The Problem With M&Ms

the problem with m m's is sitting right in front of you in this jar or in this cup or even in your hands and it is of course how many m ms am i looking at 4549 but yeah it seems like a stupid question of course we can give a rough estimator if we need an exact answer we can literally line them up and count them one by one like i did for around an hour it took a long time yet all over the world there are these jars and massive pillars filled with these things dedicated to giving you a reward if you can give the closest estimate but people including you tend to be a lot worse at tasks like these remember there are 4549 m m's in this jar and here's what people intuitively think without doing any math or calculations or anything like that in your opinion without doing any math without using any anything how many how many how many m ms are in here i'm gonna go with 490 two 1472. would say maybe four thousand two hundred and fifty six three thousand uh i'm gonna say 583 583 wait 583 that's it yeah i was thinking like way over a thousand but how many like over well now i'm second guessing i'm like i'm like 5 000 uh i have no idea no idea is good too so 500 but also at the same time five thousand we're really not that good at this in fact right now in your head try to picture what 100 m ms would look like really do try to visualize this how big would that stack be what kind of container could it fill up chances are the image you have in your head probably looks something like this give or take a few in reality though that is a thousand m ms this is what a hundred looks like it's just a small handful wait why does this happen i mean really why does this happen we spend our whole lives dealing with amounts and figures and groupings yet our intuitive understanding for real amounts or cardinality the number of elements in a set or grouping is nowhere near as intuitive as we think it is try this with me here's a line segment that will say has length x which can represent m ms hershey kisses or really whatever it is you want now in your head really do try to visualize what a segment of twice the length of length 2x would look like now of course this really isn't that hard because we're geared to handle one-dimensional scenarios like this no problem but now let's try to increase the dimension giving us a square that again can represent m ms or any other kind of object and this time let's try to double our area now this still looks pretty normal but doesn't it kind of sort of feel like we should have increased this by you know just a bit more i mean yes if we have two dimensions to work with then doubling looks like scaling on each axis by square root of two which happens to be around 41 but doesn't it feel like we just didn't really do enough even crazier in three dimensions to get a doubling we only need to scale up each axis by 26 and at this point it really does look wrong we definitely need to add in some more volume somewhere how can this cube be twice the volume of our original now if we were four even five dimensional creatures eating our four or five dimensional m m's these perceptual distortions would continue to get worse because then on each axis it would look like we're scaling by the fourth root or fifth root of two but even in our measly three-dimensional world there are more problems to deal with for example imagine that in one hand you're holding one eminem and in the other you're holding two in other words there's gonna be a difference of one m m well then visually and then also tactile if we were to feel them we could tell which hand has more m ms now if we take the same scenario but this time use 20 m m's and then 21 m m's again this is still a difference of only one m m well in this case it gets much harder to not only feel which one has more but also visually we can no longer tell and that problem only gets worse and worse the more m m's we have in other words we see in terms of ratios or logarithmically we do not see linearly this means that no matter what m m amount i show you as long as i double it you should be able to tell the difference because the ratio is still two to one but if we do like a constant offset of one m m or two m m's at that point it will become perceptually invisible so at this point it should be clear that with our eyes and brains we're really not equipped to tell how many m ms are in here just using our intuition so the question i want to ask is how close can we get without taking a single measurement so we can't do anything like cylindrical volume formulas or anything like this how close can we get without taking a single measurement i don't know the size of the m m the jar how much packing density there is how far can we take this estimate how close can we get by using only a single photo let's find out so even in a challenge like this where we only have a single image to go off of i still think the best plan of attack is going to be a ratio of volumes now this might seem like a weird way to approach the problem right we never took a measurement i don't even know the scale of this photo nor the size of the m m the jar anything but if i had a big cube and filled it up with smaller cubes in this case it's obviously an 8 to 1 ratio you can still figure out how many cubes are inside this even though i never gave you any kind of units for this kind of problem right it could have been in meters cubed feet cubed centimeters cube right units end up not mattering as long as we're dealing with ratios where the units are the same on the top and bottom and we can use this exact kind of trick on this image where we don't have any kind of scale or reference or anything like that to do this let's exploit this jar's rotational symmetry in other words this thing is going to be very easy to model and similarly let's pick some m m inside this jar and use similar modeling techniques to now get two models that are not necessarily to scale with the real world but relative to each other they're definitely to scale so incredible we have our 3d models which means we can now calculate volume again these numbers are going to be very big in an arbitrary unit but if we divide relative to each other this problem goes away and we finally get a number that is quite frankly too big how do we account for this why is that number so much bigger than the initial estimate of something like 4549 well the answer must be somewhere out there or somewhere out air because we didn't account for the gaps of air that must be in between the m m's right this isn't like completely full of chocolaty goodness no there are packets of air and we do not know the exact packing density especially when we have kind of random orientation and all this so let's try to calculate that so to calculate packing density as you probably already expect you take the combined area of your objects and just compare them to the area of the container that it's inside now in this case of circle packing because this is two-dimensional so these are effectively circles you might think that this grid or lattice type configuration is the most effective way to pack but this is not true it turns out that hexagonal packing or honeycomb packing is much more optimal at something like 91 when it comes to packing circles inside a two-dimensional region now people have been doing these kind of packing problems or optimization problems for a long time now right putting squares inside of squares or circles inside of squares stuff like this optimally but we don't care about this right we care about this random distribution kind of stuff and we want to calculate what the packing density is in this kind of random physics based scenario so that's usually done via simulation so we're just going to simulate this in three dimensions in my case instead of doing this by filling a container until it's full and then looking to see how many m ms were in there i wanted to do the opposite in other words reshaping the container until it fits our objects and then right before m m's explode out of bounds like the frame before we call this full and then take that object to container ratio from before which in this case gives us a calculated packing density of roughly 59.7 percent so am i really saying that this jar which looks completely filled with m m's is forty percent comprised of air yeah and it might be hard to believe but so is a bunch of other stuff remember this cube is twice the volume of the original we can tell the difference between one m m and two m m's but 20 and 21 no way and don't forget that 100 m ms which seems like a lot in our head only comprises a very tiny amount of a small ziploc bag so yeah it's surprising but if we now take our calculation write the big jar volume that we calculated versus the tiny m m volume that we calculated and then multiply by this new packing density right the counting for the error that we shouldn't be counting in anyways we take all these numbers and calculate them together we get 4458 m ms without taking a single measurement think about that for a moment that is crazy we got within two percent of the actual amount not knowing the scale of the image and not taking measurements nothing so maybe the main takeaway is just don't trust your intuition or maybe the main takeaway is go to your kitchen drawer take out one of these things a spoon so you can scoop away your eyeballs get rid of those things they're useless anyways but at least we'd be able to calculate how many eyeballs would fill up a jar which is cool i guess and with that thank you for watching i've been cg matter you've been you bye bye [Music] you have no idea how long i've been waiting for this day i've been looking at this m m jar for a long time making sure i don't eat any so the numbers stayed consistent and i wasn't cheating or anything even though nobody would have noticed but today i finally get to eat one of these bad boys that i've been eyeing up for a long time i'll go with the yellow and i know this isn't the normal kind of video i usually make i'm trying to branch out into new topics whatever but whatever this video was is sponsored by skillshare and skillshare is a online learning community where you can learn a whole bunch of skills like photography videography even 3d graphics a whole bunch of stuff that you'd probably be interested in and on top of having a bunch of video courses there are also online workshops you could work alongside digitally thousands of people that are also doing that workshop at the same time so that could be a good motivation to keep progressing through learning whatever skill you're trying to learn and one premium course in particular i want to recommend is how to make merch with draplin and by the way draplin is the guy who makes a field notes if you know about that i happen to have one on me but that's neither here or there this course is not only about designing stuff which is already useful but it's also about how to repackage and how to sell to a client and stuff like this so if you're interested in design and want to be a freelancer or something like this i recommend watching this premium course now skillshare premium will get you access to premium courses like this it's also ad free and has an annual subscription that's less than 10 a month but i'm here to make that deal a little bit sweeter pun intended the first thousand people to click my link in the description can try a free trial of skillshare premium so you can watch this course that i recommended or really anything that you want on the platform so that link exists in the description and i'm gonna go on to eat now 4500 i don't even remember many more m ms

Channel: CGMatter

Views: 243,439

Keywords: M&M, m&m, M&Ms, m&ms, math, 3d, blender, vfx, cg, cgi

Id: YgaCdXvyk68

Length: 9min 42sec (582 seconds)

Published: Fri Jan 29 2021

Reddit Comments

[u/CrudeDudeSteve]

This guy talks so fast that a 9 minute video felt like it was 5. Interesting video though. I would have loved to see a solution that didn't involve using Blender to calculate the total amount but that was a really cool solution!

When I was a kid, I was asked the same question. To guess the amount of M&Ms I counted the amount at the thickest point of the jar horizontally and then vertically and multiplied. I ended up getting closer than everyone else and was off by a couple hundred. Maybe I was lucky 🤷‍♂️