Can You Solve The Martini Glass Puzzle? A Simple Illusion That Fools Most People
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- čas přidán 21. 05. 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)
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 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
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
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.
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.
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%
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 😂
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.
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
*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.
People that clicked 70 percent because they thought of the martini as 2d and not 3d here 👇👇👇👇👇👇
Did you really do that? 😂
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 knew it was 80. Didn't calculate it but I could feel it when I pour and it had pissed me off for years.
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 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
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
Presh that was needlessly overcomplicated lol
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
INtuitively I said 80%, since the volume goes up the wider the glass. Nice to see it explained.
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.
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
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
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
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...
If two 3D objects are similar, the ratio of the volumes is always the ratio of lengths cubed
Wow, surface area really makes a difference
Excellent problem and solution.
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.
Very nice analysis👍👍
Without having to rearrange, the volume scale factor for a 3d object is (the length scale factor) ^3
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.
Very interesting. I solved this with integration using similar triangles before I saw your elegant solution.
Solving this problem is way more fun with trig!
That is the reason why they use this shape of glass in the bars
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 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.
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.
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.
A good reason for using calibrated shot measures.
In the graph, the axes must only be oriented toward the positive direction only.
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.)
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
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
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.
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.
80% seemed right to me just because the area of the crossections increase by a squared factor
A cone is a special form of a pyramid. Which is also the cubed route of base * height.
AD0TJ
Cuemath is doing a wonderful job, thanks for calling it out 💪
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.
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%.
My intuition was correct - just looked right; but then I immediately thought in terms of volume.
I answered this intuitively, and calculated it later by cubing the decimal values of the percentages.
I picked 80%.
Great video, thank you.
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
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
Height is proportional to the radius, so volume is proportional to the cube of height. Hence 1/2=v2/v1 = (h2/h1)^3
High school geometry says the area of similar figures vary with the square of the ratio of linear dimensions. Similarly, the volume of similar solid figures vary with the cube of the ratio of linear dimensions.
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.
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!
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%
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.
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!
Amazing work, can I know what software you are using for such a beautiful presentation as well as mathematical and graphical illustrations? Thanks.
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).
The volume grows cubicly with height and the cube root of 1/2 is about 80%.
The tricky part is that we see this as a 2D triangle but its actually a 3D cone
This is exactly the kick in the pants I had as I was watching the solution.
0:04 if you look closely it is actually a 3D figure
Yeah, the intuition of 70% is actually really close to the answer in 2d (√(1/2)=~70.7%)
Given the strength of a well made Martini, the small volume is a good thing.
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.
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
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.
I had my insight due to a jigger measure. 1 1/2 oz was almost full on a 2oz measure.
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.
Before anyone asks, whether a glass is half full or half empty (assuming it has liquud up to the halfway point) depends on the sign of derivative of the level with respect to time, dL/dt. To determine this, you must know the level just prior to it reaching the halfway point. If this is deri ati e is negative, the level is going down and it is half empty. If it is positive, the level is rising and the glass is half full.
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
Volume of a cone varies with the cube of the height, so 80% will be closest.
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.
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).
Draw a line down the middle. Square it and then measure the Area of the two squares
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.
New bar trick unlocked
As a side note, you're right that the angle doesn't affect the answer to the question at hand. What it affects is just how much it holds. If two glasses have the same height from point to rim but differing widths, the volume will vary directly as the square of the width. So, if the question had been whether you come closest to 50% of the volume with one whose rim is 90%, 80%, 70% or 50% of the original, the answer would indeed be 70%.
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.
The only thing I knew for sure right at the beginning was that the answer does neither depend on the angle nor on the volume of the glass. It's basically the same result as for why the water pressure does not depend on the area of the sea in which you are diving but only on the diving depth.
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 ;-)
very sneaky to make it look like a poll about area when it's really a poll about volume
I got shown this back when I did bar work years ago. Take a one-measure and pour into a double-shot conical glass. Now ask everyone if ANOTHER one-measure will fit or not? It LOOKS like it won't, but the names give it away. Two one-measures will always fit in a double-shot glass.
as I use 2oz jiggers alot, having the 1 1/2oz being almost nearly at the brim always baffles me. But because of this I could guess it's the 80% as that 1 1/2oz line is basically at the 90%
This result applies to every pair of same shaped 3-dimensional objects, regardless of what that shape is.
Definitely more than 70% but less than 80%.
The mind trick is to stop thinking of it as a flat image. If it were a triangle, the half height would be 25% the _area_ of the triangle, rather than 12.5% the _volume_ of the glass.
In 3D this is exaggerated so volume becomes more distorted compared with the visual assumption.
I only guessed 80% because my wife and I did a test to measure our jiggers in ounces for cocktails and it was crazy how much more volume a full jigger was to a close to full one. Jiggers weren’t cones but led my intuition for this problem.
This problem was on today's (June 10th) Lithuanian National Mathematics exam. It was on the test part.
The volume of the cone is 1350 ml. How many ml are there in the cone if it is filled to 2/3 of the height.
A 400 ml
B 600 ml
C 700 ml
D 900 ml
All those years of drinking finally paid off, knew it was 80% instantly
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.