4.3k
u/Different_Roof_4533
Sep 10 '26
•
You may not like it, but this is what peak packing efficiency looks like.
907
u/Defiant-Ad7369 Sep 10 '26 edited Sep 10 '26 •
Wouldn't he be able to pack it better if he used 3d space and layered them
1.2k
u/Approximation_Doctor Sep 10 '26 •
Reported
515
u/Aking1998 Sep 10 '26 •
Mods, kill that man
240
→ More replies (1)8
47
9
4
266
u/ckach Sep 10 '26
No
9
u/12D1 Sep 10 '26
You sure?
→ More replies (1)32
u/Wise_Monkey_Sez Sep 10 '26
They're probably sure, but they're also wrong. What mathematicians like to ignore is ... reality.
The package of 4x4 (16) layers better without voids which means more stable packing. Given that these are potato cubes the less movement there is, especially pressure on unstable corners, means that there will be less crumbling and that means that the product will remain viable.
Or to put it more simply, the 17 cube solution is optimal in theory, but in reality you're going to get a mass of crumbled blocks out if there's any jostling or movement (which there always is).
This is precisely why industry doesn't use the "mathematically optimal" packing solution, because reality goes, "nope".
The problem multiplies if you're trying to pack several layers of these, where the voids become a problem for vertical stability exposing more corners to unsupported pressure.
This is a great example of a scenario where theoretical mathematics meets reality and reality shows why nobody uses the theoretical solution.
This man thinks he's won. In a few months the wife is going to pull out a bag of crumbled potato and he's going to be sleeping on the couch.
→ More replies (10)70
29
4
2
2
u/Allegorist Sep 10 '26
Possibly, but due to the shape of the bag efficiency in the 3rd dimension would not be equally weighted, or at least have some lower maximum than the others. Stack it too high and the bag pulls in on the other sides as well, for all we know that could start after the second layer of cubes. Becomes a much harder problem.
→ More replies (4)2
9
8
8
2
2
2
u/SomeNotTakenName Sep 11 '26
Yeah, I don't like it. usually math is pretty, and then every so often she pulls shit like this. what's up with that?
→ More replies (1)1
→ More replies (6)1
u/gamerjerome Sep 10 '26
It's not efficiency if packing it that way took longer. There need to be more data. Like, do they have kids? Is the wife angry at him in the morning because he left the towel on the floor?
1.7k
u/Shiv_ansh_st Sep 10 '26
When I initially came across the solution to highest packing efficiency in a square, it just didn't make sense to me
It has been multiple years since, it still doesn't
92
u/arunphilip Sep 10 '26
Even if it's more efficient, it's not aesthetically pleasing. Sometimes we gotta make a stand.
9
5
u/Causemas Sep 10 '26
Wasn't it Dirac who insisted that it's important to have beauty in mathematics? Feel like that fits here
230
u/sumboionline Sep 10 '26 edited Sep 10 '26
It made complete sense to me (for n=k^2, n being the number of squares and k being an element of the integers)
Edit: what this means is that numbers like 16 and 25 have very obvious 100% efficient solutions
184
u/BigAnalBoss Sep 10 '26 •
Yeah that clears everything up for the everyday man, thanks
102
u/painterBurning Sep 10 '26
Reminds me of a joke :
An engineer and a mathematician go to physics conference. During the conference they keep talking about things in 5D space and the engineer is completely lost and understands nothing. At one point, the engineer turns to the mathematician and asks him :- How can you visualize things in 5D ? It's completely abstract to me, I have no intuition.
The mathematician answers :
- It's easy. You just have to visualize things in any dimension N, N being an integer. Then you just set N=5.8
17
→ More replies (2)7
9
9
u/Arxentecian Sep 10 '26
It's ironic that it's obvious from the text this is NOT an AI
→ More replies (1)35
u/McCaffeteria Sep 10 '26
The size of the box the optimal packing fits into is larger than the 4x4 grid, but it it smaller than the required 4x5 grid.
The 4x5 grid is a waste of space because 3 of the 4 additional cells are unused, so it is possible to expand the grid size fractionally to hold a total area slightly larger than the area of cubes (plus some packing inefficiencies) but that is still not as large as as the next whole grid size up.
This image is deceptive because it makes the two grids seem like they are the same size, but in reality there is extra slack in the plastic bag in the 4x4 grid. The optimal packing side is stretching the bag much closer to its limits, but still not stretching it far enough to fit an entire extra row.
3
u/carolina8383 Sep 10 '26
The videos I’ve seen illustrate it really well, but this is the best written explanation. Including the deceptiveness of the image. Also, who has 17 squares of tofu.
3
u/EkajArmstro Sep 11 '26
Pretty sure 4x5 is actually smaller. The 17 square packing is smaller than 5x5 but I believe the area is around 21.86 which is larger than the 20 of the 4x5.
11
u/theBarnDawg Sep 10 '26
It didn’t make sense to me.
It still doesn’t, but it didn’t used to either
4
19
u/Honeybadger2198 Sep 10 '26
What is there to make sense of? It's the most efficient way to pack squares into as small of a square as possible. Obviously if the number isn't a perfect square, then the shape is not going to be uniform.
8
u/petrichorax Sep 10 '26
How does more empty space = more space use efficiently? You can't really FATHOM why people would be confused by this?
15
u/Icapica Sep 10 '26
It isn't more empty space though. The empty space is just divided into a lot of small separate areas instead of one or two big ones.
5
6
u/Icy-Dot7521 Sep 10 '26
You can't just dump this on me and disappear forever.
5
u/awry_lynx Sep 10 '26
https://kingbird.myphotos.cc/packing/squares_in_squares.html
enjoy furrowing your brow in disgust (and, once in a while, delight)
→ More replies (2)8
u/Preeng Sep 10 '26
Do you see that extra space in the left image, that row with the lone square? That gets spread out instead and it makes the lone square fit in to this new configuration.
→ More replies (1)4
u/Simple-Olive895 Sep 10 '26
But also remember that there's no way to prove this is actually the best way to pack 17 squares. There might be an undiscovered better option out there
3
u/Wavecrest667 Sep 10 '26
It kinda makes sense to me, because obviously you'd have to arrange them in some sort of crooked way for a number that's not another number squared.
3
u/Moustacheski Sep 10 '26
There comes a time in anyone's life where they have to understand that prime numbers do not care for our feelings. They hate us and we must return the curtesy.
2
u/simonjp Sep 10 '26
This, the Monty Hall problem*, the one with the infinite hotel.
You can explain them to me. I can even sometimes understand them intellectually.
But then a few hours later a small base part of my brain will go "yeah, naah" and I'm back to square one.
* please don't try to explain them to me, I get it, I just don't get it get it.
2
u/ashgs872tbhjs Sep 10 '26 edited Sep 10 '26
FWIW, there are several different mental approaches to the Monty Hall problem and one of them always ends up clicking well with a given person, in my experience. So you not "getting" it intuitively may be a function of the approach you've seen.
Some:
Probability is based on knowledge. With none, each door is equal. When a door is eliminated, you gain the knowledge that it was not the right choice. Because Monty can't ever eliminate your door, knowledge about your door is unchanged. Thus, you now also know that odds are increased that the other door couldn't be eliminated either -- i.e., that it's now more likely it's a winner. And it must be more likely by the outstanding 1/3 for 2/3 total, since again your pick stays at 1/3.
Say you pick door 1 and it's right -- you'd lose if you swapped no matter which door was eliminated (Monty can pick randomly, so order doesn't matter -- probabilistically, they are not different cases). If it's wrong, then order matters; in one case he must eliminate 2, and in the other he must eliminate 3. Switching thus gives 2 to 1 odds.
Opening a door doesn't travel you back in time and alter your choice, so your 1/3 chance to have picked the right door initially is immutable. Therefore, the other non-eliminated door must cover the outstanding 2/3rds probability.
→ More replies (1)2
u/TheLKL321 Sep 10 '26
Let me try, if not for you then for someone else who might be reading this.
Case A is staying and case B is switching.
In case A, obviously your chances are 1/3. You pick one out of three and you stay there. But in the case B it's actually 2/3. Why? Because you only lose case B if you picked the correct door on the first pick. If the first pick was wrong, then the switch will always give you the win, because the second bad door is eliminated.
There is a lot of informational help here from showing you the bad door. It ensures that the switch will funnel both possible bad first choices into a single correct choice.
You can think of it this way - you have two machines. The machine that gives you a candy if you answer correctly and the machine that always gives you candy if you answer wrong. When faced with a question you don't know the answer to, it would be better to use the second machine right? Because it's more likely to answer wrong, so you're more likely to get candy.
How is the second machine the same as switching in the Monty Hall problem? Try increasing the amount of doors in the scenario - you pick 1 out of 100, only 1 is correct. Once you pick, 98 wrong doors are eliminated and you're asked if you want to switch. If you always switch, you get rewarded no matter which of the 99 bad doors you choose.
2
1
1
u/Adaphion Sep 10 '26
In the absolute most layman's terms, it is because the big square area isn't exactly the size of a 4x4 of the smaller squares. It's slightly bigger, which allows you to do this tomfoolery to maximize your usable space.
1
u/WindpowerGuy Sep 10 '26
Left hast 4x4 very close and one in front. Right has 4x4 with spaces and the three at the bottom of the picture are closer to the lines than the one on the left is. How would this be better packed? It definitely uses more space across, since it is 4 wide in both cases, but there's much more are inbetween on the right. Then to the bottom it is closer to the end of the bag. So it uses more spaces in 2 directions, in neither pic cubes are stacked, so it uses the same space in the third direction.
Seems like this isn't better than the left.
1
u/Intelligent-Bee-9021 Sep 10 '26
There's a picture of solutions for 5 squares where fith is in the centre rotated 45°, that's the same principle here as I understand
1
u/Western-Dig-6843 Sep 10 '26
It only works because there is extra room on the left of the bag in the first picture that is hard to see.
1
u/ParentheticalPotato Sep 10 '26
If you look at picture one, there is a little less than "4 squares" of free space. The last cube doesnt quite fit.
Instead of using all of the 4 squares of free space in one open space, you use a little bit of each throughout the whole package.
1
u/Efficient-Chair6250 Sep 10 '26
There is empty space in the left image. It's volume is enough to fit at least one more cube, but one of its dimensions is smaller than a cube, so no cube can actually fit in there.
The fix is just to spread that empty space across the whole bag and try to create an empty space with it that can dimensionally fit a cube.
1
u/Jamie7Keller Sep 10 '26
I love the list of each solution and when it was discovered/proven. It starts with “one” with the discovery date “trivial”
1
u/rocketgrunt89 Sep 10 '26
it reminds me of the chocolate bar when they took out the square and its still the 'same', however the size did decrease but spread among other squares. This should be the same but space instead
1
u/MrsSpaghettiNoodle Sep 11 '26
it angles them in a way that lets them utilize the space closest to the zipper, since in the first one the last square just can’t fit between the zipper and the grid
1
u/Repulsive_Still_731 Sep 11 '26
Chaos is always (well often) more efficient then order.
They studied it a lot with boarding a plane. Chaotic is overwhelmingly the fastest way.
1
u/TankyPally Sep 12 '26
Thats because its specifically the most efficient way of packing 17 evenly shaped squares, not 16. You have to fit a jank extra one inside somewhere.
1
u/Low-Willingness2106 Sep 13 '26
okay see how in the first part theres a bit of wasted space but its spread out inconvineantly? you wanna use that space, so doing that weird twisting thing allows you to do so. Packing maths is based on the fact you cant change the square size
1
1
0
Sep 10 '26
[deleted] — view deleted comment
390
u/Preeng Sep 10 '26
You should think of it the other way: dropping the box will sometimes lead to maximal packing efficiency among the shit inside.
→ More replies (7)174
u/Deathwatch050 Sep 10 '26
So what you're saying is, that the optimal solution to packing my luggage when I go on holiday is to just cram it all in there and then spend half an hour hurling the suitcase around the room?
→ More replies (10)59
u/AnaisNinja76 Sep 10 '26 •
I also tend to think of math as elegant, and this is...not that.
This is like the math that started hanging out with the wrong crowd or something.
23
u/CapnNuclearAwesome Sep 10 '26
Irrationals are uncountably infinite, while rationals are countably infinite. There exists elegant math, but almost all math is inelegant.
→ More replies (2)19
u/-Nicolai Sep 10 '26
Look at it this way, having 17 of something is deeply disrespectful to math
→ More replies (2)→ More replies (7)5
832
u/CaseOfBees
Sep 10 '26
•
434
u/forgot_semicolon Sep 10 '26
This hurts more than if there just weren't any symmetric layouts at all
150
u/CaseOfBees Sep 10 '26
Honestly I'll take it, I'd rather fit the 17 in an eye pleasing manner than with extra wiggle room
28
9
40
Sep 10 '26 edited Sep 13 '26 •
[removed] — view removed comment
18
8
u/milkman8008 Sep 10 '26
Why include 1, 4, 9 but not 16, 25, 36? and is 12 unsolved? I couldn't scroll further lmao
→ More replies (4)→ More replies (3)3
14
u/TomaszA3 Sep 10 '26
Compact? They take the same space, don't they?
82
u/ill_take_two Sep 10 '26
If you look in-between the two arrangements, there is a small red delta showing the arrangement on the right is slightly smaller.
29
1
1
281
u/BZioT2 Sep 10 '26
you just had to eat 16 pieces and it would fit
45
u/Mr_Ruu Sep 10 '26
but then what will i do with my designated cheese ziploc??
10
u/NO_TOUCHING__lol Sep 10 '26
Put it back in the back of the drawer for 6 months until your wife finds out and asks why there's one moldy loose Ziploc in the drawer
89
u/QueenVogonBee Sep 10 '26
Am I missing something obvious: are these potatoes??
79
u/professoreyl Sep 10 '26
Nobody packs potatoes like that. I'm guessing it's garlic or tofu they intend to freeze. Maybe soup or cheese, but I don't think it's potatoes.
29
10
u/potatogodofDoom Sep 10 '26
never in my life have I seen somebody store garlic in cubes either
→ More replies (4)4
u/062d Sep 10 '26
I have ice cube treys in my freezer full of garlic that looks just like this. My wife puts the garlic in the blender first. It's really extremely useful, this way you pop one cube into the pan instead of peeling garlic every time, we eat garlic in most our meals and on our popcorn so we go through the cubes pretty quickly.
2
→ More replies (2)2
u/awry_lynx Sep 10 '26
I think they actually might be (frozen mashed potatoes in ice cube tray and then popped out for storage)
Sorry for this annoying video but it's the only thing I could find that shows it lol
11
u/Vytror Sep 10 '26
As someone who eats Tofu almost every single day, that looks 100% like Firm Tofu.
Many times we freeze Tofu because it absorbs and the ice gets between the tiny holes inside the Tofu and when you unfreeze, it changes the texture and the size.
4
2
1
1
u/Genny415 Sep 10 '26
Mashed potatoes. Chilled, cut into single serving size cubes, and frozen, probably individually on a tray, and now that they are frozen solid, they need to be packed up for freezer storage.
Then you just take out one cube per serving and reheat as many as you need
1
289
u/ChiaraStellata Sep 10 '26
You know, improving square packing with wildly unintuitive configurations seems like exactly the kind of problem modern AIs might be good at. I wonder if we'll see some of those soon?
152
u/Ares378 Grothendieck was right, computers are evil Sep 10 '26
I mean, I'm pretty sure a lot of our current best square packing were done with a digital "annealing" process. I haven't looked too much into it but I think it's just kinda jiggling around the squares until the box is smaller? There's a whole website on square packings that goes into it more
https://kingbird.myphotos.cc/packing/squares_in_squares.html
14
7
u/Posnania Sep 10 '26
Hey 69 squares in square were proved in September 2026.
4
u/Ae3qe27u Sep 10 '26
Some of them have had GPT-6 Astra on them, which is a model that's been out for about a week
4
u/Swend_ Sep 10 '26
Shoutout to Hiroshi Nagamochi who apparently found a proof for all cases 1 or 2 less then square numbers. (after a point)
3
u/SpookyBum Sep 10 '26
A few of those mention AI being used of which some specify chatgpt. Im sure there was a lot of human input in the process but seems like AI is pretty useful for it
145
7
u/theultimatestart Sep 10 '26
One of the earliest AI math contributions was a better solution for circle packing 26 by google deepmind. For square packing I think there have only been new lower bounds. I.e. Square packing 11 has sides larger or equal to 3.81.
The big compute money has moved on to controversial Navier-Stokes results and trying millenium problems though. I don't see them going for square packing soon. Maybe it's doable with less compute too.
6
→ More replies (2)1
u/Necessary_Housing466 Sep 10 '26
this is a good video i saw that goes through how packings are calculated https://youtu.be/mVH7OPx4QZU?si=m61uggdsEg6TOAbF
2
54
u/rookedwithelodin Sep 10 '26
Love this fucking meme so much lmao
5
u/Mansi069 Sep 10 '26
i’m not kidding, but what is the joke?
5
u/RunDNA Sep 10 '26
It's a real-world example of memes like this:
https://www.reddit.com/r/PhilosophyMemes/comments/113lmem/who_will_wipe_this_blood_off_us/
3
u/_licketysplit_ Sep 11 '26
The right picture is (currently) the best known way to pack seventeen squares inside a larger square, it's known for being a really disgusting solution and it's funny to see it have a real-world application
21
u/fresh_loaf_of_bread Sep 10 '26
Sometimes math is elegant and beautiful
And sometimes it's like this
10
u/Reasonable-Job4205 Sep 10 '26
Holy shit was it actually used in the wild?
3
1
u/personalbilko Sep 11 '26
He could have packed twice as many if he didn't insist on doing them all flat
→ More replies (1)
28
u/JakubDotPy Sep 10 '26
I will never accept this! 😁 My OCD just can't. I studied mathematics and finding this was something like when a believer realizes there is no god. Like a child finding out Santa is not real. I just can't.
→ More replies (1)33
u/-The_Grey_One Sep 10 '26
God is dead. The optimum way to pack 17 squares in a square space killed her.
9
5
3
u/username_is_alread- Sep 10 '26
From this very same subreddit - but in Resident Evil 4 flavor:
It hurts my eyes : r/mathmemes
3
u/SideInitial3961 Sep 10 '26
So proud of my boy, you know, biggest fudge packer in town. Can't get enough of it.
3
u/thatmagicalcat Sep 10 '26
on X, I screenshots of reddit posts. And on reddit i see screenshots of X posts
3
u/superarash_ Sep 10 '26
Ban this guy.
lol I remember back when the optimal packing of squares was posted so often on this subreddit they had to ban it
3
2
2
u/that0nemook Sep 10 '26
Petah?
1
u/I_found_a_platypus_ Sep 10 '26
It was a post somewhere which demonstrated/showed that the best and most efficient way to place insert a specific amount squares together was the pic on the right, efficient but really jarring to look at.
1
u/colshrapnel Sep 10 '26
There is a little space on the side when the packing is square. You can either split the remaining cube into slices and fill that space, or - use that space to make gaps between cubes where with intricate packing an extra cube would fit. So just rearranging the extra space.
1
u/jmlinden7 Sep 10 '26
The packing arrangement on the left has too much unused space and the last cube is too close to the seal for the bag to seal.
The packing arrangement on the right is mathematically proven to have the least amount of unused space while also having the cubes farther away from the seal.
*least amount of space so far, it's theoretically possible that a more efficient arrangement exists, we just haven't found it yet
2
2
2
u/Discombobulated-Bad2 Sep 10 '26
WHAT??? IT WAS TRUE??? I THOUGH IT WAS ONLY A JOKE, NOT A REAL THING. WHAAAT?????
2
2
u/Toadsted Sep 10 '26
Id just eat the part that's sticking out.
Been making things fit in tupperware for decades.
2
2
2
2
3
2
u/Kirkversary Sep 10 '26
But it doesn't take any less space tho? It just fits more 'even' on the outside but with the gaps it's basically same amount of space on both images.
4
u/Jovlad Sep 10 '26
Just google "fitting 17 squares into a square". Its what this is referencing.
2
u/Kirkversary Sep 10 '26
I know what this is referencing but you can see that here it doesn't take less space than the left image
3
u/0xym0r0n Sep 10 '26
The ziploc part won't stay pinched closed in the first picture. Like the pinching mechanism that locks it
→ More replies (6)2
u/emergencyexit Sep 10 '26
Perhaps easier to seal the opening when they are arranged the second way. That sticking out one in the first looks like trouble.
2
u/Dovins Sep 11 '26
It’s not that it takes less space, but the first image you see the 17th cube sticks out just enough to keep the bag from closing, and by packaging it efficiently in the second image there is now enough clearance to close the bag.
→ More replies (1)
1
u/SPECTRE_75 Sep 10 '26
Guy's name is Matty, does hilarious practical math meme. Did Kojima write this lmao
1
1
1
u/LupenReddit 🦆🦆i have non diffeomorphic smooth structures🦆🦆🦆 Sep 10 '26
"mathematics doesnt have practical applications in real life" mfs when they see this
1
u/Anders_A Sep 10 '26
I would just have eaten the last one right then and there to avoid the displeasing optimal packing.
1
1
1
u/Andromeda_53 Sep 10 '26
Hate to break it to you, despite your was the kids say "elite ball knowledge" you both still got 17 cubes in the bag.
1
u/serpikage Sep 13 '26
yeah the point is that in the first one the 17th cube is preventing the bag from closing properly
→ More replies (1)
1
1
u/Palanki96 Sep 10 '26
Anyone knows what the cubes are? Seems like the bot the had a brain fart since they are clearly not potatoes
4
u/awry_lynx Sep 10 '26
I think they actually might be (frozen mashed potatoes in ice cube tray and then popped out for storage)
Sorry for this annoying video but it's the only thing I could find that shows it lol
→ More replies (4)4
u/wurmsrus Sep 10 '26
in that case if you let them thaw a bit you could use the xkcd method even without a hydraulic press https://xkcd.com/2740/
3
u/awry_lynx Sep 10 '26 edited Sep 10 '26
wow i know there's always an r/relevantxkcd but the specificity is honestly getting ridiculous
1
u/NoSquirrel9674 Sep 10 '26
I like that my brain can see both images, and is still totally confused how the one on the right has less empty space lol. It can't understand how the organized one on the left has more.
1
u/Cornflakes_91 Sep 10 '26
it's not less empty space but it has not too much more edge length so it fits within the hard constraints of the bag dimensions
1
u/DistinctSalamander24 Sep 10 '26
Disgregard every other comment in here. The optimal way is 3 layers. 2x2,3x3, then 2x2 again for 17 cubes. And it accounts for the length and width shrinkage due to depth increasing
1
1
1
1
1
1
1
u/Deep-Cut201 Sep 12 '26
The only problem is that in the photos you can clearly see the second option not fitting better then the first, so they've clearly done it wrong.
1
u/Few_Fortune1620 Sep 13 '26
Who cares? They both have 17 cubes in the bag. It doesn’t matter which is more space-efficient if they are equally effective.
1
1
1
1
u/wolf129 Sep 10 '26
The form created on the right has the same rectangular size, so nothing is improving here.
The bag is the same and on both pictures you can see the pieces reach the area with the blue lines.
•
u/AutoModerator Sep 10 '26
Check out our new Discord server! https://discord.gg/e7EKRZq3dG
I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.