The solution exists. But the problem was never solved. A true mathematician.

@Jamifa007

5 жыл бұрын

A haiku: "The answer exists. But the problem wasn't solved. Mathematicians.."

@bestopinion9257

5 жыл бұрын

That single cut exists, as you saw in this clip. But you do not know where. So, it is proved that that single cut exists. It is not proved where.

@Nightriser271828

5 жыл бұрын

This was literally what I predicted about 30 seconds in.

@Darcy783

4 жыл бұрын

But if the problem has never been solved, how do you know that the solution exists? Wouldn't there have to be a solution found in order to prove that such a solution exists? You can't just *say* that such a line exists. That's not proof.

@anthonynorman7545

4 жыл бұрын

@@Darcy783 umm...it's actually not that rare in math

Math gives you a solution to a problem you never had.

@Septimus_ii

6 жыл бұрын

gyes99 it proves that the solution exists, but never tells you what it is

@floggyWM1

4 жыл бұрын

@@Septimus_ii is that how AA meetings work

@karolbomba6704

3 жыл бұрын

@@floggyWM1 Anonymous Alcoholics?

@floggyWM1

3 жыл бұрын

@@karolbomba6704 yes

@karolbomba6704

3 жыл бұрын

@@floggyWM1 ooh, I used to attend the zoom ones but got kicked often :/

Put sandwich in blender, use measuring jug. Solved.

@machiavelli326

5 жыл бұрын

Think smarter not harder

@joshandrews8913

4 жыл бұрын

@@machiavelli326 It is harder, though, if you think about the difficulty of actually consuming that sandwich sludge.

@TheZooropaBaby

4 жыл бұрын

yeah because putting blender is.....cutting each piece only once?

@missionpupa

4 жыл бұрын

@@machiavelli326 Wrong. doing that actually does not make sense. You now have a more difficult problem. You cant tell how much ham you have when you split it up because you just blended them together with the sandwich.

@tooljockey2777

4 жыл бұрын

@@missionpupa if you blend it enough it will be mixed the same

you know you have the correct audience base when about 50% of the comments are talking about Hannah and the other 50% are talking about how you MISPELLED BANACH

@ElTurbinado

6 жыл бұрын

Plus now we know it's possible to cut all the comments in half with one cut.

@WujuStyler

6 жыл бұрын

And Hannah said one of the researchers was called Turkey while his name on the screen was Tukey

@Serfuzz

6 жыл бұрын

Wonder if it's the "Turkey" of Fast Fourier Fame.

@edwardfanboy

6 жыл бұрын

They also mis-pronounced "Tukey".

@petros_adamopoulos

6 жыл бұрын

The misspelling is unacceptable.

1990 - i bet we will have flying cars in the future 2017 - Ham Sandwich Problem

@nathana.4467

6 жыл бұрын

2014 - I bet these copy and paste comments will disappear and people will start to gain originality. 2017 - A wild Wojciechowski appears.

@CraftQueenJr

6 жыл бұрын

Wojciechowski aye... depressing, isn't it?

@GlobalWarmingSkeptic

6 жыл бұрын

This was proven in 1942, then Pearl Harbor was bombed. Coincidence? I think not.

@MIbra96

6 жыл бұрын

+Global Warming Skeptic Thanks for the laugh mate! xD

@nicemelbs

6 жыл бұрын

I'm guessing you don't know about the Hairy Ball Theorem. minutephysics made a video about it.

As one of 5 children we discovered the way to precisely divide a ham sandwich (or more often a dessert dish) in half. It is the "You cut, I choose." principle. Given a suitably delicious item, the precision of halving goes up to near 100% accuracy. (The knowledge that the next halving exercise would be to have the "chooser" be the "cutter" on the next round, diminished the tendency to put your non-cutting hand all over the larger of the two sides while cutting.)

@loganfisher3138

5 жыл бұрын

This works until one party realizes that they can claim that they don't want very much, leading the cutter to cut unequal pieces, prompting the liar to then take the bigger piece.

@beeble2003

3 жыл бұрын

The "point" here is that the ham sandwich theorem says that it actually is possible to have a fair cut. It's conceivable that there would be no fair cut, which would give the "chooser" an inherent advantage because the "cutter" would be somehow forced to always produce a big slice and a small one.

@NoriMori1992

3 жыл бұрын

@@loganfisher3138 You ignore that person. As the cutter, your role is to cut into two pieces that _you_ consider equal. What the chooser considers equal does not enter the equation until you have finished cutting.

@NoriMori1992

3 жыл бұрын

You invented math (as 3B1B would say) without even knowing it! That's cool!

@Talaxianer

2 жыл бұрын

The problem is your judging precision is higher than your cutting precision, so the chooser will always get the bigger piece. (Except if the chooser's judging precision is in the range of the cutters cutting precision)

I'd love to hear bed time stories narrated by Hannah. Great voice for audiobooks also.

@matthewchampion8214

3 жыл бұрын

@Roger Loquitur This dude supposed to see your reply huh?

@chriswebster24

2 жыл бұрын

Pervert

@jacobschiller4486

2 жыл бұрын

@@chriswebster24 Got it; complimenting one's voice is a perversion. What's next, their intelligence?

I really like the style of these videos and how they explain it to someone else rather than just talking straight to the camera

Hannah's voice is so lovely

@bunpeishiratori5849

6 жыл бұрын

There's something sexy about a British accent!

@brianmiller1077

6 жыл бұрын

The accent is similar to the one Michael McKean used as Davis St Hubbins in Spinal Tap

@gasser5001

6 жыл бұрын

Hannah is so lovely*

@whozz

6 жыл бұрын

DJ Deckard Cain Of course

@toferj7441

6 жыл бұрын

I completely agree. Smart women are beautiful. :)

Or you could just do the Banach-Tarski thing with the sandwich and have two identical ham sandwiches...

@lucamuscarella4085

6 жыл бұрын

Jack Carton size this one needs go viral

@ISenjaya71

6 жыл бұрын

*vsauce music intensifies*

@cOmAtOrAn

6 жыл бұрын

Only if you take the Axiom of Choice.

@gremlinn7

6 жыл бұрын

That's how I do my shopping. I buy my goods, go all Banach-Tarski on them at home, and then take half back for a refund. The axiom of choice is free with Amazon Prime.

@mandolinic

6 жыл бұрын

I'd rather have two identical Hannahs!

4:44 "There seems to be a lot of mathematical problems that center around things that happen in lunch rooms and tea breaks, isn't there?" I don't like imagining how the hairy ball theorem fits in there... I hope it's kiwis...

@cricketknowall

3 жыл бұрын

"Lunch" might have been a more fun, perhaps disgusting proposition for that one.

@NoriMori1992

3 жыл бұрын

Never heard of that one, I'll have to look it up.

Yay more Hannah!

@cbarre9937

6 жыл бұрын

Hello

Just imagine going on a date with Hannah and ordering a sandwich and cake would be like.

@HappyBeezerStudios

6 жыл бұрын

It would be fair share.

@want-diversecontent3887

6 жыл бұрын

Vinay And what if you decide to eat pancakes one day?

@misterhat5823

6 жыл бұрын

Sorry... That would be the last thing on my mind on a date with Hannah.

@argh1989

6 жыл бұрын

Just imagine going on a date...

@livedandletdie

6 жыл бұрын

Just imagine...

This video is so adorable and interesting at the same time

@happylittlemonk

5 жыл бұрын

It all depends which side your bread is buttered ;)

@Triantalex

7 ай бұрын

WYSI

“Do you want to share this sandwich with me?” *proceeds to touch every part of the sandwich*

"Stone & Turkey" sounds like a really bad sandwich.

@jotabeas22

6 жыл бұрын

It sounds like a documentary on how kurds are treated.

@jotabeas22

6 жыл бұрын

Says you, who was clearly the one who started getting aggressive. Hey, politics are everywhere so if a bit of un-sided satire hurts you, tough luck.

@thomasyates3078

6 жыл бұрын

It's Tukey, not Turkey. As in Tukey's procedure to eliminate familywise error rate when making all possible comparisons between three or more means.

@DreitTheDarkDragon

6 жыл бұрын

Could be also name of punk band

@agr.9410

5 жыл бұрын

It’s the hexaflexagon bois

MORE HANNAH

I love that Brady's reaction is just "who comes up with this?"

Thank you Hannah, I haven't had this much fun with a challenge for a long time. At first I was very confused when you said something about 3 objects anywhere, as I was expecting to be able to place the objects on a cutting board and make the cut...

Hannah is just plain full of awesome.. and apparently sandwiches..

2:54 Surely you mean Cooking and _Frying_ with Hanna.

@sofarky

6 жыл бұрын

omg

@unreal-the-ethan

6 жыл бұрын

Ba dum tss

4:43 They totally missed a Turkey Sandwich joke. Also, a Bananach Joke

@davemarm

6 жыл бұрын

Isn't his name Tukey, not Turkey?

@guillaumelagueyte1019

6 жыл бұрын

I was surprised to not hear a rebound on that name

@st3435

6 жыл бұрын

With stone ground bread

@karlmuster263

6 жыл бұрын

Because that would be disrespectful.

@teomanyalcnkaya5072

6 жыл бұрын

CcC Türkler geliyor

Ohhh that's clever! And brilliantly explaned, as always, by Hannah. Thanks a lot!

From now on, every breakfast will be full of my thoughts to Hannah Fry. What a wonderful beginning of a day!

I think the only problem here is that there's only ham in that sandwich.

@Sam_on_YouTube

6 жыл бұрын

DHGameStudios But if you add cheese, then you need to slice it in the 4th dimension.

@DHGameStudios

6 жыл бұрын

@Sam Make it so.

Hannah you’re so great! I love numberphile with you in it:)

I read about the ham sandwich problem (and this solution) in "The Mathematical Experience," by Phillip Davis and Reuben Hersh, a book I was awarded winning a high school math contest back when it was first published. Great problem, and a great book showing how dynamic and diverse mathematics could be, which served as an inspiration for me to be more curious. Now, almost 40 years later, I've got Numberphile (and a few other like channels) still doing the same for me.

Dr Hannah always finds solutions to problems I never thought existed

I think Dr Hannah just did an ASMR video instead, because this made me feel so relaxed.

Hannah, Brady, I'm sorry but this video is absolutely pointless. When you have a ham sandwich in your possession, there are no problems in the world whatsoever.

@insightfulgarbage

6 жыл бұрын

-Ken M.

@paulgoogol2652

6 жыл бұрын

until you eat it and realize you are out of sandwiches. a human is a fascination problem generator.

@mistletoe88

6 жыл бұрын

well you do when you have to share it with someone else and they want exactly the same portion.

@tomhoffs8209

6 жыл бұрын

rusty_frame knock them out and eat the whole sandwich yourself. Problem solved easily.

That was one of the first Numberphile videos I completely understood without glazing over or rewinding or hearing the twilight zone intro in my head.

It's like Foghorn Leghorn & the Cat trying to cut the Worm in half.

Yes, more of the lovely Dr Fry always love her explanations for things :)

A man named Tu(r)key helping us eat ham sandwiches. What a time to be alive

I love this topic. Thanks to all you Numberphile team.

Hannah helping us again with fairness in food cutting problems!

I'd like to see the illustration of the three ingredients' bases being co-planar, i.e. resting on a table, arranged in a triangle. They have different thicknesses, so it can't be a horizontal cut. Interesting to see how different the pieces end up based on how they're placed apart from each other.

@2bfrank657

2 жыл бұрын

I suspect the theory allows three planes to be each divided in half, but not necessarily three volumes. If this is the case, then the sandwich is not really the best example of or name for the theory. When cutting a sandwich in half, you're not just worried about the area of each half, you're trying to get equal volumes.

@ZayulRasco

2 жыл бұрын

For this case, the cut bisecting each slice perfectly in half (volume-wise) would be almost horizontal, with a slight yaw so that it passes through the center of mass of each ingredient. This accounts for their different thicknesses.

@MichaelDarrow-tr1mn

Жыл бұрын

@@2bfrank657 no, it is three volumes

I love how they are both laughing through this because in reality, trying to seriously make this sandwich and split in half is ridiculous xD

Hannah Fry, is owesome and one of the best.

All Hannahs I know of on youtube have now had a cooking with Hannah segment! (And some of you now know what I used to watch on KZread)

Something about this problem really reminds me of the intermediate value theorem... But generalized...

@xavierstanton8146

2 жыл бұрын

I believe the Intermediate Value Theorem is used in the more general proof.

There is something i don't understand. You can find a cut on the first piece of bread for angle but are they going trough a unique point? If not, how can you guarenty the continuity of the quantity of Ham on each side of the cut ? And if it is not continuous then how can you prove that there is an angle which cuts the Ham in half? Thanks

@tom4794

4 жыл бұрын

I'm not sure [EDIT: I was wrong, see below!], but I think yes, all those "halving cuts" pass through a unique point, which is the geometric center (the center of mass, the balance point, if you will). Which means that you can find this point by taking the intersection of two arbitrary (non-identical) halving cuts. And any line that passes through this center point is a halving cut. (Thus we can do the rotation needed to find the half-cut of the ham.) Intuitive proof: it's clear that you can balance any 2D shape on a point. Now imagine a cut through that point. If either half was larger than the other, the whole shape would have tipped over in that direction rather than being balanced. Thus each half must have equal size. This generalizes to more dimensions (e.g., the center point of a 3D object like the slice of bread). [EDIT: That was wrong. The two halves don't necessarily have equal area: I didn't consider leverage. So all of this is wrong!] The entire ham sandwich problem then is "just" finding the center points of the three objects, then cutting along the plane containing these three points. (This plane is unique unless all three points lie on a straight line. With four objects, the cut would no longer be possible in general - only if at least two of them happen to align like that.)

@leif1075

2 жыл бұрын

@@tom4794 yea but when you alogn the knife so the ham ks cutnin half it changes the alignment kf yhe bread so it can't cut both the bread and ham in half..see what I mean??

These two people have amazing chemistry.

The most precious cuts were made when my sister and I both wanted something. One of us got to cut but the other chose first!

@FlyingSavannahs

3 жыл бұрын

Hannah is in a video explaining the "Everybody's happy" algorithm that refines the "you cut, I'll choose" method for more than two people. I think it's a Numberphile video as well. Shouldn't (Shan't?) be too hard to locate. Worth the effort.

I love Hannah!

@davidwuhrer6704

6 жыл бұрын

Lucky sod.

Let me just point out, without drawing any conclusions, that this video is in "Women in Mathematics" Numberphile playlist and it involves making a sandwich. I'll say no more.

Hannah once again solving a problem I didn't know I had

Nice production value on this video.

Banach, not Banarch!

@dan-gy4vu

6 жыл бұрын

Banach turkey

@laptok

6 жыл бұрын

Jednak już ktoś zauważył wcześniej :)

@jakisid

6 жыл бұрын

I bet the first to notice the typo were the Poles =) ps. myślałem, że będę pierwszy / thought I'd be the first

@jareknowak8712

6 жыл бұрын

O, ktos juz zauwazyl. A to znaczy ze ktos z Polski to oglada. Oglada i rozumie. Rozumie czyli zna angielski. Oglada, zna angielski, rozumie. To znaczy ze nie jestem jedynym myslacym Polakiem! A juz stracilem nadzieje! Co mieliscie dzis na obiad? U mnie smazona kura, drob to nie mieso wiec nie szkodzi ze w piatek... :)

@PerseEki69

6 жыл бұрын

I think it's Barney, or was it Branagh?

Two notes: 1. The "solution" in this video is obviously incorrect or at least incomplete. As soon as you start rotating you no longer evenly split the bread. The real proof for the 2-dimensional case is a bit more complicated but similar at least. However, the 3-dimensional case is quite a bit harder and is normally reduced to the Borsuk-Ulam-theorem. This problem isn't as trivial as the video makes it out to be. 2. A lot of people in the comments think you can just take the centers of mass of the three objects and take the plane that goes through them. However, a plane through the center of mass does not necessarily divide an object in half (neither by volume *nor by mass*). In fact, this already fails for simple triangles in 2D. For a convex body of uniform density you can get up to 1 - 1/e on one side of a hyperplane through the center of mass (in the limit as the dimension goes to infinity).

@iamcurious9541

6 жыл бұрын

I was trying har to come up with an counter example. Thanks for the tipp. Your comment is actually helpfull

@jbinmd

6 жыл бұрын

Regarding Point 2, I assume each slice is homogeneous in the z dimension (aka the food plane). We then arrange the slices so they're coplanar and then slice. If homogeneity in z doesn't hold, the solution is to put the sandwich in a blender and then divide.

@EebstertheGreat

6 жыл бұрын

You are misunderstanding the 2-D argument here, which is trivial and makes use of the intermediate value theorem. Given _any_ angle in the plane, there exists a cut at that angle that divides the bread in half. Moreover, if the bread is bounded by a Jordan curve, the function from the angle to the line for the given cut is continuous. Now either all of those cuts (that evenly divide the bread) also divide the ham in half, in which case the theorem is satisfied for all angles, or at least one cut leaves more than half the ham on one side, while another leaves more than half the ham on the other side, in which case the theorem is satisfied for at least one angle by the IVT. For the 3-D case, things do get more complicated. The idea is to use rotation about another axis, but assuming the bread and ham either have finite thickness, do not occupy the same plane, or both, you can't just use the same line you did before with a new angle to define your new cuts, so the IVT is insufficient. As you say, the Borsuk-Ulam theorem is necessary.

@TKNinja37

6 жыл бұрын

Of point 1, it's argued that for any angle of the knife, there is a line that halves the first slice. So if I took two such lines and find where they intersect, it's a point. Shouldn't, then, any angle of line through that point evenly halve the slice? (I'm assuming the bread is of even thickness and a single, perpendicular cut, as one normally does.)

@smithmcscience4526

6 жыл бұрын

I agree, and now I'm confused. Are they thinking of the bread and ham as 2D objects? Because this clearly does not work if they have a volume.

There is a HAM SANDWICH theorem... Wow, I don't think Graham's number blew my mind as this one did. 4:32 - BANACH, Stefan Banach, great Polish mathematician!

I'm a simple man. I see Hannah Fry, I click on the video.

The fact that it works for up to three is due to the three ways to reposition an object in 3D space -- pitch, yaw, and roll. Does this mean that in 4D space, there is always a way to cut 4 objects exactly in half?

I’m suppose to be studying for finals...how did I end up on this sandwich video?

@maxguichard4337

5 жыл бұрын

What if this is on the exam! BTW how did they go?

I think its pretty clever to precisely divide things given the limited tools to measure with, the more things you put in the more precise it becomes, its measuring itself!

Hey i listened to your interview on NPR! Loved it!

This only applies to spherical sandwiches in a vacuum...

So in 4d you can add a lettuce In 5d you can add some sauce etc.?

@Selektionsfaktor

6 жыл бұрын

o O 0 Anyone got a link to where I can order a 4-dimensional knife?

@frechjo

6 жыл бұрын

Selek, I heard in the last kitchen cabinet in Hilbert's Grand Hotel there are ndimensional knives. There's a countable infinite number of them, so if you take one they might count them and notice one missing though.

@davidwuhrer6704

6 жыл бұрын

I think it only works for up to three dimensions.

@alexwang982

5 жыл бұрын

No, sauce can be spread out, you can do it in 3 dimensional

02:52.00 Hannah: ''Welcome to Cooking with Hannah!”

So completely in love with Hannah Fry...

From 3:25 how can you just rotate the knife about a point? It won't keep the sandwich in half every time. Won't the point of rotation will keep on changing? Please help.

@dixie_rekd9601

6 жыл бұрын

it wont cut the sandwich in half every time, just 1 time... its pretty clearly explained in the video , practically its pretty much impossible noones hands are THAT steady but theoretically its pretty trivial

@VAFFANFEDE18

6 жыл бұрын

To explain it better we can agree that a cut is a function of 3 varaibles 1) position 2) plain rotation 3) angle rotation as they showed in the video Every object cut in half gives us an equation so 3 object=> single solution (same logic in every dimention I think)

@misterhat5823

6 жыл бұрын

Federico Is it three variables? It takes two to describe the position. You'd need X and Y from a given center point.

@LechuvPL

6 жыл бұрын

The point of it that is you don't just rotate it about a point, but as you're rotating it you need to slightly move it, to the point, where your bread is divided exactly in half. Also, you need to make sure that it's continuous - teleporting the knife is forbidden, but it's logical that when you rottate it a little bit, an area changes so little that you need to move it also a little bit

@GarbageGamer74

6 жыл бұрын

You're correct, rotating about a point does not work. This video unfortunately doesn't explain the mathematics properly. The proof in the wikpedia article is also incomplete. Both fail to consider that the cut lines for object 1 might "orbit" a region of the plane (in fact it cannot, but the arguments fail to show that). I came up with a better proof but KZread comments are an inadequate medium. :) Briefly, in 2D, for a given cut angle theta, define x1(theta) as the displacement of the half-cut line above the origin for object 1, and x2(theta) as the same for object 2. Crucially, x1(0)=-x1(180) and x2(0)=-x2(180). This is enough to show that there exists a theta such that x1(theta)=x2(theta).

but if you change the angle of the cut doesnt that mean the cut through the other bread and the ham is going to be at a different place which then doesnt necesseraly cut it in half?

@jaykoerner

6 жыл бұрын

yes, that would be true if the cut wasn't able to move also

@VAFFANFEDE18

6 жыл бұрын

To explain it better we can agree that a cut is a function of 3 varaibles 1) position 2) plain rotation 3) angle rotation as they showed in the video Every object cut in half gives us an equation so 3 object=> single solution (same logic in every dimention I think)

@D4rKminer

6 жыл бұрын

Federico Mangano thank you

@jaykoerner

6 жыл бұрын

Federico Mangano the easiest way to simplify it is three spheres in space, no matter the position they still have a plain that cuts all three in half

@VAFFANFEDE18

6 жыл бұрын

Of course, the one passing through the three centres

Numberphile do love their ham sandwich problem. Pleasure to watch Hannah though.

A lot of my fav things in this video

A strong contender for _most underwhelming Numberphile vid ever._

@solstice2318

5 жыл бұрын

Yes, but it does give you the rare opportunity to evaluate the Hannah attraction effect, which is quite as important as the whole program though you could never estimate it before This rare footage.

So in 4 dimensions, could you have a sandwich with 4 ingredients (including the bread) and still cut it equally. Or perhaps it will be easier to imagine a 2 dimensional sandwich with only 1 piece of bread

@allylilith5605

2 жыл бұрын

well, 2 dimensional with 1 piece of bread and 1 piece of ham is bascially just everything in the video before they start cutting in an angle. and I assume that 1 more dimension means 1 more layer, yes

Arthur Stone is credited with the discovery of flexagons when he was a student at Princeton. His friends John Tukey, Richard Feynman and Byrant Tuckerman became interested in flexagons and formed the Princeton Flexagon Committee. Stone and Tukey wrote a paper on the Ham sandwich theorem a couple of years later.

I love their chemistry

I feel bad for the Mathematician whose legacy is "the ham sandwich theorem"

@Czeckie

6 жыл бұрын

three of the mentioned mathematicians are actually all time math superstars. Steinhaus, Banach and Tukey. Stone is famous as well, but surely less known.

@Oneiroclast

6 жыл бұрын

Would you prefer the Hairy Ball Theorem to be your legacy then?

This seems very similar to the intermediate value theorem

@uchihamadara6024

6 жыл бұрын

I love that theorem, it's so elegant yet simple

@KenCubed

6 жыл бұрын

You're right, the intermediate value theorem is used in the formal proof.

@ColoredScreens

6 жыл бұрын

That's the logic you use when deducing the part of "here, all the bread is on one side, and here it's all on the other side so there must be a place where it's equal", which you technically don't know without confirmation that the value is differentiable at all points in that range.

@uchihamadara6024

6 жыл бұрын

Colored Screens Does it have to be differentiable? Or just continuous?

@TheManxLoiner

6 жыл бұрын

@Uchiha. Differentiable => continuous => intermediate value property. So continuous is sufficient. (Note, however, that continuity is not *necessary* in order to have the intermediate value property. However, a function which is not continuous but which does have the intermediate value property is pretty weird.)

I genuinely learned something from this

I'm so totally in love with Hannah. What a class act. :)

Banach, not Banarch :-(

@SchiwiM

6 жыл бұрын

Barnach?

@PerseEki69

6 жыл бұрын

Barney?

@shayan_ecksdee

6 жыл бұрын

Barnard?

@ICECREAMan2991

6 жыл бұрын

Ed?

@IllidanS4

6 жыл бұрын

Hanach

Are those supposed to be planes or each slice of bread and ham are volumens?

@xway2

6 жыл бұрын

Since they angle the cut as the last step, I think it's safe to assume they are meant to have volume.

@fyermind

6 жыл бұрын

Haven't read the paper yet, but it looks like you can prove the existence of a plane which bisects N N-dimensional objects in N-space and this is the rough description of the proof for N=3

An intuitive way to think of this is the following. For each piece off bread/ham, find it's center of mass. Then, any cutting plane that goes through the center of mass will exactly cut that piece of bread/ham in half. Now you just have to find the cutting plane that goes through all 3 of the center of masses. This can always be done because a plane can always be defined by any 3 points.

@siosilvar

4 жыл бұрын

I thought the same thing, but it's not true that any cut through the center of mass guarantees an even split. Take a hammer, for instance - even with a weird, uniformly dense hammer, there's cuts through the center of mass that don't divide it evenly.

the comment about mathematicians spending tea/lunch break time hyper-productively was quite on point Banach was one of the profesors of the famous polish Lviv School of Mathematics, from which originated the Schottish Book - famous book of math problems & solutions created over years of math profesors and students scribiling them down as challenges (with prizes) for each other in a shared notebook at the "Schottish" caffe (where they all spent unreasonably large ammounts of time - for some reason it became a hotspot for brainstorming and disscussing maths at all times of day everyday)

I really want a ham sandwich right now

@nerdbot4446

6 жыл бұрын

I guess you have a Ham Sandwich Problem ( ͡° ͜ʖ ͡° )

@stumbling

6 жыл бұрын

No.

@wouldhave4998

6 жыл бұрын

I really want a Hannah right now

@brewbrewbrewthedeck4138

6 жыл бұрын

I really want Hannah’s ham sandwich right now ifyouknowwhatI’msayin’ ...

@XenophonSoulis

5 жыл бұрын

I want half a ham sandwich.

I'm not questioning the theorem, obviously, but I didn't follow the argument at around 3:36. Assuming we have found a position for the knife which divides the bread in half, it doesn't follow (does it?) that if we then rotate the knife round an arbitrary axis it will still divide the bread in half. For some axes this will obviously not be the case (e.g. if the axis is near one of the corners). If there a proof that there must be *some* (at least one) axis for which it is the case?

@VAFFANFEDE18

6 жыл бұрын

I think that the thorem works this way we can agree that a cut is a function of 3 varaibles 1) position 2) plain rotation 3) angle rotation as they showed in the video Every object cut in half gives us an equation so 3 object=> single solution (same logic in every dimention I think)

@williamrutherford553

6 жыл бұрын

I believe the issue is you're assuming the point of rotation is constant, when in fact it is arbitrary. Instead of thinking of a knife, think of a line that moves infinitely in both directions, and changing it's position over the sandwich. Given a line that divides the bread in half, there must be some point on that line (in the center) where a rotation maintains half on each side. The idea of a "corner" case doesn't exist, because you could just pick a rotational point closer to the center.

@DavidB5501

6 жыл бұрын

+William Rutherford I'm only 'assuming the point of rotation is constant' because that is how it is presented in the video. 'Given a line that divides the bread in half, there must be some point on the line (in the center) where a rotation maintains half on each side'. Maybe, but that seems to be the main thing needing proof. I guess it would be some kind of fixed point theorem. Also, I doubt that the axis of rotation would always be at the center of the line. Consider a circle with a long thin rectangle projecting from its circumference. A line drawn from the center of the circle through the center of the rectangle would bisect the combined figure, but another line rotating round the midpoint of that line would not in general bisect the figure.

@williamrutherford553

6 жыл бұрын

It doesn't require a fixed point proof because the point isn't fixed. It's arbitrary. Draw a line where the bread is all on the left. Draw a line where the bread is all on the right. The point where those lines intersect is the point of rotation. Therefore, you can rotate the first line to be the second line, and at some point it must divide the area equally. You don't need to prove that. It's just a fact that two lines intersect at a point.

@DavidB5501

6 жыл бұрын

+William Thanks. Re-reading what I said earlier, I think I explained my concern badly. I don't dispute that around any point as an axis we can rotate a line so that at some point in its rotation it divides the bread equally. That is almost self-evident, though I dare say a rigorous proof would require a bit more argument. My concern is that, according to Hannah in the video, we can rotate that same line round some point along it in such a way that the rotated line *simultaneously* bisects both the bread and the ham. This is far from obvious (to me, anyway). By assumption, before rotation that line bisects the bread, and by assumption, after rotation it bisects the ham, but what we need is a proof that it still bisects the bread as well. I can see that in some figures that would be possible. Most obviously, if the bread is circular, any line bisecting it must pass through the geometrical center of the circle, and if we rotate it round that center, it will continue to bisect the circle. So all we need to do is to rotate it round the center of the circle until it also bisects the ham! The same, I guess, would be true for other regular figures like a square or even a rectangle. But it is not obvious (to me) that the same would always be true for asymmetrical and irregular figures, like a piece of bread with ragged chunks torn out.

A similar problem came up for me on a picnic with a Scotch Egg. We thought about it for a bit and decided that the instinct of one of my friends was right-- you should be able to bisect N ingredients in N dimensions with one straight cut. But what if the ham isn't all connected? A better example might be currants in a cake since ham usually is fairly connected. You have a cake with say currants and chocolate chips, and you want two people to each get equal amounts of cake, chocolate and currant with one straight cut. You can cut through the chips and currants, and they can be distributed anywhere in the cake, which can be any shape. Or you could even say: forget about the cake, nobody cares how much cake they get but we want equal amounts each of chocolate, currants and blueberries-- that way the cake is just background and all three ingredients can be disconnected. I think you still can always do it in these cases.

This was so cool

Hammah

Easy answer. One slice of bread each then fold the ham in half and cut along the crease.

@poisonoushallucinations3168

6 жыл бұрын

MrID36 I’m sure they would much rather not put in the effort of taking apart the sandwich and remaking it, and would prefer trying to find out how to cut the sandwich in half using just one slice

@MrID36

6 жыл бұрын

The Changing Mob My point is that there are two slices of bread - one for each person; there's no need to cut them. Just cut the ham in half.

@Zooxheth

6 жыл бұрын

Not all slices of bread are created equal.

@FrostDirt

6 жыл бұрын

They said using ONE cut

@MrID36

6 жыл бұрын

FrostDirt My solution uses one cut.

I saw this immediately in terms of the centre of gravity of each bit. If you cut an object in half through its CoG each bit will of equal mass. Each CoG is a point in 3D space. There will always be a plane in 3-D space which will have all three points on. Job done.

@matteoschiavone5459

Жыл бұрын

I'm not sure that if you cut an object through its centre of gravity each half will of equal mass without other assumptions on its shape.

@jugbrewer

Жыл бұрын

that’s true, imagine a birthday card open at a 90 degree angle. the centre of mass is somewhere in the empty space between the two pages. you could center your knife on that point, then rotate the knife so that it was parallel to one of the pages. since the knife plane and the page plane are parallel they can’t intersect, meaning the knife won’t cut through one of the pages at all and it would cut through the other one. So one of the resulting pieces would have to contain all of the page that wasn’t cut, plus some amount of the other page. the two pages would be different sizes.

I like that its more visual than auditory.

The next 'cooking with Hannah' should be Thai fish in a bag xD

Ironically, the Generalized Ham Sandwich Theorem is solved by Turkey. So it's a Turkey Sandwich Theorem?

@oliverkolossoski1434

5 жыл бұрын

Stone and Turkey sandwich

I'm fully in support of Dr Fry's food dividing corner.

Those people are amazing!

"You'd almost think that maybe that was the most productive part of a mathematicians day" Awesome! (Even if I've misquoted by accident)

I wish I lived in 4d so I could split the butter equally as well!

🌷 awesome thank you!

The "magic slice" is just the plane that passes through the centroid of each ingredient.

Bananarch Tarski?

@jakeroosenbloom

6 жыл бұрын

Tobbzn yes

@christoffer1917

6 жыл бұрын

Watch Vsausce's video about the Bananarch Tarski paradox

@Krakkk

6 жыл бұрын

Banach not Bananarch lol

@sadhlife

6 жыл бұрын

Niemy007 _that's the joke

@elkiensad7003

6 жыл бұрын

Bananarch turkey

4:34 it's Banach not Banarch

@numberphile

6 жыл бұрын

Yes. Mistakes happen in the animation process but we’re ever grateful that we have a vigilant and vocal audience to remind us again and again and again.

@gfixler

6 жыл бұрын

The Banarch-Taski Paradox is a theorem in set-theoretic commentary, which states the following: Given a correction in anonymous space, there exists a decomposition of the correction by an infinite number of disgruntled subscribers, which are then put back together in different ways to yield infinite identical copies of the original correction.

Cute smart redhead + ham sandwich + Math concept = winning combo

I really liked this video.

I noticed that each additional object requires manipulation of the knife along one additional axis. Does this mean that a knife being manipulated in four spatial dimensions would be able to evenly cut four objects with any given overlap?

Hannah is the strongest waifu

@astherphoenix9648

6 жыл бұрын

William Morgan woo

@anjopag31

6 жыл бұрын

Profile picture checks out.

Waiting for episode 2 of cooking with hannah

This sounds like a great way to generally problem solve…