basically where because you don't have to I speak it mostly maths fluently I don't know so you've got numbers either increasing or decreasing in all directions give out never increasing and decreasing as you go along so it's always going to go from high self wrote I mean up and down in that so it's exactly the same as my structure is exactly same yeah exactly the thing but do you have any other strategies for playing this game not really apart from swiping and combining tie rules and things like that yeah so this is the thing if you're if you're programming a computer to do AI you need to tell them about those obvious stuff as well like try and combine times like this right obvious but you have to compare them so one of the rules for this AI is that you taught me you want to try and keep as many of the squares free right yeah which means combining tiles and the other one is try and get tiles of a similar value closer together which is basically the thing before combined so you know I want to set my swiped are for example that's good because I've get these aids together you know and so that's that's a move that I could make and that would be valued within this AI so these strategies are programmed in but the way this strategy works is it's something called minimax which I will do in another video and it but basically what it means is you assume that your opponent is going to make the worst move for you and you're going to make the best moves for you and then you look at different options for what you can do and score them and you score them in this case according to those different strategies so this is like two players yeah right they're playing against each other yeah exactly yeah you make a choice they make a choice yeah and it's like some sort of big tree diagram of choices yeah and each decision gets a score exactly but you can only go so deep because with a tree diagram it gets more and more complex until you computed anymore so I don't know how far ahead you thinking again only can think a few moves ahead I can start to plan and make a chain yeah yeah so it's the same with the AI you analyze the different options maybe three or four moves deep and then you work out the best option and it's based on those different idea but you you give the different weightings it's like how important is it to have numbers decreasing and increasing but not going up and down how important is it to get numbers together how important is it to keep the ball clear so it is the AI strategy the best well I don't think it is and the reason I don't think it is is because of this minimax thing you're you're basically saying I assume my opponent is going to make the worst move for me but your opponent in this game if you think about the moves you make you make a swipe move and then your opponent yeah ads in your client pewter right as I'm so randomly and but if you use minimax then you're going to work out the worst possible place for the time to go or you're going to assume that your opponent will put in the work place but that's not the case too random yeah so really you should be doing something where you work out the average of what your opponent is likely to do and then based on move on and I think that's the way humans play is that this point the way you play yeah well I try and sort of hope for the right style in the right cell yeah so I guess a AI doesn't cross its fingers yeah this you're gonna play the game ten times you might as well cut your fingers one time it's going to work okay and so so the strategy I think those end up being a bit different there's a brilliant outcome of this which is that someone looked at the AI that was built by this guy Matt oven and he worked out a version of the game he coded a version of the game where the computer puts the new tile in the worst possible place yes it's a horrible game the link is in the description you're welcome I don't know if this is one of those games where you tend to dream about it if you play you like Tetris you have tetris tricks I've started having dreams about you know Tullus wiping around in 2048 but this hard version of the game it's like all nightmare yeah yeah yeah so Steve very kindly simple that to me as a high schooler not very much I got to 128 which is pitiful because he's playing against you the game is playing the game stood it's a really really high ball but that's what this is interesting you this there are loads of other versions of the game yes a lot so my favorite version of 2048 is the Doctor Who fleshing have you played the Doctor Who fan I haven't played on TV but I've seen it on your phone the Doctor Who version where they wear each tile instead of a number is one of the doctors from Doctor Who so you start with your heart knows to Hartnell's make a trowel turn of course they don't residence in transoms make poetry so you try to get to and you're trying to get to Matt Smith because Matt Smith is the eleventh Doctor and this was perfectly accurate accurate up to about October 2013 so you want to get to Matt Smith the eleventh Doctor and that works because at 2048 the goal in 2048 is 2 to the power 11 there are 11 steps in the game and in the doctor home version there's 11 steps so the games are the same in some so then the get the same game you've got the tiles are exactly the same you just changed the labels right so the game plays exactly the same and that's called isomorphic there are isomorphic gamers the 2 to the power K tile is the cave doctor ok ok good I said this I've played the DOS version the game that's nice and walking as well so you're just going through different doge memes where you just combining the doctor took the dogs so you just combined the same you have to come by the same though those meme yeah to get to the legs so stairs me and every time you do it a word up here saying such number or wow how do you know how to know what other dude well that's how that's what I think it's not isomorphic because I know that to 240 equals rotate 16 so on but I don't know that to doges with their eyes moving equals a dough's with a pizza in the background but you know for what you do combine the same type to make the list yeah leaves you in a sequence so if you know the sequence it's isomorphic yeah that sounds isomorphic to me as opposed to the Fibonacci version of 2048 where you have to join two adjacent numbers in the Fibonacci sequence yeah which is not isomorphic because you're not combining two tiles of like type rice as a different game to play so the mechanics of the game fundamentally it's fundamentally and what about flat ledo's 2048 yeah that's nice a morphic what is the maximum score the best score that you can get so my best score so far has been 2048 that's the highest time I've got well there's there's that's the thing there are two games that you can play here right so some people are just played to get the 2048 time right but on the other hand you can play to get the best squawks there's a score as well going along when you play the game if you don't know the way it works is if you combine two two tiles together like this you combine them they would merge you then create a four time and you get four points for that so you get 4 points for creating a 4 tile if I join to four tiles together I would create an eight tile I get eight points for making the eight tile so you get the number of points on the tile that you just made yes and it's all a dinner together and it's Google running toes you got the running total so what's the best score you can get when you finish the game and you finish games when you get mm oil is it okay yeah so to explain the best score and this actually they'll formula for this but to explain that will keep it easy for now we're going to look just look at how to make the 16 so okay we're going to look at what's the maximum score if you go to 16 yeah all right so to create the 16 tower what do I need to create 16 type to it yeah it's like two eighths to make the 16 now for each of the eight tiles what you need to fours so you need two fours to make each of the eight tiles semm get those in and to make each floor tile what would I need you need two twos example exactly I need a pair of twos to make each 4 tile so I'm gonna end up with lots they got us neither for I needed and lots of twos running off the edge of the border all right okay how about that V so I need another pair to about eight Tuesday yeah so what you made is a pyramid so you start with so you've got eight two tiles on the bottom level yeah to make four four tiles to eight towers and it all accumulates to make the 16 tire at the apex of the pivot and you get you get points for each one of these levels yeah so for each level you actually get 16 points so for the top level here you combine two eights to make the 16 your 16 points yeah for this level here you combined to force to make eight two fours make eight again and this scores you 16 this level scores you 16 points and it's the same here all these twos will combine to make 4 or 4 4 and a 4 and these will add up to 16 points but you didn't make this bottom row yes so these are created for free right so you don't get any points for these yeah but you get points for these these in these yep so each level of the pivot is 16 points so three levels yes those three times 16 so it's 48 points to make the 16 tile altogether you also forty-eight points to make sixty so how do we generalize that because we want to get to 2048 exactly so 16 in Matt's world isn't just 16 as where we are this is where we are as 16 is actually 2 to the power 4 right and notice this pyramid has four levels oh yeah all adding up to 16 apart from the bottom one so because it's super powerful we've got four levels yes if you do it in general yeah and have 2 to the power K I don't be a general tire of this type yeah then he will make a pyramid with K levels the last one doesn't count but they each level will then add up to 2 to the power K so BK minus 1 lakhs of 2 to the power K but we want to know the score for getting 2048 which is 2 to the which is 2 to power 11 level so that would be a 2 is 11 exactly a pyramid with 11 rows it'd be 10 lots of 2011-1 is this see I knew there's a reason 10 lots of 2048 so to 20,000 480 is the score you would make for creating the 200 point so that's that's the minimum score that well not quite it's certainly it's the best score you can make if we were only generating 2 so that's not the case is it thousand ten percent of the time you don't get twos you get four tiles which means 10 percent of the time I get four time I can remove actually 20 percent of this row at the bottom Oh at least come for three here actually removes this from your score so if you want to get the 2048 tile it takes out about 400 points okay so the best score or good score a good game should be about 20,000 points when you say best cover when I've got 2048 I had a load of other tiles on the board so I had a 512 and yeah this this is where this where I disagree we have some people playing it I think the best score is your lowest winning score I think the most efficient yeah okay is your best game because I think the most efficient game would be ending up with just one 2048 tile so this is a screen grab of James's best score what what James thinks is best score which is his lowest score with a 2048 tile on the board yeah you can see you've hardly got anything else there yeah so under 20,000 the cool kids get only the 10,000 they only 20,000 Club yeah yeah now what is the rise the maximum score which is another question so we know you know you've gone beyond 2048 so you can go beyond you can get the next how the 4,000 tile okay you can get the 8,000 tile in theory we've looked at the the game which has the hardest game one yeah gives you the hardest tile the AI that puts the tile in the worst in the worst position right if you had a game that was really nice to you and just put a ton of the best players put them in the best places so it all worked out nicely I reckon the best tile you could get the highest achievable tile would be 2 to the 16 all right and then you would fill this board with 2 to the 15 to to the 14 to 2002 to power 12 2 to power 11 2 & 4 8 that to power 10 9 8 7 6 5 4 3 2 2 to the power one which does two which we would end with and we wouldn't be able to create any more tiles do we need another two time to merge eyes and there's nothing there's not like it emerges anymore that's about as much as you can do and to make this well we know what each tile scores oh yeah never so the total score for this board big number it's 1 million eight hundred and thirty-five thousand and twelve and I reckon then must be the maximum score is that way again two's all the time now we're scared now it's assuming that it's just - okay but I'm generating so if I generate 10% of those or fours at that then I'd have to knock some points off for that so I think the highest achievable if as much working your way would be this one here the 65536 we didn't say why this is the highest tile that you can get but there is some reasoning behind that keep an eye on Singh banana YouTube channel to see James going green explanation now you looked at the best the highest score on the leaderboards and the high score was two billion no bit over okay so how have they done yeah so which is a little bit higher than my warrior median and something because they hacked it oh yeah because the highest score on the leaderboards you may notice is a 2 to the power 31 minus 1 which is the highest achievable score you can make in a game programmed with 32 bits and with the worst scores of around about a hundred thousand and that is the score that I would predict you would get if you managed to get up to the two to the 13 which hires which is the 8,000 tire which is my a Oh then someone sent you an image yeah screengrab and we were wondering whether it was fake or not and we can verify whether it's fake alotta we have evidence whether it's fake or not because the score will should match what I predict it will be we know what each tile on the scoreboard or on this grid here will score yeah we can add all those scores together and it agrees with the score down on the screenshot so either the guy that seen in the screenshot isn't cheating and as yeah isn't cheating fantastic and I hate him yeah oh he's worked out all this mess as well he's done the same maths after stood at he's faked his score yeah okay let's see let's give'em better and because you I mean you've got to the top for that all right I'm so close I am so close I've had the 4000 tile with the 2000 tile yeah with $1,000 I think I've got 500 tile and all lined up and then I messed it up Fran I'm so close I'll get there so if you've been playing this game for a long time and you've completed it there is still things you can do like getting to 8,192 before James Graham gets there you can also play the hard version of the game where he puts the tile in the worst possible place and you can play the game that this was inspired by with this game called threes these available on iPhone and Android and and you can also play flat Beados 2048 rubbish anyway thanks for watching subscribe I will kill you did bother you either so I've been playing the game 2047 liked it safe okay let's

Channel: Steve Mould

Views: 595,955

Rating: 4.8755507 out of 5

Keywords: 2048, strategy, maths, james grame, singingbanana, AI

Id: OO4tA5i7X9g

Length: 24min 29sec (1469 seconds)

Published: Thu Apr 10 2014

Reddit Comments

[u/dnspies]

I thought new tiles could only spawn in the same row or column as another tile moved out of. (ie

2  ->  ->  -|
-  -   -   -
-  -   -   -
-  -   -   -

can only spawn a tile in one of

X   X   X   -
X  -   -   -
X  -   -   -
X  -   -   -