Ham Sandwich Problem - Numberphile

[@buzz4162]

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

[@Jamifa007]

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

[@bestopinion9257]

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]

This was literally what I predicted about 30 seconds in.

[@Darcy783]

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]

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

[@revblade]

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]

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]

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]

@@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]

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

[@Talaxianer]

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)

[@leahandtabi1]

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

[@MLB9000]

Put sandwich in blender, use measuring jug. Solved.

[@machiavelli326]

Think smarter not harder

[@joshandrews8913]

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

[@TheZooropaBaby]

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

[@missionpupa]

@@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]

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

[@GlutesEnjoyer]

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]

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

[@WujuStyler]

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

[@Serfuzz]

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

[@petros_adamopoulos]

The misspelling is unacceptable.

[@brokenwave6125]

British people have a speech impediment so they can't really tell when there is an R in the word or not.

[@PerseEki69]

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

[@matthewchampion8214]

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

[@chriswebster24]

Pervert

[@jacobschiller4486]

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

[@tc98826]

I'd love to have bed time with Hannah.

[@gyes99]

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

[@Septimus_ii]

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

[@floggyWM1]

@@Septimus_ii is that how AA meetings work

[@karolbomba6704]

@@floggyWM1 Anonymous Alcoholics?

[@floggyWM1]

@@karolbomba6704 yes

[@karolbomba6704]

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

[@ristopoho824]

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]

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

[@NoriMori1992]

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

[@whozz]

Hannah's voice is so lovely

[@bunpeishiratori5849]

There's something sexy about a British accent!

[@brianmiller1077]

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

[@gasser5001]

Hannah is so lovely*

[@whozz]

DJ Deckard Cain Of course

[@toferj7441]

I completely agree. Smart women are beautiful. :)

[@kev_whatev]

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

[@MrJaponesdoido22]

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

[@nathana.4467]

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

[@CraftQueenJr]

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

[@GlobalWarmingSkeptic]

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

[@MIbra96]

+Global Warming Skeptic Thanks for the laugh mate! xD

[@nicemelbs]

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

[@thinkbolt]

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

[@jotabeas22]

It sounds like a documentary on how kurds are treated.

[@jotabeas22]

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]

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]

Could be also name of punk band

[@agr.9410]

It’s the hexaflexagon bois

[@CARTONofSUKI]

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

[@lucamuscarella4085]

Jack Carton size this one needs go viral

[@ISenjaya71]

*vsauce music intensifies*

[@cOmAtOrAn]

Only if you take the Axiom of Choice.

[@gremlinn7]

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]

I'd rather have two identical Hannahs!

[@cubethesquid3919]

Yay more Hannah!

[@cbarre9937]

Hello

[@AalbertTorsius]

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

[@sofarky]

omg

[@unreal-the-ethan]

Ba dum tss

[@Frosty-oj6hw]

I guess this has something to do with degrees of freedom? The first degree of freedom with the knife is position in the plane. The second degree is its rotation relative to that plane, and the 3rd is angle of slice.

[@joyitadarling5815]

This video is so adorable and interesting at the same time

[@happylittlemonk]

It all depends which side your bread is buttered ;)

[@Triantalex]

WYSI

[@douglasdunkle1592]

There is a flaw with this reasoning though. The angle needed to cut through the upper level of bread would also slice through the lower levels at an angle. The initial cuts were thought of as being perpendicular to the cutting surface. A cut at an angle would no longer guarantee the lower levels are cut in half.

[@kol2456]

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

[@davemarm]

Isn't his name Tukey, not Turkey?

[@guillaumelagueyte1019]

I was surprised to not hear a rebound on that name

[@st3435]

With stone ground bread

[@karlmuster263]

Because that would be disrespectful.

[@teomanyalcnkaya5072]

CcC Türkler geliyor

[@orbital1337]

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]

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

[@jbinmd]

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]

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]

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]

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.

[@9time007]

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

[@HappyBeezerStudios]

It would be fair share.

[@want-diversecontent3887]

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

[@misterhat5823]

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

[@argh1989]

Just imagine going on a date...

[@livedandletdie]

Just imagine...

[@cmmp5510]

Quantic answer: replace the sandwich by a chips packet (discret quantities) and divide the total by two.

[@SwiftGames_]

MORE HANNAH

[@acoupleofschoes]

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]

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]

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]

@@2bfrank657 no, it is three volumes

[@pierre-emmanuelwulfman104]

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]

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]

@@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??

[@Rakii27]

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

[@citricdolphin]

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?

[@theMG174]

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]

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.

[@allie-ontheweb]

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

[@benc8386]

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.

[@yuvalco]

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

[@xavierstanton8146]

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

[@pleindespoir]

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

[@Sir_Irwin]

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

[@KatieK-OnYt]

I prefer to think of it through physics; each of the three objects have a ceter of mass, which is a point. Any cut through that point, no matter the angle, separates the object into two parts of same mass. If the density is consistent throughout, then that means the volume of each half is the same too. With any three points you can always set up a plane, that passes through all of them.

[@Utube4chuck]

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...

[@delaroyas]

My solution: each of the three sandwich parts have a center of mass (a point where any cut in any 3d orientation will produce 2 equal parts). Their is always one plane passing trough the 3 points, cutting each of the three parts in half.

[@tiberiu_nicolae]

My thought process as well

[@YTAnihilati]

Banach, not Banarch!

[@dan-gy4vu]

Banach turkey

[@laptok]

Jednak już ktoś zauważył wcześniej :)

[@jakisid]

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

[@jareknowak8712]

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]

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

[@mjones207]

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.

[@ampPLrant]

I'm really unsatisfied with the argument in 2D. The issue is that it seems like you claim you can pick some well defined point in the first slice of bread and then rotate, and I don't think that's true. Don't you need to pick the angle and the position of the cut at the same time since a motion in either will affect the fairness of both the bread and the ham? And doesn't this same argument extend to 3D?

[@agapiosagapiou]

It seems that it have limitations. I see that for each object that added on the stack is also one axis of freedom needed to use. One slice:move the cut on one axis only Two slices: one axis and rotation Three slices:one axis, rotation and tilt. I don't know if is possible to use more.

[@DHGameStudios]

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

[@Sam_on_YouTube]

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

[@DHGameStudios]

@Sam Make it so.

[@dragoncurveenthusiast]

My intuition is that the cutting plane should be the plane through the centers of mass of the three objects. For each object (the two bread slices and the ham) the cutting plane will go through its center of mass and thus cut the object into two halfs of equal mass.

[@dragoncurveenthusiast]

This way it's also easy to extend into n dimensions: In n dimensions, you cut along the hyperplane through the n centers of mass of the n objects. Each object will be cut into two halfs of equal mass, because the cutting (hyper)plane went through its center of mass.

[@kungfurabbits]

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

[@aaronboor2818]

All objects have a center of mass (or volume). For three objects, those three centers exist on a single plane. That plane equally bisects all three objects.

[@katowo6521]

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

[@Selektionsfaktor]

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

[@frechjo]

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]

I think it only works for up to three dimensions.

[@alexwang982]

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

[@gnembon]

I think the theory also tells how to define this cut. In an N dim space, any object has its center of mass defined as a point in this Ndim space. Any N-1 dim projection that includes that point divides that object into two equal halfs ;) (Physics and center of mass theorem? maybe). To define a N-1 dim hyperspace, you need N points, which means that any N objects that center of mass lies on these N points will be evenly cut with this manifold. With the 3d example, if we find the centers of masses of both slices of bread and the ham slice, these 3 points define a 2D space which is the cut we need.

[@keel1701]

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

[@planktonfun1]

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!

[@ThomasGodart]

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

[@MrAlRats]

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.

[@D4rKminer]

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]

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

[@VAFFANFEDE18]

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]

Federico Mangano thank you

[@jaykoerner]

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]

Of course, the one passing through the three centres

[@Robi2009]

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!

[@alfiechenery4146]

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]

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

[@MisterItchy]

If you tilt the knife like you did, you don't get half of each slice of bread.

[@DavidB5501]

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]

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]

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]

+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]

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]

+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.

[@miri.mayhem]

The cut should be easy to construct: Each piece has single point - a center - where all cuts halving it goes through (if you choose to define halves by weight, it's the center of gravity). 3 Pieces give us 3 points/centers. The cut to be constructed is the plane going through those 3 points.

[@baap2499]

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.

[@Great.Milenko]

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]

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]

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

[@LechuvPL]

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]

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 KZbin 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).

[@blaegme]

To make a strait cut through the half way point of 3 object is basically defining a plane that intersects 3 points which is one of the simplest ways to define a plane. However while doing this problem you don't start with Where those 3 points are.

[@IllidanS4]

Banach, not Banarch :-(

[@SchiwiM]

Barnach?

[@PerseEki69]

Barney?

[@shayan_ecksdee]

Barnard?

[@ICECREAMane1017]

Ed?

[@IllidanS4]

Hanach

[@AFR0PR1NC3]

Does this apply to more than 3 objects? For instance, if you added a slice of cheese to the sandwich and had 4 things to cut. Or, does this principal only work with 3 or less objects because we can only interact with 3 dimensions of space?

[@dannyarcher6370]

I don’t care how it’s sliced, there’s no butter on it.

[@SocksWithSandals]

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.

[@Blobcraft13]

This seems very similar to the intermediate value theorem

[@uchihamadara6024]

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

[@KenCubed]

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

[@ColoredScreens]

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]

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

[@TheManxLoiner]

@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.)

[@93lozfan]

you could calculate the center of geometry of each object, or mass if you're using a non-homogeneous material, and using those 3 points calculate the plane you need to cut on.

[@MrID36]

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

[@poisonoushallucinations3168]

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]

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]

Not all slices of bread are created equal.

[@FrostDirt]

They said using ONE cut

[@MrID36]

FrostDirt My solution uses one cut.

[@benjaminkatz4839]

Quick question: how do you know that there is a point where all lines will cut the bread equally in 2? For the ham and 1 slice, it is assumed that you can put the knife at one point and all lines will cut the bread in 2 equal. Therefore we can find and angle where it will cut the ham as well in 2. But why does such a point exist? The same question would occur for the line in 3D for the 3rd piece of bread.

[@henridelagardere4584]

A strong contender for _most underwhelming Numberphile vid ever._

[@solstice2318]

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.

[@herethere2091]

This sounds like a great way to generally problem solve…

[@Jonafridge]

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

[@lewismassie]

Well you just find the center point of each object, then mark those points in space. Draw a triangle between those points, then tesselate out into a flat plane. That is your cutting point. Though that does assume that they are of uniform thickness which is unlikely

[@deanst98]

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

[@SerioeserName]

1st slice toast: There is a point we can rotate the knife around and it will always cut it in half. Ham: Same " 2nd slice of toast: Same " So if we connect all three point through a plane it’s guaranteed that it slices them all in halves

[@Roxfox]

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?

[@diogeneslantern18]

Dr Hannah always finds solutions to problems I never thought existed

[@tylerkerr4043]

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

[@maxguichard4337]

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

[@ericgomis1041]

If I'm not wrong, we should assume that the sandwich is piled up, because if the pieces can be on the side without touching one another, that perfect cut it can be a way that it does not exist (sorry for my bad English)

[@halt1931]

Since the two pieces of bread are (theoretically) around the same, you could just cut the ham in half, give each person a slice of bread, and wrap their piece of bread around the ham to make a half-sandwich. Just rip and tear at the OTHER bread to make it identical to the other one.

[@WeArePharmers]

Is this assuming that all three planes are in parallel? Or does it work if the planes intersect?

[@maxnullifidian]

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

[@zorm_]

In any n-dim space, a hyperplane is uniquely defined by n points. The cut will be the only hyperplane that passes through all the centers of mass of each of the n objects

[@karotix5]

Hannah is the strongest waifu

[@astherphoenix9648]

William Morgan woo

[@anjopag31]

Profile picture checks out.

[@andrewkvk1707]

Every object has a center of mass. If you cut thru that center at any angle it will be half on one side and half on the other. If you have three points(in this case the centers of mass of three objects) in space; there will always be at least one plane that they all line up on.

[@fonno_]

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]

-Ken M.

[@paulgoogol2652]

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

[@mistletoe88]

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

[@tomhoffs8209]

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

[@dannyarcher6370]

A video made for Jews and Muslims.

[@ffggddss]

In n dimensions, it says there's a hyperplane [ (n-1)-dimensional ] that bisects the volumes of n, n-dimensional solids, no matter how they're placed in (Euclidean) n-space. There are constraints on the solids - something like that they have to be compact (topologically) and have "nice" connectedness properties.

[@WyandWombat]

You should be able to find the plane to cut through, by throwing each item in the air two times and filming with a high speed camera. They should all be rotating around an axis through their centre of mass and doing it twice you should be able to find the centre of mass for each item. You can then arrange them any way you want and cut through the plane defined by those three points. You could probably even build a machine that does it for you. This will however give you two sandwiches with the same mass, not neccessarily the same size.

[@michaelgill6989]

Actually though, one should be able to find the center of mass (as you describe :)) or the of area (if you want the same size halves) for each of the three pieces. A plane defined by each center must divide all three pieces in half. I think you're on to something...

[@McDinoh]

Solve the problem by laying the ingredients out on the board, cut each individual part equally, then assemble.

[@AlejandroBravo0]

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

[@xway2]

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

[@fyermind]

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

[@logicalfallacies3352]

What constitutes a sandwich for this problem? We could assume it to be three squares of material arranged in an xyz field, such that the second square shares at least one set of xy coordinates with the first and the third shares at least one set of xy coordinates with the second i.e. the squares are stacked one atop the previous. However, we could then construct a theoretical sandwich where the middle slice shares a singular xy coordinate set with the bottom slice at one corner and shares another singular set with the top slice at an adjacent corner. This would result in a sandwich with a requisite cut along the edge of the middle slice - which could not be made to bisect all three slices. And if we assume that all three slices must share a set of xy coordinates i.e. each is either above or below the other two, we can still construct a counterexample by arranging the slices such that they are fanned radially around the shared xy set and the center xy coordinate of each slice is 120 degrees from the center xy set from the next slice.

[@pooya130]

I love Hannah!

[@davidwuhrer6704]

Lucky sod.

[@mrjbexample]

I can't see how rotating the knife will keep the same proportions of both the top slice of bread and ham. The image at 4:09 shows what I mean, I think one will have to change.

[@JAN0L]

4:34 it's Banach not Banarch

[@numberphile]

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]

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.

[@SashaBitbroyt]

Hannah Fry, is owesome and one of the best.

[@XxPlayMakerxX131]

I really want a ham sandwich right now

[@nerdbot4446]

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

[@stumbling]

No.

[@wouldhave4998]

I really want a Hannah right now

[@brewbrewbrewthedeck4138]

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

[@XenophonSoulis]

I want half a ham sandwich.

[@tostkon8067]

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)

[@mattsadventureswithart5764]

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

[@thebuzzah]

Hannah helping us again with fairness in food cutting problems!

[@evanchandler508]

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

[@jadenruanes1858]

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

[@maxhaibara8828]

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

[@oliverkolossoski1434]

Stone and Turkey sandwich

[@Dawwwg]

There's way more to cutting the sandwich in half, like the breadcrust-crustiness distribution and perhaps the same with the ham-texture distribution and what about diagonal vs rectangular slides preferences ?