Can You Solve The Martini Glass Puzzle? A Simple Illusion That Fools Most People
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- เผยแพร่เมื่อ 21 พ.ค. 2024
- Just how full is your cocktail glass anyway? The answer will surprise you.
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I am half way through the video and forgot about the third dimension
Yep, me too.
That was my exact mistake
Same. 😅
Was thinking between 70% and 80%, but closer to 70%.
Yea I guessed that is what most people though. That is why I guessed 80% as it was a cone.
But first, we need to talk about parallel universes.
Part of why so many people's intuition was 70% is that in the 2D case, the answer would be 1/sqrt(2), which is 70.7%. And although the wording was very clear that we're interested in the 3D case, the pictures themselves are naturally 2D, which colors our intuition.
Yeah, my brain was doing maths with triangles because that was the visual I was presented with, instead of cones. It's another fun way to think about how easy to the mind is to trick (and often does the tricking itself).
Pictures are not naturally 2d. They are artificially 2d to mislead and confuse.
@@StRanGerManY What's the shape of the screen you're viewing this on?
@@trimeta you don't know how to draw a cone on a piece of paper?
edit: I took a closer look and it's drawn like a 3D cone
@@talkingbirb2808 Without doing a full-on animation (which arguably does add a third dimension -- time), it's hard to draw a cone that doesn't look like a triangle.
as a previous bartender, i knew it was 80% instantly. if you were to pour it into a glass of the same volume but with a cylindrical shape, it would be at the halfway mark.
another round, please
I was going to say the same thing! lol Real world experience is priceless!
@@spacewolfjr another martini, Paul?
What if a mathematician shows up there?
@@abhishankpaul 2^(-1/3)
I think that 80% of the height = 50% of the volume is not as mindblowing as 20% of the height being 0.8% of the volume
Without using a calculator I realized the answer was the cube root of one half. Then I needed a calculator.
I had it memorized from when I used to to teach physics
I approximately cubed 1.2 and 1.3 (I figured working with the reciprocals was easier - that may or may not have been true) in my head, realised it had to be somewhere between them and guessed 1.25, which corresponds to 80% so went with that.
Given the initial question was multi-choice, you could've worked backwards from there.
80% is 4/5, so you can just multiply 4/5 by itself twice to get its cube (4/5, 16/25, 64/125). 64/125 is very close to 1/2.
@@ThatFoxxoLeoI like that explanation. Thank you.
512 is 8^3, so .512 is .8^3, so .8 is close to cbrt(.5)
I guessed 80% bc i’m a bartender.
Literally no calculation, just from working with martinis. Then you started proving it with math and I got so confused haha.
Love your content. Thanks for everything!
i'm not a bartender and i didn't calculate it but still guessed it right, i just figured it was common sense given the understanding that thinner volumes hold less fluid than wider volumes. martinis are cone shaped, so i figured they'd exaggerate the effect quite a bit, so my initial guess was 80%
@@AiNaKa yea that's what I thought
I guessed 80% because it looked like that was the closest to half full. Not a bartender or anything.
Technically 100/2^(1/3)% = 79.37%
The angle doesn't matter cause stretching the glass horizontally and depth-wise will preserve the proportions.
Cone is really good shape of glasses for bars’ owners😀
And cubes. Don't forget cubes...Ice cubes. Plenty, plenty ice cubes.
If they don't fill it to the top, it's a sham!
„Con” is literally in the cone name so
There's an intuitive explanation for the % volume = (% height)^3 formula: as the glass is filled, the shape that the water makes is scaled larger and larger with the tip of the cone anchored in place. Since a cone is three-dimensional, the volume of the cone is scaled in accordance with the cube of the scale factor of the lengths, thus, % volume = (% height)^3.
I just used pretty much that, r is proportional to h , let's call r = ah, so rewriting that we have V = 1/3 a^2 h^3 since a is a constant, for V' to be 1/2 V, h' would just be 1/2^(1/3) h; it feels like we are just overthinking the problem.
@@martianunlimitedbut both of you are doing calculations 😂
All the glasses are 100% full. It's just a matter of what they're full of.
my glass is filled with 90% water and 10% bose-einstein condensate
"It's just a MATTER of what they're full of"
Pun intended?
@@user-pr6ed3ri2k Mine has 10% quantum foam
@@fatmccat1513 what's the other 90%
One thing about eyballing it is that when doing the poll with just eyballing it, I am naturally inclined to pick the triangle where the orange area is half of the volume rather than remembering that the triangle represents a cone which is what we're actually supposed to eyeball. When I eyball the triangle, 70% is definitely closer to half the area of a triangle, but since the triangle is supposed to be a cone, he 80% actually wins out.
As a former bartender I do have to make a minor correction in that a Martini glass actually should be filled to nearly the rim. The purpose of the glass is to give the volatile aromatic molecules a large surface area to evaporate into the air but not to enclose and capture them, so that when sipping you get a strong aroma from the drink only at the moment you first bring it to your face to sip, not throughout the motion of tipping the glass to drink. The glass should be filled almost completely to the rim, like 95%. Even 80 or 90 will give too much room for the aromas to collect and the intended effect of the first sip will be list, as the drinks nose will be full of hot alcohols and terpenes
A much simpler way to approach this is to observe that both cones are similar. So you want to scale down the larger cone in order to half its volume. Scaling a body does not depend on the shape at all, it can even be done with a cube. If you scale a body in 3D by factor s in every dimension, the volume increases by a factor of s^3. We want to know the scale factor x so that the volume factor is one half: 0.5 = s^3. Therefore, x = 0.5^(1/3) which is almost 0.8 and the answer to the question.
No pi, radius, graphs or complicated formulas needed.
That's the way I did it
Its easy to assume wrong when you're presented with a 2-d image for the options, in a question that wants an answer based on 3 dimensions. What everyone learns in school is to go with what information is presented if the problem doesn't specify any detail.
I know some people will try to be cheeky and say they top of the drawing may imply something, but we all know that can just be there as an artists choice for making any generic 2d cup.
Yeah, I didn't think about the diagrams enough and made a simple mental calculation based on 2D cups and got 70%, but in 3D it gives 80%.
Ah yes a 2D glass
I’ve never heard of a 2D liquid. The question implies that it is in 3D.
I'd love you to pour me a 2d drink! How many ml go into a 2d martini glass? 😂
@@Questerer When 60% assumes something "wrong" based on how the question is presented, then there's absolutely something wrong with the question, especially since their answer is correct when calculating the area in 2D.
*Remember that the real-life counterpart of that diagram is 3-dimensional...*
Yeah, that was what got me, lol. 2d diagrams are a nasty trick.
The misleading part of this question is that it is presented as a 2D problem visually, when in fact it is a 3D problem.
I fell into the same trap.
In 2d 80% is still the closest answer, right?
@@MrJoerT I think it is 70.7% for 2d case. Solve the 2d version, 1/2 = x^2.
Underrated.
Visually 2d? There's an oval at the top, an oval at the bottom, glasses irl are clearly 3 dimensional, 2d glasses don't exist (closest you'll get is a very flat prism glass), etc
1:20 Assuming the inside is not a truncated cone, i.e. it's pointy all the way down, the angle should not matter. My intuition says it's around 75%, but it wouldn't surprise me if it's closer to 80%.
That explains why my iPhone thinks 80% battery level is considered full!!!
it only goes to 80% to preserve battery life overall
@@Temporary_yesyes trust I know and for full performance and capacity as well!
this video explains the opposite
If the height of the liquid is 80%, the glass would only be about half full.
If the glass is 80% full, the height of the liquid would be about 93%.
Rofl
I knew the answer immediately.
I came across this in the early 1980s. I was in a bar (in a place called Klerksdorp in South Africa) when two people ordered liqueurs. One wanted a single and the other wanted a double. The barmaid poured their drinks (by eye) into the same size glass. The guy with the double complained because he felt the guy with the single was getting a better deal (more than a single). She got a tot measure & poured a single into it and topped the glass (with the single) up. It came to exactly the same level as the double (she was pretty good at her job).
Exactly the same thing happend to me when I worked behind a bar,
A 'double' served in a proper martini glass is NOT 'twice the height of liquid in the glass'.
That's why I always used a measure and poured the drink into the glass in front of the customer.
I'm a martini glass half full kinda person
So are you 1/8 full or 80% full?`
Maybe @justdilka is in a quantum superposition of both?
I'm a " Martini to the half way point"
I remember me and my dad figuring out where you would have to cut a cone into 2 perfect pieces. We also got ~79.4%
surely you cut the cone vertically down the middle....
@@noomade lol this is what we thought when seperating a cone into 3 equal pieces
People that clicked 70 percent because they thought of the martini as 2d and not 3d here 👇👇👇👇👇👇
Did you really do that? 😂
Presh that was needlessly overcomplicated lol
I knew it was 80. Didn't calculate it but I could feel it when I pour and it had pissed me off for years.
This is why there's usually something like (20 cl) next to the wine descriptions in restaurants. Mixed drinks are usually sold by the shot, as well.
I'm even more impressed by the fact that the 50% height is only 12.5% full.
Very nice analysis👍👍
INtuitively I said 80%, since the volume goes up the wider the glass. Nice to see it explained.
Very interesting. I solved this with integration using similar triangles before I saw your elegant solution.
Excellent problem and solution.
Wow, surface area really makes a difference
Solving this problem is way more fun with trig!
You don't need the formula of a cone, for many very difficult with pi etc.
The glass can be any form
It's enough if the partly filled glass is an image of the full glass.
The factor is 0.6 or 0.7 and it works in 3 dimension to the 3rd power.
0.8 * 0.8 * 0.8 is approx 0.5
A good reason for using calibrated shot measures.
Now hol up. Who said it has to be a cone? What if we're looking at a cross-section of a triangular prism?
The math works out to be the same, since volume still scales with height^3
EDIT: NVM, mixed up prism and pyramid. If it's a triangular prism then yeah 70% should be the answer here.
Doesn't make a difference. Any prism is 1/3 cubed route of height.
A cone is just a many-sided prism.
AD0TJ
Than it wouldn't be a martin glass
@@deathpacito8702the volume of a rectangular prism is triangular area (which is the cross section we see) * the length of the glass in the Z axis.
Since the length of the glass at any given moment is unchanged, wouldn’t the answer change to 70%, as only two dimensions change instead of three and it’s related to the square of the height instead of the cube?
@@protoman1365 Ah my b, mixed up prism and pyramid
Without having to rearrange, the volume scale factor for a 3d object is (the length scale factor) ^3
Great video, thank you.
I knew the answer because I recalled doing a similar thing in 1st semester calculus 40 years ago. I think we had to calculate the rate of volume change vs height of liquid change. I also thought of the pizza size vs diameter optimization here and knew since it was 3d, it would be a cubic instead of a square.
Cuemath is doing a wonderful job, thanks for calling it out 💪
i’ll keep this in mind for my bladder
If you have one of those measuring containers with similar shape you can notice how often the value changes at the top and how rarely it changes on the bottom, that is why I was one of the 23%
I saw this demonstrated in tv education program back when I was a child. They did it for several shapes and demonstrated the relation of shape to certain graph shapes. Taught me more about mathematics than any two of my school teachers.
we can `sense` the intuition with the volume formula: V = pi*r^2*h/3, that the r is a power of 2 with delta{h}. Therein, delta{h} in increment of %, can lead to a power of 2 increase.
80% - I had a good intuition on this. As someone who makes a pot of pour over coffee every single morning in a cone filter, I have a very keen awareness of when the pot is almost done brewing.
I usually solve your riddles from the miniature, then I just skip to the answer and like. Not sure if that helps the algorithm, but in any case thank you very much for all those riddles!
70% makes sense for our eyes because we intuitively see 70% as half the glass, looking at it we see a 2-dimensional picture - and indeed, the area of the smaller triangle at 70% height is 49%, because it is a square relationship. Cubic relationship is more extreme, so 80% feels right...
In the graph, the axes must only be oriented toward the positive direction only.
Amazing work, can I know what software you are using for such a beautiful presentation as well as mathematical and graphical illustrations? Thanks.
YES I figured 80% because of videos I've seen about shot glasses showing how much alcohol you can lose out on, 70 was my first guess but it seemed low to me
If you work with a measuring cup a lot, you would kinda have an idea to what height it might be, i know its not the same shape but the similar concept.
As soon as percentages were introduced without the x100 factor when one side of the equation was cubed, it makes the equation out by a factor of x10000. Very surprised at Presh for incorrectly converting to percentages rather than sticking with ratios.
Height is proportional to the radius, so volume is proportional to the cube of height. Hence 1/2=v2/v1 = (h2/h1)^3
I used the disk method from calculus for this problem, and found the distance from the origin at where I would get half of the full volume of a cone.
If two 3D objects are similar, the ratio of the volumes is always the ratio of lengths cubed
A cone is a special form of a pyramid. Which is also the cubed route of base * height.
AD0TJ
There's actually a shorter way to do the calculation. The cross sectional area of the cone shaped glass is equal to pi*r^2. r is directly proportional to the hight because it's a cone so the area has to be directly proportional to the hight squared. Then we can simply integrate this h^2 term with respect to the h and we get the volume to be directly proportional to 1/3*h^3. The factor of 1/3 can be ignored like any other constant factors thus far. That way it becomes clear that the answer has to be close to the cube root of 1/2 which is 79,37% or nearly 80%.
8:03 @MindYourDecisions He said " v/V = % of the volume of the large cone ". No, it is only the percentage like ( y %). Value with %... So the relation is y % = (x %)³
Bartender here. Drink recipes are measured independent of the glass. Then you consider "wash lines" - the point at which a given glass is visually "full", while still allowing it to be carried comfortably without spilling. With final volume of the drink and volumes of various glasses at their wash lines all known, an appropriate glass can be selected - or recipe and price adjusted.
very sneaky to make it look like a poll about area when it's really a poll about volume
I immediately concluded: volume of cone => 1/3 base area times height; Base area is proportional to the square of the height => volume proportional to height cubed => for 1/2 volume, height is cube root of 1/2
The number of comments (on the poll. Not this video) who were saying that it depends on whether the glass is 3d or 2d, astounded me. How anyone could fill a 2d glass with liquid is beyond my understanding.
Can work tho with 2d drawing animation I guess
@@mofprailes7140 The question was clear. It said LIQUID.
Because they forgot what the problem was, since the graphic is 2D with a very hard to see 3D glass rim, especially on mobile. It is pretty easy mistake to make if not paying attention
In 2 dimensions I can color the surface of the glass with a marker.... That is liquid.
But hey 😂 I am joking here 😉
My guess was based on 2d as well... i just saw it as a triangle
I sat down and mathed this out for the fun of it, but if you asked me this question at a party, I would just get a book (or other hard, flat object), firmly cover the end of the glass, and turn it horizontal to see which one had a liquid level that reached the point of the cone. This is called solving by brute force, and while it's not mathematically efficient when there are many examples to be tested, for a set as small as this (particularly considering that I had already eyeballed the 80% one as looking right so probably would have began there) it is good for making your point quite aggressively
I didn't do calculations, but I did think of it as a triangle, and then visually cut it in half and made it into a rectangle. It didn't give me the correctvolume, but it gave me the closest answer.
This feels related to the cube-square law, which demonstrates that there is a lag between how much the surface-area of a given solid increases when you double its volume. Said surface-area only increases by the square of the cube-rt of 2 (e.g.: 2^(2/3)). Conversely, when you double the given solid's surface-area, you more than double its volume
Whew! Glad to be in the 28% that selected 80%! When I saw the results leaning so heavily to 70%, I assumed I’d totally botched it. Great puzzle Presh!
New bar trick unlocked
I answered this intuitively, and calculated it later by cubing the decimal values of the percentages.
I picked 80%.
My intuition was correct - just looked right; but then I immediately thought in terms of volume.
Next time you want a shortcut to use for any general shape, just use the square cube law. or in this case the line cube law (I think I made it up).
What I did first is taking the formula of the volume and substituted r with its expression through h and the angle. Now that h became the only variable - I solved for the ratio of h, which would half the volume.
80% seemed right to me just because the area of the crossections increase by a squared factor
Had a head start thanks to Numberphile's video yesterday on the exact same problem. One could construct an Euler diagram of the audience/viewers/subscribers of this channel and that one
The area of the cross-section at any height is proportional to the square of the height. The volume swept by the cross-sections from the bottom to the height is the integral of the area vs height function, so it is proportional to the cube of the height. If the height of the top is 1, the volume could be represented as 1*1*1*K = K (K is just the actual volume of the glass). Similarly, the volume at 50% would be 0.5*0.5*0.5*K = 0.125K, which is 12.5% of K. If we want the height that is 1/2 of the volume, we need the cube root of 0.5 which is 0.793701 (or 79.3701% of the height).
All those years of drinking finally paid off, knew it was 80% instantly
That was killer.
great puzzle
I remember seeing a video somewhere, discussing the volume of a cone. I am also a math student so I instantly guessed it was 80% without calculation, as the lower part would contain lesser volume of the drink. Most of the volume of the glass lies in the frustum of the cone.
How wonderful, I will keep that in mind every time they give me a glass of wine.
I'm so happy for my cat for giving me the correct answer
Or to put it another way: the liquid in the glass is a 3-dimensional volume in which the x, y, and z sizes (height, width, depth) all vary proportionally together. Volume is x*y*z, so, the answer is the cube root of one half.
That is the reason why they use this shape of glass in the bars
I did the calculation, but not because I wanted to cheat, but I enjoy doing calculations - and this is the reason I like this channel. I did the calculation to see whether my guesstimate was right, before I un-pause your video, just like I usually do with your videos ;-)
As already pointed out in several reactions, the calculations can be simplified knowing that the ratio of the partial volume to the cube of the height is fixed. This is the case when the bottom of the volume is a point and the top surface area of any partial volume is some constant times the square of the height. This provided, the shape of the top surface does not matter, so it might be square, triangular or any weird shape you might imagine. If the orientation and general appearance of this "horizontal cut" shape stay similar throughout the height, the "sides" of the volume can be described as a bundle of straight lines from the bottom to the top edge of the volume. E.g., simple square or triangular pyramids fall in this category.
5:49 Isn't a "right-angled cone" one return a side angle of 45⁰, i.e. where base diameter equals height.
I believe that what is discussed here is a "right cone", i.e. one whose tip is is perpendicularly above the centre of the base circle.
Happy to be corrected.
(Disclaimer: I'm in the UK. Terminology may differ across the pond.)
Based on the crossection of the glass being a triangle, I instinctively thoughts something around 75% and considering the 3rd dimension, I hedged my bet on 80%
Given the strength of a well made Martini, the small volume is a good thing.
I think the easier approach to calculating this is to use the intercept theorem. If you scale up (or down) any geometric object any given area will change with the square of the scaling factor (because areas are two-dimensional) and any given volume will change with the cube of the scaling factor (because volumes are three-dimensional). This is because all measures scale up (or down) with that same scaling factor.
Because, if held upright, a cone standing on its head (as is the case with this Martini glass) filled up by x% will result in another, similar cone, the volume of this cone of liquid will be (x%)³.
All that said, even though I knew this, my first impulse was to say 70%. It really doesn't look like two 80%-full glasses would (almost) fit into one full glass!
You could use a similitude.
The two cones are similar to each other. The ratio of volume is the cube of the ratio of height(in 2d the ratio of area is the square of ratio 10:02 ).
So( r1/r2) ^3=v1/v2.
And we get ,for sure,same result.
7:57 I need more math in my life I was so close to this before I watched this part. Getting the ratio of volumes was the part I missed because I was trying to solve a specific case instead of the general.
Draw a line down the middle. Square it and then measure the Area of the two squares
When I saw the thumbnail I did the calculations in my head, then wondered why I managed to cancel out the relation to the angle, so I watched your video where you made the exactly same calculations and still can't picture (but I understand the maths behind) how this can not depend on the angle...
Note that the exact taper of the glass matters. Any self-similar shape, that is, shapes which maintain their relationship to a cylinder containing them as they fill, will fill with an exponent in accordance with its fractional volume of a cylinder.
A cylinder will fill linearly.
A paraboloid will fill quadratically
A cone will fill cubically.
A long tapered shape taking only a quarter the volume of its bounding cylinder will fill quadratically.
The angle of the taper DOES NOT MATTER as long as the sides are straight.
A cone implies straight sides
It doesn't matter what the angle of taper is, what matters is that it is 0 at the bottom, and goes linearly to some positive value at the top.
Without watching, unlike most solids, filling a cone from the point up is equivalent to scaling it along a single dimension, so with the square cube law, if you want 1/2 the volume, you need to scale by (1/2)^(1/3) ~= .794
So 80%
The volume grows cubicly with height and the cube root of 1/2 is about 80%.
Yay! I guessed it right!
So I tried to visualize an imaginary line on the two dimensional glass extending from the top-right corner to the middle of the left leg of the triangle, cutting it into two even right triangles (And thus create a visual for what 50% full would look like from a different perspective). Then I sorta imagined tipping each of the pictured glasses until the liquid was at the top right edge, just before spilling out. Then I just tried to visualize which of the options put the water level at that 50% point on the left leg of the glass, and the 70% image felt like it would come up too low, but the 80 landed just right.
It definitely depends on an angle, as smoe kind of ratio of H & R determinates the angle. 90° filled to 50% height is 25% volume, and 75% height is 58,(3)% volume. 60° angle gives us 50% volume at 66,(6)% of height.
That's not true
I had my insight due to a jigger measure. 1 1/2 oz was almost full on a 2oz measure.
Damn, I was one of the small percentage that guessed 90% on the earlier poll. I thought it would be similar to the question about lily pads covering a lake by doubling each day. (That one is 1/2 covered on the second to last day.)
I did the math 'wrong' in that, in my head, I knew the volume was 1/3*area*height, and I just thought that I would multiply that by 1/2 being the 50% height mark, but intuitively the halves aren't 'equal' like they would be in a cylinder, but more volume in the widest (upper half). Thus roughly 1/6 (1/3*1/2) of the 'fat' side of the cone height is where the 50% should be and that is about 17%, thus 83% 'full' is halfway there going from the other direction. Not correct, but a close estimate without dealing with roots and whatnot. Glad you pointed out my error (and I always love this stuff because I've been out of school for decades).
i paused this video at 1:34. It's obvious to me that the correct answer is 80%, since (0.8)^3 = 0.512 is the closest of the given values to 50%. Note that the shape of the liquid in the half-full glass is similar to the shape of the full glass regardless of the slope of the glass, whence the constant of proportionality of the linear dimensions of the full glass to the half full glass is the cube root of 2, or approximately 1.26.