I like how you sneakily add the bit in about sigma notation. Sigma is one of those scary math symbols that non-mathematicians tend to assume is way more complex than it actually is.
You're totally right; when I taught it in high school, my students would get intimidated by it pretty much instantly because HOLY MACKREL A GIANT GREEK LETTER (not gonna lie; this might also have been my reaction when I first learned it :) ) I hope this helps alieviate that for people. :)
If the right tiles appear in the right places, then well, zero. Assume, player only moves left and right and that the tiles that appear are always different from what's to the left or right in the row where it appears. We don't see this happen often in practice, since 90% of the new tiles are 2s and it's likely the adjacent tile in that row is also a 2. But given the rules applied in the video, a zero score is possible. More complicated is: What is the lowest possible score if the player always moves in the correct direction to maximize scoring chances, but is just really unlucky? Is it guaranteed that at least one tile combining vertical swipe will always come available?
He doesn't have a ton of subs because he's talking about math and not blowing stuff up while making exaggerated faces. People don't tend to want to learn when on RUclips. They typically try to be slap-stick entertained. I appreciated this tidbit of information and learned what a sigma is and how its useful. Am I ever going to use sigma in my life? Probably not, but it is still something new I learned and I'm sure I'll be seen as a nerd some day for being able to name a Greek symbol.
For those that cannot wait, the answer is 3,932,100. :) Edit: Yes, you need a just-in-time-4 for all the tiles. Explanation below. (tl;dr: pause the video at 3:32 to see the answer) I didn't do a great job of explaining why you need a just-in-time-4 for all of the tiles less than 2^17 on the final board; the animation happens quickly, and I didn't verbalize much. Let me try to remedy that. Consider the 2^16 tile. Previously, you'd been able to make a 2^16-tile with a snake that had two 2-tiles at the end, and was sixteen tiles long. But, when that collapsed into a 2^17-tile, there were only fifteen tiles left! This means that you don't have room for those two 2-tiles when you go to make another 2^16 tile; you need two 4-tiles. You can pause the video at 3:32 to see the board setup. Thing is, when you complete the 2^16 tile, you are now down TWO spaces to make your next snake, meaning you can only make a 2^15 tile -- but it'll still end in two 4-tiles. This pattern continues all the way down to the 8 tile. Hope that helps!
9gag.com/gag/a9MMVyo I did this. I'm guessing the discrepancy is due to the number of times a 4 spawned instead of me creating one out of two 2s. This amounts to 16130 4s, which either means that my hypothesis is wrong, or I should be glad that I still have fingers after all that swiping.
This is the type of content in which I would gladly watch the commercials in its entirety to make sure you get compensated properly. Thanks for making these videos! 🙏 👍
I agree, this is a great question. :) While the spawn value probability part is probably straightforward to figure out, the spawn location probability part seems a LOT harder!
The Taylor Series It is probably ~1-10%. You calculated by saying that all tiles show up when we need them. Because there is a higher chance of 2 to show up, (90%:10%), and 4 is more used here, it will most likely be less than 5%. Maybe less than one. I’m only grade 6 though, so I can’t do it ;-;
@@317pboqyc2c6oemvkkcwcncckykq nice thinking, but lets consider only the chances of spawning only 2s until he gets a single 2ˆ16 tile. To get 1 2ˆ16 tile you need 2ˆ15 times the tile 2, and the chances of a single 2 tile spawning is 9/10. But since we need it to happen 2ˆ15 times, you have to multiply it by itself 2ˆ15 times: (9/10)ˆ(2ˆ15) and that is so small that my calculator couldn't calculate. But it`s pretty much zero. Hope I helped :D
I remember a Numberphile video from a few years ago, where various professors and mathematicians were reacting to 2048. Their reactions were mostly along the lines of "It's a silly game, not really my thing, maybe my wife/daughter would like it". Really annoyed me at the time that they seemed to _only_ look at it in terms of "do you like this game", and didn't actually look at the game's _structure._ I mean, Minesweeper has been proven Turing-complete, FFS.
@@TheTaylorSeries If I'm understanding the proof correctly, any given Turing machine program can be converted to (and from) a variant of Minesweeper's rules played on an infinite grid with a specific initial configuration of mines/symbols, such that the given Turing machine will eventually halt _if and only if_ the corresponding pattern of "mines" (or symbols) cannot be extended infinitely under the given variant rules. (Conversely, the source Turing Machine will _never_ halt if the resulting mine pattern _can_ be extended infinitely.) It's an extension of an earlier work which showed that Minesweeper was NP-complete, by using mine configurations to simulate digital logic gates. web.mat.bham.ac.uk/R.W.Kaye/minesw/infmsw.pdf
That whole counting->generalizing segment from 4:00 to 6:04 is something I wish I watched before taking my probability class, because that's exactly the kind of thinking it requires. You explain it so well!
Vid quality is too high for only 11k subs. I don't often sub to channels that make me use my brain, but you deserve this. Lets get this handlebar mustache to 100k!
That's a good question! I probably could have been clearer about that. Yeah, you'd need it for every tile. To see it, consider the 2^17 we made that required 16 tiles to accomodate the 'snake' needed to make it. If you look closely, when I go to make the 2^16 tile, you can see it needed another 'just in time 4', because it had one less free tile on the board (it'd been taken up by the 2^17 tile). It turns out that's a pattern, and every time you try to pack the next highest tile on the board, you need that just-in-time-4. :)
@@TheTaylorSeries Thank you for explaining that. Now I see that you need the just-in-time-4 only the last time you pack the 2^n tile, but you need it for each one.
@@therealEmpyre Every other tile also needs to be a 2, and it takes 2^n-1 tile to make a 2^n tile (then take off 2 to account for the 4 required). We already start with 2 2 tiles so we need 2^2+2^3+2^4+2^5+2^6+2^7+2^8+2^9+2^10+2^11+2^12+2^13+2^14+2^15+2^16-32 2 tiles, which comes to 131036, and 15 4 tiles. Using the percentages shown of getting each tile the odds of getting those tiles in the required order are 0.9^131036*0.1^15. shame no calculator I found online could give me an actual number.
@@darkraidisciple8717 It is even more complicated than that. Not only do you need the right number at the right time, but also at the right place. Usually, there are more than one spot that would work, and there is not a constant number of empty spots. Since I am not a mathematician, I have no idea how to formulize those odds.
I really enjoyed watching this video, thanks! To take it one step further: the score for a 16-tile equals 2xthe 8-tile score + 16 (when you only get 2's). Taking that in account, you can use this formula to calculate the maximum score of any 2048 grid: (n-1)2^(n+2)-4(n-1) where n equals the number of squares in the grid. A 4x4 2048 game has 16 tiles: 15x2¹⁸-4x15=3932100.
one of our professors had us try to come up with proofs about the win conditions for the take away game nim. There are alot of variations of the game, for example you could only pick up one or two or three, or any number of any size pile or you'd take any number of sticks from just one pile. Win condition would be whoever takes last stick wins or loses. I think this would be a nice game to talk about.
Haven't you heard of it? Check out Nim with 2 piles where you can take any number from either pile or the same from both. It leads to a surprising optimal strategy involving the "spectrums" of phi and its square. The spectrum of a number, k, being the series given by floor(k*n) with n being the positive integers.
@@Xnoob545 I found a website that as you go on, the numbers get higher and higher so like it starts at 2 Thebesthen 4 and 8 and goes higher. I've gotten the 8096 tile before with it. Basically cheating.
I just wanted to add this comment because when you read the highest possible score it reminded me of my 5th grade teacher harping on “and” in numerical values. According to my teacher “and” is reserved for the decimal place. I appreciate someone using it for its general use in public instead of the technical purpose. Loved your video and thank you for a breakdown of 2048! 🙏
This is pretty much how I play games. I learn about the mechanics and things like damage/defense calculations, then I start solving for theoretical maximums and minimums. I can't tell you what a ball it is to go down this rabbit hole in a games like Warframe or Pokemon. The wikis have most of the formulas, so I luckily don't have to derive those equations usually lol.
Too bad Game Theory rarely does those kinds of videos anymore, and the physics ones (my favorite ones) are usually done by Austin now instead of Mat Pat. Austin’s voice is obnoxiously annoying
I recently did that with AoE 2 and not that long ago with Mario Kart 8. I even have a Mario Kart spreadsheet with kart stats. My friends thought it was way overboard. lol
When I was a kid I loved doing this with Paper Mario! I remember having lots of sheets of paper trying to calculate everything (and probably failing with my terrible kindergarten math skills). Nowadays I don't see this level of nice counting problems. Closest I get is numerical approximation software and programming. Looks like its time to visit AoPS for my fill!
I love this question! It's very hard to answer tho. There's two random elements to a tile spawning: the value and the location. If we only consider the value probabilities, we can likely figure this out in a pretty straightforward fashion. The location one, though, seems a LOT harder, and I have no idea how to approach it. :)
I had a version of the game that had an undo button. I used that to reach this score. You could only ever undo just the previous step, but the next tile would appear randomly every time.
This channel is so unnoticed, it kinda reminds me of early vsause videos before they got all their views. I wouldn't be surprised if this guy's channel really boomed.
This is actually slightly low! In the early stages of the game, we don't need the "just in time 4" since the board isn't filled up yet, so we can afford to have all 2s spawn. I'm not gonna do the math, but I'm guessing the answer you gave is low by around a hundred or two.
I used to play this game in my way to school, and sometimes reached some incredible scores. This video made me go back to those days, it was joyful and interesting! keep the good work
There was a version I used to play where you could undo your last move and redo it and repeat. Some of us actually reached that score (It only took weeks of playing and not messing up too bad)
"Σ" sounds like "S" you're right!! (I'm Greek.) "Taylor Series" = "Ταιλορ Σερεις" ("ρ" is "row" & sounds like "r") and ₁₇ ∑ 2ⁿ(n-1)-4 could = S 2ⁿ(n-1)-4 ⁿ⁼³
Woah. Great job! A lot of math explanations assume prior knowledge or just give facts instead of working through them. This video (and your general style) require very little math background, but they still encourage math reasoning-which is exactly the thing that makes math more interesting and less scary. I hope this channel continues to grow-I’ll keep watching!
Haha, I like the cheeky comment that flashed on the screen at 5:11, and I TOTALLY AGREE!! That's a frustration of mine when textbooks could have just added a few more sentences - or sometimes just one more phrase! - and it would make the explanation so much clearer. Why?? If it's just that textbooks are just trying to minimize the printing costs by condensing explanations, please don't! I'LL PAY FOR THE EXTRA PAGES!! This was a good video! I'm glad I watched it! I love analysis of theoretical math scenarios like this!
I knew exactly how to do all of this (I love math :D) but listening to the guy explaining it is so relaxing and it made me want to watch the whole video. I love the production!
I have just found this channel and after reading the comments I have figured that the channel just bloomed coz most of the comments are just a few days old. Keep doing the good job buddy hoping for some great videos from u in the future
I hope you get a lot of subs really quick you deserve it your work is fantastic keep it up! Btw you were correct the Σ (sigma) it's just a capital letter for the Greek letter s
Found this video, never before seen you, and I love it! You took something that would have taken me a few months to solve, and gotten it simplified where it would take me a few minutes to solve. You are better than most other math RUclipsrs, because you make it something that only a Harvard student would understand, turned into something that a grade-school student would understand. Others would just give how they solved it, and give the solution, and that would be it. Not you, though. Thank you for that.
4:00 The tiles are... 2 to the 3rd: 8 2 to the 4th: 16 2 to the 5th: 32 2 to the 6th: 64 2 to the 7th: 128 2 to the 8th: 256 2 to the 9th: 512 2 to the 10th: 1024 2 to the 11th: 2048 2 to the 12th: 4096 2 to the 13th: 8192 2 to the 14th: 16384 2 to the 15th: 32768 2 to the 16th: 65536 2 to the 17th: 131072 More tiles: These are designed for 5x5 and bigger. 2 to the 18th: 262144 2 to the 19th: 524288 2 to the 20th: 1024576
There was a version that allowed for infinite undos. After solid 2 weeks of playing I finally managed to get the biggest combination on screen. I would have probably played twice as long if I was aiming for the maximum score as well.
I'm happy that I found your channel. Good production value, great and thorough explanations, awesome voice. Turns out that RUclips recommendations aren't complete rubbish once in a blue moon :)
BTW, it may be a good example to show some series sums can be computed, first with the geometric series and then using a dummy variable and the derivative (though I don't think you've dealt with derivatives yet). The shorthand answer to this sum comes out as rather compact - > 15*(2^18 - 4)
Calculus is too advanced for most audience... also, you don’t need calculus to reduce it... you only need to times each term by 2 and cross subtract to reduce it into a geometric series...
@@TheTaylorSeries Totally understandable, I meant this as a possible suggestion in case you ever wanted to make videos about the subject of summation of series.
I remember years ago doing practice mode for months redoing 4s to never appear until I needed them for the maximum tile values. I actually managed to get the maximum score and felt weirdly proud of it.
Excel is not required. The sum of powers of 2 (a geometric progression) is just 2^(n+1) - 1. Geometric progression formula is (1 - a^n)/(1 - a). Taking the derivative w.r.t. 'a' can yield a formula for the sum of n*(a^n). Anyhow answer can be arrived from (17*2^19 - 18*2^18) - (2*2^4 - 3*2^3) - 2^18 + 2^3 - (18 - 3)*4 = (17*2 - 19)*(2^18) - (2*2 - 4)*(2^3) - 60 = 15*(2^18) - 60 = 3932100.
I didn't expect to see a measly ~4.6k subs given how clean and well-done your videos are. You could give folks like Mathologer a run for their money. Good stuff, man! Subbed, and can't wait to see what you do next!
Taylor Series >>> T series
Ahh a 9 year-old I see
T series+aylor
Aylor=infinity
*EXPOSED*
That’s literally exactly what I thought when I looked at the channel name lol
T-gay
I like how you sneakily add the bit in about sigma notation. Sigma is one of those scary math symbols that non-mathematicians tend to assume is way more complex than it actually is.
You're totally right; when I taught it in high school, my students would get intimidated by it pretty much instantly because HOLY MACKREL A GIANT GREEK LETTER (not gonna lie; this might also have been my reaction when I first learned it :) )
I hope this helps alieviate that for people. :)
Just like your channel icon :P
@@Dorumin Just so. :)
I wish it was called "big addition".
@@alan2here Sum day, it may be.
My 2048 question would be “what is the minimum possible score that you can fail at?”
If the right tiles appear in the right places, then well, zero. Assume, player only moves left and right and that the tiles that appear are always different from what's to the left or right in the row where it appears. We don't see this happen often in practice, since 90% of the new tiles are 2s and it's likely the adjacent tile in that row is also a 2. But given the rules applied in the video, a zero score is possible.
More complicated is: What is the lowest possible score if the player always moves in the correct direction to maximize scoring chances, but is just really unlucky? Is it guaranteed that at least one tile combining vertical swipe will always come available?
Flyin' high
probably the score that I get
2424
4242
2424
4242
I got 132 as my lowest
my lowest minimum is 32
Teaching summation through the game 2048! Genius
This is really professional for 11k subs, how don’t you have more
Still pretty new by RUclips standards. :) Thank you for the kind words!
He doesn't have a ton of subs because he's talking about math and not blowing stuff up while making exaggerated faces. People don't tend to want to learn when on RUclips. They typically try to be slap-stick entertained. I appreciated this tidbit of information and learned what a sigma is and how its useful. Am I ever going to use sigma in my life? Probably not, but it is still something new I learned and I'm sure I'll be seen as a nerd some day for being able to name a Greek symbol.
Because it's math related.
@@dannygjk true dat
For those that cannot wait, the answer is 3,932,100. :)
Edit: Yes, you need a just-in-time-4 for all the tiles. Explanation below. (tl;dr: pause the video at 3:32 to see the answer)
I didn't do a great job of explaining why you need a just-in-time-4 for all of the tiles less than 2^17 on the final board; the animation happens quickly, and I didn't verbalize much. Let me try to remedy that. Consider the 2^16 tile. Previously, you'd been able to make a 2^16-tile with a snake that had two 2-tiles at the end, and was sixteen tiles long. But, when that collapsed into a 2^17-tile, there were only fifteen tiles left! This means that you don't have room for those two 2-tiles when you go to make another 2^16 tile; you need two 4-tiles. You can pause the video at 3:32 to see the board setup. Thing is, when you complete the 2^16 tile, you are now down TWO spaces to make your next snake, meaning you can only make a 2^15 tile -- but it'll still end in two 4-tiles. This pattern continues all the way down to the 8 tile. Hope that helps!
nice
I actually reached that before, playing the option of undoing 1 move back. Took me a freakin month
Thank you
cool
9gag.com/gag/a9MMVyo
I did this. I'm guessing the discrepancy is due to the number of times a 4 spawned instead of me creating one out of two 2s.
This amounts to 16130 4s, which either means that my hypothesis is wrong, or I should be glad that I still have fingers after all that swiping.
This is the type of content in which I would gladly watch the commercials in its entirety to make sure you get compensated properly. Thanks for making these videos! 🙏 👍
I can't believe RUclips recommended me a video I am actually interested in watching it!
Can't wait for your next upload
i found this channel recently and i love it! please keep making videos, they are gold :)
I will, promise. :)
I learned more about math in this one video than my entire time in middle school
What kind of school did you go to...
sounds like you went to a REALLY shitty school
You were either the most distracted kid in the whole school, or the school actually didn't teach shit. If that's the case, that's sad
I found this awesome channel starting with the tetration video.
Love the content!
Same
Same, that vid blew my mind
Ditto! Delightful, simply delightful!
Same
I'm pretty sure a lot of people saw that tetration video before any of his other videos
These are some cleannnnn animations. BIG Upvote
Now calculate probability of this happening 😍😍😍
I agree, this is a great question. :) While the spawn value probability part is probably straightforward to figure out, the spawn location probability part seems a LOT harder!
The Taylor Series It is probably ~1-10%. You calculated by saying that all tiles show up when we need them. Because there is a higher chance of 2 to show up, (90%:10%), and 4 is more used here, it will most likely be less than 5%. Maybe less than one.
I’m only grade 6 though, so I can’t do it ;-;
@@317pboqyc2c6oemvkkcwcncckykq nice thinking, but lets consider only the chances of spawning only 2s until he gets a single 2ˆ16 tile.
To get 1 2ˆ16 tile you need 2ˆ15 times the tile 2, and the chances of a single 2 tile spawning is 9/10. But since we need it to happen 2ˆ15 times, you have to multiply it by itself 2ˆ15 times: (9/10)ˆ(2ˆ15) and that is so small that my calculator couldn't calculate. But it`s pretty much zero.
Hope I helped :D
Close to 0 without undo ngk its probaby at lwast 1/10^40 ish
I remember a Numberphile video from a few years ago, where various professors and mathematicians were reacting to 2048. Their reactions were mostly along the lines of "It's a silly game, not really my thing, maybe my wife/daughter would like it". Really annoyed me at the time that they seemed to _only_ look at it in terms of "do you like this game", and didn't actually look at the game's _structure._ I mean, Minesweeper has been proven Turing-complete, FFS.
Wait, how is minesweeper Turing complete?
@@TheTaylorSeries If I'm understanding the proof correctly, any given Turing machine program can be converted to (and from) a variant of Minesweeper's rules played on an infinite grid with a specific initial configuration of mines/symbols, such that the given Turing machine will eventually halt _if and only if_ the corresponding pattern of "mines" (or symbols) cannot be extended infinitely under the given variant rules. (Conversely, the source Turing Machine will _never_ halt if the resulting mine pattern _can_ be extended infinitely.) It's an extension of an earlier work which showed that Minesweeper was NP-complete, by using mine configurations to simulate digital logic gates.
web.mat.bham.ac.uk/R.W.Kaye/minesw/infmsw.pdf
@@HaloInverse That is AWESOME. Thank you; I now have tonight's reading. Thank you!
"anyone with an intelligence greater than that of pond slime will immediately see" I'm so dead
Dont forget other math famous phrases "it's trivial that..." and "the reason for this is left as an exercise for the reader"
@@FadkinsDiet Proof by complete intimidation ...
This is so underrated
“Oh wait... that game was in in 2014...?” **slowly closes RUclips app and uninstalls 2048** “I was totally aware of that”
Hahaha. :)
I still play the game in church because I can do that while paying attention. Same with drawing. Not the same while texting though.
Ha
you put 2 ins
Why did that cause u to uninstall it?
That whole counting->generalizing segment from 4:00 to 6:04 is something I wish I watched before taking my probability class, because that's exactly the kind of thinking it requires. You explain it so well!
you trying to pronounce doge was enough for me to subscribe
Wait a minute, you have only 10K subs......???
You look like a million subs RUclipsr !!!!!
You should be deserving like a million subs !
It is pronounced DOGE!
['doudj]
In spanish would be dohe.
OH, YES! I LOVE TAUTOLOGIES!
@@Noam-Bahar Yo ur so dumb it pronounced "Doge" not "Doge" XDXDDXDXDXD 1v1 cod u nub
Lol
"anyone with an intelligence greater than that of a pond slime will immediately see"
LMAOOOO IM WHEEZING
That was so worth slowing down to 0.25 speed and pausing !!!!!
we found vsauce v2
Very high quality videos, I don't get how you don't have more subscribers. Keep it up!
Vid quality is too high for only 11k subs.
I don't often sub to channels that make me use my brain, but you deserve this.
Lets get this handlebar mustache to 100k!
Just found this channel. The content is amazing, and I'm glad to be one of your first 1000 subscribers!
I certainly didn't expect such high quality content from a channel with so few subs.
Subbed right away! Great stuff!
Do you need that -4 in every step (for each value of n)? It looks to me like you would need that just-in-time-4 only once, or did I miss something?
That's a good question! I probably could have been clearer about that. Yeah, you'd need it for every tile. To see it, consider the 2^17 we made that required 16 tiles to accomodate the 'snake' needed to make it. If you look closely, when I go to make the 2^16 tile, you can see it needed another 'just in time 4', because it had one less free tile on the board (it'd been taken up by the 2^17 tile). It turns out that's a pattern, and every time you try to pack the next highest tile on the board, you need that just-in-time-4. :)
@@TheTaylorSeries Thank you for explaining that. Now I see that you need the just-in-time-4 only the last time you pack the 2^n tile, but you need it for each one.
@@therealEmpyre Every other tile also needs to be a 2, and it takes 2^n-1 tile to make a 2^n tile (then take off 2 to account for the 4 required). We already start with 2 2 tiles so we need 2^2+2^3+2^4+2^5+2^6+2^7+2^8+2^9+2^10+2^11+2^12+2^13+2^14+2^15+2^16-32 2 tiles, which comes to 131036, and 15 4 tiles. Using the percentages shown of getting each tile the odds of getting those tiles in the required order are 0.9^131036*0.1^15. shame no calculator I found online could give me an actual number.
@@darkraidisciple8717 It is even more complicated than that. Not only do you need the right number at the right time, but also at the right place. Usually, there are more than one spot that would work, and there is not a constant number of empty spots. Since I am not a mathematician, I have no idea how to formulize those odds.
@@TheTaylorSeries ahhhhhh, yea I was thinking this and that explained it
dude thank you for showing me how amazing math is and what i missed out back in school :D
I really enjoyed watching this video, thanks! To take it one step further: the score for a 16-tile equals 2xthe 8-tile score + 16 (when you only get 2's). Taking that in account, you can use this formula to calculate the maximum score of any 2048 grid: (n-1)2^(n+2)-4(n-1) where n equals the number of squares in the grid. A 4x4 2048 game has 16 tiles: 15x2¹⁸-4x15=3932100.
you just made math great for me. Thanks!
Finally, a great educational video that doesn't lead up to an ad.
It's nice knowing RUclips played for you! Great vids, keep 'em up, and I'm still waiting for the inverse of Tetration!
Someday ... :)
I had thought of this question just a few days ago and was going to try and solve it myself before I saw this video. Great job, very well done.
one of our professors had us try to come up with proofs about the win conditions for the take away game nim. There are alot of variations of the game, for example you could only pick up one or two or three, or any number of any size pile or you'd take any number of sticks from just one pile. Win condition would be whoever takes last stick wins or loses. I think this would be a nice game to talk about.
Huh! That's kinda cool. I just wikipedia'd Nim. Thanks for bringing that to my attention. :)
Haven't you heard of it? Check out Nim with 2 piles where you can take any number from either pile or the same from both. It leads to a surprising optimal strategy involving the "spectrums" of phi and its square. The spectrum of a number, k, being the series given by floor(k*n) with n being the positive integers.
You seem so much like my brother. Sees random thing, instantly starts math.
You have a cool brother!
And now we can put a filter on all scores higher than this and declare them hacked.
Yup
I got a high score of 16,512,956. And another time, I got an even higher score. ^^
@@sunlakestar2549 what
@@sunlakestar2549 IVE BEEN WAITING FOR 4 MONTHS PLZ ANSWER HOW
@@Xnoob545 I found a website that as you go on, the numbers get higher and higher so like it starts at 2 Thebesthen 4 and 8 and goes higher. I've gotten the 8096 tile before with it. Basically cheating.
I just wanted to add this comment because when you read the highest possible score it reminded me of my 5th grade teacher harping on “and” in numerical values. According to my teacher “and” is reserved for the decimal place. I appreciate someone using it for its general use in public instead of the technical purpose. Loved your video and thank you for a breakdown of 2048! 🙏
Also now I’m conditioned to not use “and”when verbally discussing a numerical value unless it contains a decimal.
Subbed before 1M
i seriously thought you had at least 4M subs
Wow, very better than the average content on youtube!
I got a 16384 once, but I got so lucky with my tiles that game
and i did inspext element
Anyone who goes for over a 4k tile knows the game is REALLY RNG based lol
I’ve got the 4096 tile
My best is 65k
I got the 16384 tile with a score of over 240k
Well done, sir. You are one step closer to finding a non-critical line zero on the zeta function ;)
Oh my. :) Wouldn't that just be hilarious? I can see the headline now: "2048 helps solve Riemann Hypothesis" :)
This is pretty much how I play games. I learn about the mechanics and things like damage/defense calculations, then I start solving for theoretical maximums and minimums. I can't tell you what a ball it is to go down this rabbit hole in a games like Warframe or Pokemon. The wikis have most of the formulas, so I luckily don't have to derive those equations usually lol.
Truth. I played Factorio for a while; that one is a very deep rabbit hole.
Too bad Game Theory rarely does those kinds of videos anymore, and the physics ones (my favorite ones) are usually done by Austin now instead of Mat Pat. Austin’s voice is obnoxiously annoying
I recently did that with AoE 2 and not that long ago with Mario Kart 8. I even have a Mario Kart spreadsheet with kart stats. My friends thought it was way overboard. lol
When I was a kid I loved doing this with Paper Mario! I remember having lots of sheets of paper trying to calculate everything (and probably failing with my terrible kindergarten math skills). Nowadays I don't see this level of nice counting problems. Closest I get is numerical approximation software and programming. Looks like its time to visit AoPS for my fill!
r/iamverysmart
Haha. Loved that bit about the sigma. In Greek, sigma is the "s" sound, and that specific symbol is a capital S.
What would be the probability of getting the maximum possible score using the snake strategy?
I love this question! It's very hard to answer tho. There's two random elements to a tile spawning: the value and the location. If we only consider the value probabilities, we can likely figure this out in a pretty straightforward fashion. The location one, though, seems a LOT harder, and I have no idea how to approach it. :)
@@TheTaylorSeries 64k
I legit just discovered this just as I'm learning the sigma stuff, cool video
Just-in-time-2048 video. :)
easy to digest math and cool animations, darn doink it you just sniped yourself a subscriber
I’m glad I found this channel I’m subing
Btw Taylor RUclips loves you now xD
this video is simply awesome. it's interesting and educational without being condecending or skipping important stuff
You kind of look like thanos
No, Thanos looks kind of like *me*. :)
1st like
No
You've heard of Thanos, now you've seen Epistemos. Or Logos, I never know.
I had a version of the game that had an undo button. I used that to reach this score. You could only ever undo just the previous step, but the next tile would appear randomly every time.
Ooooh, that's useful. You could actually get this score with that game.
6:06
Mental buffering
*the entire video*
Mental buffering
30 years from now, all possible questions could get answered
If we are lucky, there will always be questions that require exploration and discovery. :) To my mind, anyway.
I am from Greece and yes Σ = S THX FOR DA HEART
This channel is so unnoticed, it kinda reminds me of early vsause videos before they got all their views. I wouldn't be surprised if this guy's channel really boomed.
This is actually slightly low! In the early stages of the game, we don't need the "just in time 4" since the board isn't filled up yet, so we can afford to have all 2s spawn. I'm not gonna do the math, but I'm guessing the answer you gave is low by around a hundred or two.
i didnt know a game as simple as 2048 could be explained in such a technical and mathematical manner
Doge: Dog + e as in best, not e as in equal. 😂👍
I used to play this game in my way to school, and sometimes reached some incredible scores.
This video made me go back to those days, it was joyful and interesting! keep the good work
Right now I am at a 65k i think I have been playing this on and off for about a year or two 😅
Steven Nikolov I’ve been playing for about 2 months and I’ve got 60k 2wice
Not even 2 minutes into the video and I'm subscribed. This is great.
You're so close the the camera. AHHH! Nice vid though.
Yeah, my wide angle lens earns its keep in my studio. :)
Since RUclips is now making this channel to everyone, I’ll just be leaving my mark here before this amazing channel gets its deserved recognition!
Very well made video
old dad trying to be cool with the memes, i luv this guy
Only the dankest my friend, only the dankest. :)
@@TheTaylorSeries hahaha
6:15
thank me later
There was a version I used to play where you could undo your last move and redo it and repeat. Some of us actually reached that score (It only took weeks of playing and not messing up too bad)
Who is playing 2048 in 2048?
I've already planned my tribute video, and it will be released on the 2048 SpaceRUclips platform. Because we'll be in SPACE.
6:47 As a Cypriot (here in Cyprus we speak Greek), I can confirm that what you said about the greek letter Σ is correct.
"Σ" sounds like "S" you're right!! (I'm Greek.) "Taylor Series" = "Ταιλορ Σερεις"
("ρ" is "row" & sounds like "r") and
₁₇
∑ 2ⁿ(n-1)-4 could = S 2ⁿ(n-1)-4
ⁿ⁼³
This video was so thorough; it was just amazing. Thank you for making such a great video!
The graphics in this video deserve more recognition! VERY cool!
Woah. Great job! A lot of math explanations assume prior knowledge or just give facts instead of working through them. This video (and your general style) require very little math background, but they still encourage math reasoning-which is exactly the thing that makes math more interesting and less scary. I hope this channel continues to grow-I’ll
keep watching!
Haha, I like the cheeky comment that flashed on the screen at 5:11, and I TOTALLY AGREE!! That's a frustration of mine when textbooks could have just added a few more sentences - or sometimes just one more phrase! - and it would make the explanation so much clearer. Why?? If it's just that textbooks are just trying to minimize the printing costs by condensing explanations, please don't! I'LL PAY FOR THE EXTRA PAGES!!
This was a good video! I'm glad I watched it! I love analysis of theoretical math scenarios like this!
You have perfectly encapsulated the feeling I have whenever I read that in books. Well said. :)
What an enlightening explanation. Thank you!
I knew exactly how to do all of this (I love math :D) but listening to the guy explaining it is so relaxing and it made me want to watch the whole video. I love the production!
Wow this was super satisfying and fun!
I have just found this channel and after reading the comments I have figured that the channel just bloomed coz most of the comments are just a few days old. Keep doing the good job buddy hoping for some great videos from u in the future
I hope you get a lot of subs really quick you deserve it your work is fantastic keep it up! Btw you were correct the Σ (sigma) it's just a capital letter for the Greek letter s
Wow, such a small channel with actual good content.
Mind blowing.
Found this video, never before seen you, and I love it! You took something that would have taken me a few months to solve, and gotten it simplified where it would take me a few minutes to solve. You are better than most other math RUclipsrs, because you make it something that only a Harvard student would understand, turned into something that a grade-school student would understand. Others would just give how they solved it, and give the solution, and that would be it. Not you, though. Thank you for that.
thanks helping procrastinateing, while googling for "the Taylor Series". xD
You're welcome! :) If you need help with the proper Taylor Seires (which I haven't gotten to yet, alas), 3Blue1Brown has some good stuff.
I actually learned how Σ works. Thank you.
4:00 The tiles are...
2 to the 3rd: 8
2 to the 4th: 16
2 to the 5th: 32
2 to the 6th: 64
2 to the 7th: 128
2 to the 8th: 256
2 to the 9th: 512
2 to the 10th: 1024
2 to the 11th: 2048
2 to the 12th: 4096
2 to the 13th: 8192
2 to the 14th: 16384
2 to the 15th: 32768
2 to the 16th: 65536
2 to the 17th: 131072
More tiles: These are designed for 5x5 and bigger.
2 to the 18th: 262144
2 to the 19th: 524288
2 to the 20th: 1024576
Great stuff content wise and interesting to take on an older app, keep up the good work :)
There was a version that allowed for infinite undos. After solid 2 weeks of playing I finally managed to get the biggest combination on screen. I would have probably played twice as long if I was aiming for the maximum score as well.
i love when people trick kids into learning something
I'm happy that I found your channel. Good production value, great and thorough explanations, awesome voice. Turns out that RUclips recommendations aren't complete rubbish once in a blue moon :)
I just found out your channel, and just wanted to say that is pretty awesome!! Thanks for joining the math RUclips community!!
Thanks! Glad to be here. :)
Thanks to RUclips for suggesting me this video!
6:50 you're correct
BTW, it may be a good example to show some series sums can be computed, first with the geometric series and then using a dummy variable and the derivative (though I don't think you've dealt with derivatives yet). The shorthand answer to this sum comes out as rather compact - > 15*(2^18 - 4)
True! I just kinda wanted the introduction to the sigma notation to be a gentle one in this video, but you are quite right.
Calculus is too advanced for most audience... also, you don’t need calculus to reduce it... you only need to times each term by 2 and cross subtract to reduce it into a geometric series...
@@TheTaylorSeries Totally understandable, I meant this as a possible suggestion in case you ever wanted to make videos about the subject of summation of series.
I remember years ago doing practice mode for months redoing 4s to never appear until I needed them for the maximum tile values. I actually managed to get the maximum score and felt weirdly proud of it.
I never understood reiman sums this video really helped
Excel is not required. The sum of powers of 2 (a geometric progression) is just 2^(n+1) - 1. Geometric progression formula is (1 - a^n)/(1 - a). Taking the derivative w.r.t. 'a' can yield a formula for the sum of n*(a^n). Anyhow answer can be arrived from (17*2^19 - 18*2^18) - (2*2^4 - 3*2^3) - 2^18 + 2^3 - (18 - 3)*4 = (17*2 - 19)*(2^18) - (2*2 - 4)*(2^3) - 60 = 15*(2^18) - 60 = 3932100.
I have so many hours on this game and finally got max score
Underrated
I didn't expect to see a measly ~4.6k subs given how clean and well-done your videos are. You could give folks like Mathologer a run for their money. Good stuff, man! Subbed, and can't wait to see what you do next!