What's the color density of Minecraft?

Wanna learn about the 2D puzzle?
Here's the Quanta article:
www.quantamagazine.org/the-nu...
And for those who dive off the high board, here's the actual paper that proved 15 colors works for any 2D shape (and is the smallest amount of colors that works for any 2D shape):
arxiv.org/abs/2301.09757
Timestamps:
00:00 intro
0:38 2D puzzle explanation
3:05 3D puzzle explanation
4:17 the challenge
5:56 3D solution sneak peek

[@whitestonejazz]

June/July update: I started Summer classes today for my PhD program📚! Looks like I won't have time to make the recap video anytime soon. I've been at my new apartment a month now and haven't even been able to set up my desktop PC 😬 So at the very least, here's the 🎮high scores🎮 from all the puzzle solutions people sent in: 4x4x4: 62 blocks out of 64 (by Rayne)* 8x8x8: 413 blocks out of 512 (by Tanner) 16x16x16: 3068 blocks out of 4096 (by Tanner) 32x32x32: 23799 bloks out of 32768 (by Tanner) *Tanner sent this solution in too but Rayne sent it in first! So the percentages filled, from smallest cube to biggest, goes like 97%, 80%, 75%, and 72%. That means the Color Density of Minecraft, all infinitely much of it, is probably sitting around 70% somewhere. Also, a viewer named Conner sent in a solution generator on github💻: github.com/Fish-o/color-density/releases/tag/v0.1 And ✨honorable mentions ✨ for: Ppaatt: 397 of 512 blocks on the 8x8x8 cube Sixthstep: 61 blocks out of 64 on the 4x4x4 and 402 blocks out of 512 on the 8x8x8 (which were both records at the time they were sent in!). Thanks to all who participated. Going forward, I'll have to get better at predicting which projects I'll actually have the time and interest to properly follow through with. Till then 👋

[@_Sixthstep]

I still think about this a lot, this challenge revitalised my passions for both maths and minecraft, and when I'm feeling burnt out with either of those, this is what I come back to to remind me why I love them. So thank you, your content is some of the coolest I'm aware of, and I hope your studies and life in general are going well :)

[@thederpydude2088]

This is a question I've never before in my life thought to ask, yet I am now fascinated about what the answer might be 😆

[@dabdotnet]

A tip for people trying this in game: Pressing both left click and right click at the same time will replace the block you are breaking with the block you are holding simultaneously. This can be done on a single free floating block without needing to make scaffolding

[@TammonOG]

How do we send in our solutions?

[@whitestonejazz]

You can email the channel ([e-mail]). The best method would be to upload your Minecraft world or a schematic of the solution to a file sharing site (Planet Minecraft, Dropbox, or whatever) since this is probably most secure for both of us. Then you can send the download link to the email, to the instagram (@whiteboardjazz), to the twitter (@OnnoMawCain

[@whitestonejazz]

You could also even just share detailed screenshots of 4x4x4 and 8x8x8 solutions too. I just need to be able to verify that a solution is correct reasonably quickly.

[@TammonOG]

@@whitestonejazz alrighty sounds good

[@whitestonejazz]

1 month later update: Today was my first day of classes going back to school for my PhD! Heather and I moved into a new apartment and I haven't even setup my desktop yet. With a full class load and research projects, I might not get around to displaying the solutions people sent in. So at the very least, I'll post the "high scores" for the different sizes here in this pinned comment: SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER

[@jukmifggugghposer]

Gotta say, I just discovered your stuff through the animals walking uphill video, and I am really loving all this. I've never seen anywayone else using Minecraft like this, and it turns out that it's actually a really great way to show off/communicate this kinda stuff. Looking forward to see what you do next.

[@hbelam]

My thought (I’m too lazy to scratch it out on paper) to solve it for n-number of volumetric cube and x-number of colors would be the brute force way of generating first the patterns in which the Xth color was valid within the n-cube, storing them as “puzzle pieces” a la your bit at the end, going all the way down to the last (first color) doing the same thing (note: you would need to account not only for the most restricted but also the least restricted puzzle piece “shapes”) then collecting the universe of pieces and brute force comparing all possible legal constructions to determine the percent filled with “color” (note: one could consider the absence of color as a color, too). That’s sort of brute forcing it from the end backward, using some filtering. Or you could do it by basically starting from the corner of one cube, each space has x (or x+1 depending on how you define color and count for the null) options, and each space within that cube has all those available options. As both n and x increase the universe of options expands (far more than) exponentially, and once you get past 10 I guess you start working in hexadecimal, etc. each cube would be filled and then you would have n^3^x^x cubes (or some such, it’s too early to even attempt to do math with my limited skill set). Constructing each possible cube from the ground up, then checking to make sure that it’s settings are valid (less than half will be, something tells me as the number of colors increases, the number of valid constructions exponentially decreases). You’ve basically got a constrained version of the tiling problem (n length cubes) along with the map-coloring problem (x colors/x+1 “states”). Something tells me working with modulo would really help as well for the repetition aspect in some attempt at solving this puzzle a certain way. Now if only I’d had linear Alg and beyond to even figure out how to notate these things correctly and work at an attempt to solve, but I’ll just sit over here in calc three land watching everyone else using abstraction to reach a solution. Not to mention the people who can then code it into some sort of computational logic that a computer could use to tally. I really like how you use Minecraft to marry the physical and the mathematical and it’s so cool to watch.

[@dootnoot6052]

[4x4x4 puzzle spoilers~] proof (kinda) that the best possible number of colors is 18 (using the most efficient 1-lattice): n-lattice = color with a gap of n blocks, and a optimal n-lattice is one with as many blocks as possible in aa 4x4x4 region (when i say "the 1-lattice" i mean the optimal 1-lattice with 32 blocks). for any 2 blocks spaced a even number of blocks apart, exactly 1 of them overlaps the 1-lattice. therefore, no 2k-lattice is better than a 2k+1-lattice unless the 2k-lattice contains more than twice the blocks (this never happens). the smallest 2-block 9-lattice is too big to fit in a 4x4x4 space, so only lattices 1, 3, 5, and 7 need to be considered (with 2 of the 3-, 5-, and 7-lattices). a 4x4x4 cube has 8 corners. 4 of these are covered by the 1-lattice, and each of the 7-lattices require 1. this leaves 2 corners for the 3-lattices and 5-lattices. there is a 18-color solution with 2 6-block 3-lattices with 0 corners, 1 3-block 5-lattice with 2 corners, and 1 2-block 5 lattice with 0 corners. there is no 3-block 5-lattice that doesn't use 2 corners and no 4-block 5-lattices at all. for there to be an improvement, there would need to be a 7-block 3-lattice that doesn't use any corners, which doesn't exist, or a 8-block 3-lattice to replace the block lost from the 3-block 5-lattice, which also doesn't exist. there's no room to squeeze an extra block out of the 7-lattices, so 18 is (unless i missed something) the best possible. the only way a 17-color (or fewer) could exist would be to not use the 1-lattice, which would allow the use of more efficient even lattices, although the improvement would have to be at least 4 blocks to balance out the less efficient 1-lattice. otherwise, the max efficiency is 31/32 or 96.875% of the cube filled.

[@1224chrisng]

reminds me of the Sugarcane pattern. I'd wonder if it could be done with just a regular grid where each block is spaced (1, 2) or (2, 3) blocks apart, and just translating those grids

[@Cuestrupaster]

Isn't it better to start backwards? By the color with the longest distance? Or at least try to make it like a Jigsaw puzzle, where you don't go in order but try to fit the colors as it seem easier to fit...

[@MrRyanroberson1]

i find it strange that there isn't some greedy algorithm that starts with the last color and fills in the maximum density before moving on to the next color and so on

[@sayaks12]

wouldn't the density approach 100% as the number of colors goes to infinity?

[@cptCrax]

Not necessarily if you're looking at an infinitely large area, especially in 3 dimensions. Think of how large a space each new color *can't* do you any good in after you place the first of that color. I haven't convinced myself one way or another but it at least seems plausible.

[@sayaks12]

@@cptCrax but surely it is at least monotonically increasing for all of 3d space. and we know that it's possible to fill all of 3d space with infinitely many colors. this doesn't necessarily mean that it will approach 100% obviously but it seems very intuitive to me that it would.

[@fakestory1753]

if you are talking about density of finite space, yes if you are talking density about infinity space, no you can't

[@sayaks12]

@@fakestory1753 for infinite 3d space you can with infinitely many colors

[@fakestory1753]

@@sayaks12 no, after you add infinity color, you still got holes, they are just infinity far from your view first color fills 1/2 of 3d space, second color only do 1/10, the density of new color drop in a way that they can't add up to 1 you can always add 1 more color to fill up the hole in front of your view, but it doesn't help filling up further out, because their density is too low

[@coollwell_the_great]

Love this video

[@elnico5623]

i once got nerdsnipped by 4d road logic and demostrated that in a minecraft server, basically the problem was to design a 3d road intersection, rather than a 2d road intersection in our 3d world, it had some neat symetries

[@nemo-x]

Yes of course a brand new development in theoretical mathematics is going to immediately make its way into minecraft. What did i expect? xD

[@whitestonejazz]

Ideally, we could just save time by developing the theoretical mathematics in Minecraft to begin with

[@thehighground4920]

@@whitestonejazzhahaha

[@enbyd]

Hmm... packing a field with tokens spaced apart by gaps of arithmetically increasing size... if this is a coincidence I will be very surprised

[@TheMCEnthusiastPlays]

this seems like a great problem to explore using wave function collapse

[@arielamine5662]

It's an exercice i've done for this so yeah x)

[@cubicinfinity2]

Sugar cane planted on that 3D array grows into the 4th dimension.

[@SyntaxDaemon]

So just to be clear, the challenge is to pack 16 colors as closely as possible? And the difference cubes are just spaces we can use, but don't necessarily need to.

[@whitestonejazz]

Yes, that's correct. The challenge is really whatever variation of the puzzle you wanna try. But since the puzzle might become a competition, I thought it would be better to have specific sizes so that solutions could be compared more easily. Because fairly comparing infinite solutions or solutions for random shapes seems difficult. You could think of it like a juggling contest where you can really juggle anything you like. But to keep things fairly comparable, the judges only pay attention to how well you can juggle 4, 8, 16, or 32 balls.

[@_Sixthstep]

@@whitestonejazz Is 16 colours definitely possible? I've been lazily writing python code to try and solve this for the last few hours and pretty much immediately hit a brick wall at a 20 colour solution for the 4x4x4 cube, and now I'm at a crossroads as to whether to dedicate countless hours of my life to this puzzle or not lmao

[@whitestonejazz]

@@_Sixthstep It might be that the best you can do with 16 colors on the 4x4x4 cube is only like 95% density or something like that (say 55 to 60 of the 64 possible blocks filled in). You might have made the best possible solution already. The next step would be proving mathematically that 100% isn't possible with 16 colors. For my own part, I've already lost at least 2 work days to this puzzle 🙃

[@_Sixthstep]

@@whitestonejazz ah I totally overlooked searching for not 100% dense solutions, I've basically just been trying different ways of filling the cube fully and trying to minimise the number of colours used, will have to rethink my approach. Thanks for clearing that up!

[@3Ppaatt]

@@whitestonejazz Truth. My workday destroyed >.

[@solsystem1342]

Not a solution but, thinking about this as an n dimensional puzzle as we take the limit as n goes to infinity the percentage of spaces filled drops to 50% (I think). Could be wrong but spacing things out by 2 or more in 3D and beyond is going to quickly be unable to fill much space. With say your hundredth color in 1 million dimensions it's a rounding error. If you scale up a dimension (looking at just one color) one of two things can happen. Either it stays the same or it shrinks. Idk how to prove for all colors after the first have a shrinking sequence but it seems intuitively it should be? Idk

[@dootnoot6052]

assume that the maximum n-dimensional packing of some color has density x split the maximum n+1-dimensional packing up into n-dimension layers (a 1×k×k×k...×k section) by the first assumption, every one of these layers must have at most density x, therefore the entire thing has at most density x

[@danielrhouck]

Iʼm not able to think about this problem much right now; Iʼll come back to it throughout the month. But, is the doability in 2d but not 3d related to all the various other things that 3d opens up, like all graphs being representable in 3d, or random walks no longer being guaranteed to return to the origin?

[@whitestonejazz]

It seems to come from the fact that the volumes of octahedrons grow cubicly whereas 2D squares grow quadratically. Practically this means that the sum of the best possible densities for each color converges to a value less than 1. In English, even in perfect conditions the colors could never color 100% of 3D space -- well maybe. There's some weird facts about infinity that have to be sorted out to state this last part technically correctly

[@danielrhouck]

@@whitestonejazz So the sum would converge in 2D as well (but not 1D, where itʼs $\sum 1/(2n+1)$), but they converge to more than 1 so itʼs theoretically possible, and then someone showed that it can actually be done? Makes sense, except for how itʼs late and I canʼt figure out what the actual closed form would be in dimensions greater than 1, so I canʼt figure out what they converge to. Iʼll read the article in the morning, probably. EDIT: wrong sum for 1D; that wasnʼt overlapping the excluded areas correctly. Itʼs actually $1/(n+1)$.

[@OM_NERO]

You are the best math teacher ❤ am from mogswamp

[@The14Some1]

Chris, i think, for 15 colours there should be at least 15 different perfect 2d solutions. And the 3d-solution should be assembled of 15 layers of these perfect 2d solutions. This is only assumption, though.

[@whitestonejazz]

I tried it but couldn't get this "paper stack" approach to work

[@flippos]

now do this for 4d hypercubes

[@hellrangerboi2178]

u getting popular wow

[@whitestonejazz]

turns out a lot of people like wildly nerdy Minecraft videos

[@vrtr1]

4d color packing when

[@skreal211]

I wish you were my math teacher

[@The_femboy_lover]

How do you make math interesting?

[@whitestonejazz]

color, motion, physical objects, and metaphors to common experiences

[@ianerickson2210]

Do you think in the future you would start a Survival series on the channel? Not sure if you're already a whiz at redstone, but I feel like your survival world would be a sick place to see.

[@kikivoorburg]

There is a survival series! The playlist is “Permaculture in Minecraft” or something similar to that

[@whitestonejazz]

My plan is to blur the survival videos and math videos together

[@debug8377]

mogswamp brought me here

[@arandominternetuser5614]

:3

[@Kitana___]

:3

[@SlayerOfSmaug]

:3

[@christianwolirdeng4766]

:4

[@JoaBro]

:3

[@dootnoot6052]

:3

[@Gentelman8697]

Have you got a maths degree?

[@whitestonejazz]

yah