Equally sharing a cake between three people - Numberphile

Audible (30-day trial, free audio book): www.audible.com/numberphile
More links & stuff in full description below ↓↓↓
This video features Dr Hannah Fry.
More videos with Hannah: bit.ly/hannah_vids
Hannah's website: www.hannahfry.co.uk
Her book mentioned is "The Mathematics of Love".
Envy free cake cutting (the paper mentioned): arxiv.org/abs/1604.03655
A clarifying note from Hannah: n^n^n^n^n^n is the maximum number of cuts in the n person case.
Something we could have made clearer... Bob's trimmed piece still carries the status of his "first choice", so if Charlie leaves it, Bob will (must) pick it... otherwise he'll pick the one that was his original "second choice"... The upshot of this is that there's no chance Alice will be left with the trimmed piece after everyone takes their first piece. And Bob remains envy free under these conditions too.
A trilogy of mathematical cake cutting videos... bit.ly/CakeCuttingTrilogy
Editing and animation in this video by Pete McPartlan
Discuss this video on Brady's subreddit: redd.it/72jyww
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumberphile
We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science.
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[@numberphile]

Try www.audible.com/numberphile and check out The Mathematics of Love by Hannah Fry Envy free cake cutting (the paper mentioned): arxiv.org/abs/1604.03655 --- A clarifying note from Hannah: n^n^n^n^n^n is the *maximum* number of cuts in the n person case.

[@rosefeltch6313]

14 minutes ago?

[@ozdergekko]

[@johnox2226]

Time travellers!

[@johnox2226]

Numberphile What about all of the crumbs after making 'n' cuts? Is there then a new system for dividing these up evenly, seeing as no cake can be wasted?

[@niceyoutubecommenter6840]

Can I ask something? Why not just have Alice cut the cake to thirds, Bob cut each third in half, and then Charlie picks two sixths, then Alice and then Bob?

[@shubhankarpathak1786]

I used this and everyone left...I got the whole cake !!thank you❤️

[@fabiofdez]

No envy either! More contempt if anything haha

[@decCoulissante]

Haha

[@RanaRitmica4995]

based

[@nsnopper]

Respect.

[@Triantalex]

??.

[@albertb8999]

No friends = No problems

[@stefanofiorani1408]

Albert Böschow = whole cake babyyy

[@branflakes2600]

"...no man, no problem." ~Joseph Stalin

[@Ray-mw1fx]

"no men, no problems" ~Feminists, Joseph Stalin = SJW.

[@proallnighter]

== Bigger waistline

[@KuraIthys]

Well, Stalin led a communist society that was originally formed because people felt the old system was unfair... So... Sure. ;p

[@Nacho-dj3zm]

that is the saddest cake i've ever seen

[@zacharyjune7510]

It's just a loaf of bread.

[@rachelx04]

The cartoon cake looked better 😂

[@eoghan.5003]

I quite like a fruit cake

[@user-qh5jk1mn5i]

me: :) cak: :((( me: :(

[@coldfya6739]

it's a cake from uk, what do u expect

[@ojaimark]

My favorite part about this is that it doesn't seem like it but it could actually be practical for settling children's siblings disputes. Not because the kids will appreciate the fairness and mathematical/logical beauty, but because they'll get so fed up with the long process they'll be more than happy to drop the squabble and just take a piece of cake.

[@gerardocedillo7735]

Plot twist: with all that cutting, you lose 1/3 of the cake in the crumbs that fall out

[@krzysztofbandyk168]

Well if 3 ppl get 0 cake then everyone got equal amount of cake

[@theXXskater]

just cut all the crumbs into even tinier thirds-of-crumbs

[@Adierit]

they also not accounting for the fact that the cake could have an unequal distribution of cake due to air pockets in the cake itself

[@skylimitua]

you don't, because when your friends go home, you can collect them all and eat

[@derikWG]

that's how communism work

[@WannaComment2]

I prefer the classic method: - cut however - argue over who deserves the big piece - spew childish insults - engage in mortal combat - winner takes it all

[@Madsy9]

I prefer the MAD method. - All participants get their own button trigger connected to a bomb located next to everyone - Who gets to slice the cake is determined by fair dice roll - Immediately after slicing the cake, if someone is unsatisfied, they have the option of using their button, blowing everyone up

[@freddieelohopea6961]

The russian roulette method is by far the best: -Everyone pulls the trigger -Winner takes it all

[@mr.gentlezombie8709]

Cut the cake. Realize pieces are unequal. Realize that giving yourself the biggest piece would seem rude. Realize that you now have an incredibly small piece, with everyone else getting more.

[@robs.5901]

In soviet russia cake slices you.

[@deldia]

Steal a Mercedes badge and thrust it into the top of the cake

[@philip2772]

Haha great idea actually

[@excitedbox5705]

Instructions unclear. Now there is a hood ornament sticking out of the top of the cake.

[@OALM]

Geniuzzz!!!

[@PavloOrynchak]

Alas, too dependent on precise placement at the center.

[@TSMC123]

😂👏👏👏

[@AbdulHaseeb-ze7pu]

my loved ones: please abdul we want to eat the cake it's been an hour. me on my birthday: so now you cut the trimmings-

[@BlobVanDam]

This is why you never invite a mathematician to a birthday party.

[@sansyboy4181]

but if you don't invite them, the birthday problems never get solved.

[@tahirul8578]

BlobVanDam what?! Lol, This is why you always invite one.

[@numberphile]

They won't want cake anyway... They're obsessed with Pi!

[@grmph]

BlobVanDam that explains everything

[@mikejohnstonbob935]

lol expecting a mathematician to solve a practical problem... pure mathematicians pride themselves on non practical problems

[@ar9n]

Let's use the dryest, shittest most crumbly cake in a cake cutting experiment shall we?

[@yogzoth]

Made me laugh imagining the producers actually having that conversation.

[@firefish111]

If that was 10 people... the cake would do the trimmings of trimmings of trimmings for you.

[@rymerws1]

Rude. Aran isn't getting any cake guys! More for us.

[@Insightfill]

They're going to cut up pudding in the sequel

[@somedudecalledben]

Mister kiplings manor house cake is legendary! Proper British classic

[@GiskardRevenlov]

This has to be one of the most confusing and convoluted way to cut a cake

[@keir92]

T O' no one said envy free would be easy

[@mr.rabbit5642]

Imagine proposing this at the party and everyone is like "wtf is your problem dude"

[@bnoel12345]

In fact, the process is so convoluted that it's easy to miss the fatal flaw. If Bob realizes ahead of time that he will have a free choice after Charlie picks, he could potentially sabotage the whole process to make sure Alice gets as little cake as possible. All he has to do is claim the smallest piece to be his "first choice", then cut off a huge chunk as his "trimming", and ultimately "change his mind" at the end and take one of the other pieces instead. By the time the oversized trimmings are distributed, both Bob and Charlie would get well more than their fair share, and Alice would end up with little more than scraps. For this reason, Bob should be forced to commit to his original rank order prior to trimming. That way if Charlie doesn't want the trimmed piece, then Bob will have to take it. Under no circumstance should it be possible for Alice to end up with the trimmed piece.

[@bnoel12345]

Presh Talwalkar also made a video worth looking up about how to split a cake fairly. He talks about using the Last Diminisher Method, which is a little easier to follow and works well for any number of people because the process can be applied recursively. I can't leave a link because my comment will be censored, but it's easy to look up. Once you find it skip to 2 minutes, 11 seconds for the three-person procedure.

[@arnavrawat9864]

Envy Free* That's a big deal

[@spaceddoggo3647]

Split it into 4, split the last 4 into 4, split the last 4 into 4, keep doing it and you'll have infinite food. World hunger solved.

[@amanduswestin9211]

That's the only thing in the bible that kind of makes sense :P

[@rock00dom]

Big brain

[@livesh684]

split the last into 4*

[@freds2150]

@@amanduswestin9211 what

[@googavo1d]

how do you know that 4 pieces are identical ?!

[@greendogg83]

Lesson learned, never share a cake with more than one mathematician, it'll take far too long

[@limetimemusic]

So this is what people who write exam questions spend their time doing

[@atlucas1]

You have no idea. Source: Write exam questions

[@DH-og5yr]

Govt grants hard at work.

[@Triantalex]

false..

[@moara4144]

Easy solution: buy more cake. More cake is always the right answer

[@jesseMadoo]

So, the mathematical formula for how many cakes you would need for any number of people N, would be N cakes. So much easier. And if anyone doesn't like it, the formula is N - F, with F being the number of peopke who can go get f****d.

[@mazharmumbaiwala7451]

I'm sorry to break it to you but, The cake is Lie!

[@yjrzidovvjsmmra9300]

@@jesseMadoo actually no, because Someone Can Argue Another Cake Is Bigger

[@billnguyen4308]

No need to get complicated. Just starve them, who needs cake? They all get 0 cake, that is equal.

[@freds2150]

Heck you can buy three cakes and give everyone one of them. Problem solved.

[@Rishkar-Peema-Pants]

Why would someone even consider sharing a cake?!

[@k.g.b.1150]

Because Communism

[@jimmykeffer7401]

Moral: never share cake with a mathematician

[@celtelf]

I would like, but I would mess with the 333 likes and ruin this video, so here goes my 1/3 like

[@celtelf]

@Random Person oh excuse me if I made a joke with mathematical division after another joke. Sacarms apart, all problems as far as I know are problems for living things so psychology, unless mathematics is just the foundation of universe simulation, then every problem would be indeed a mathematical one. In any case you just say something wrong or obvious and didn't contribute with this topic.

[@Grassmpl]

It's ok with me. I'll just eat a cake all by myself

[@Triantalex]

false..

[@markk9446]

After making so many cuts, most of the cake remains crumbled on the plate

[@metawarp7446]

Depends on the cake

[@AstroTibs]

This is the solution for three people and one ant colony.

[@ylette]

Divide each crumb using the same algorithm as for the whole cake.

[@liesalllies]

Obviously the best way to share the cake is to liquefy it in a blender and then portion it out into three equal portions. It's not that much less appetizing than this mess.

[@Skarpo89]

@@AstroTibs 😂

[@NP-gb5su]

You guys forgot the ultimate source of dispute: Who gets to eat the creamy flowers in the corner, or the piece of chocolate in the center? Great video! Thanks!

[@JN-lq4hs]

Then you do the same thing with each of the flowers and chocolates 🤣

[@scotte4765]

The first person just cuts the pieces with those features to be smaller, so a small piece with a flower is of equal value to a large piece without one.

[@NP-gb5su]

@@scotte4765 but my kids all want the largest piece with the largest flower and the largest piece of chocolate. I will talk about your suggestion next time

[@jacobwhkhu]

That’s the whole point of this problem, they did not forget, it’s you who did not understand. Since everyone values fairness and values differently, the proportion may look very different for each person, and this algorithm shown here in the video guarantees nobody envies the other based on their perceived values for each slice.

[@nikolayyurchenko5075]

Actually, there's a way to fairly split something between 3 people with just from 3 to 6 cuts. Explained in russian math magazine "Kvant". Better to talk not about cake (you can use geometry to split it equally) but about 3 outlaws splitting their loot. It may be money plus clothes and other stuff which cannot be simply equally split. So, first how to split it between 2 persons so that nobody has excuse to complain. The first person splits loot into two halves which seem equally valuable for him and the second person chooses the half seems more valuable for him. Second, splitting between 3 persons. The first person splits loot into 3 parts which seem equally valuable, as before. Then second and third persons choose most preferred part and least preferred part. There may be three options. 1. Second and third persons like to take different parts. Then they take these parts and the first person takes the last part left. Each one is happy. 2. Second and third person dislike the same part. Then this disliked part is taken by the first person. Left two parts are joined and divided anew between second and third person using described above rule for two people. Everyone's happy. 3. The most complicated case is when second and third persons like the same part and dislike different ones. We may sort parts by their appeal for each one of them (first is most preferred, last is least preferred). The second person may prefer 1-2-3, and the third may prefer 1-3-2 (other variants would be the same after renumbering). We take part 1 most perferred by both second and third person and let them to divide it using rule for two people. We're left with parts 2 and 3. Second person dislikes part 3, so he leaves this part for third and first persons to divide part 3 between them. Third person dislikes part 2, so he leaves this part for second and first persons to divide part 2 between them. Everyone is happy. So there's no need for n^n^n^n^n splits to make everyone happy. At most, you need just 6 splits for 3 persons.

[@YoniMek]

Why so few likes?! This is simply genius!

[@ExplosiveBrohoof]

Situations 2 and 3 don't lead to envy free divisions. In situation 2, person 1 will almost always be envious of one of their partners. The piles that were originally equally valuable to person 1 got split differently, meaning that person 1 does not consider the new piles to be equally valuable anymore. One of the piles will be more valuable, and therefore it will be more valuable than the pile that person 1 received (since the piles were originally evenly split). In situation 3, suppose pile 1 is split in a way such that person 1 feels that person 2 got a better deal than person 3 (so person 2's split of pile 1 is better than person 3's split, by person 1's standards). Then person 1 may end up envious of person 2's pile in the end, since person 2 got a better part of pile 1 and an even part of pile 2, while person 1 got an even part of pile 2 and an even part of pile 3.

[@yonimaor1005]

@@ExplosiveBrohoof Hmmm... you are correct. But the solution given by Nikolay still achieves Proportionality (i.e. each gets at least 1/n of the the value of the cake).

[@connorjensen9699]

You can’t un cut a cake though

[@numberphile]

Something we could have made clearer... Bob's trimmed piece still carries the status of his "first choice", so if Charlie leaves it, Bob will pick it... otherwise he'll pick the one that was his original "second choice"... The upshot of this is that there's no chance Alice will be left with the trimmed piece after everyone takes their first piece.

[@shotguntornado]

Sounds like there's a lot of trust in this Bob fellow.

[@fUtal1mistake]

This solution doesn't work, because if Charlie had chosen the trimmed piece and Bob was left with untrimmed one and then picked the biggest trim of the trim, then Alice could say that now Bob's total piece is bigger than hers. So the task is infinite, they would just keep trimming that poor piece forever. UPD: I've looked it up. The fact who picked originally trimmed one defines who will then cut the remainder and the order of picking, then it's all fine. You've just mentioned one possible outcome w\o the other in the video.

[@pokechatter]

What if Bob's first choice is the smallest piece?

[@fUtal1mistake]

Selfridge-Conway procedure - google this. As the result, everyone is left with what they at least consider not less than any other total piece. If Alice would make them uneven intentionally and Bob would think they are fair, then Charlie would just pick the best one. In this riddle we consider people doing things THEY think is fair, that way it can be solved as envy-free.

[@Chris-jo1zr]

This all looks like a poor solution for selfish people who can't cut cake.

[@cicci0salsicci0]

What about the crumbs?

[@achu11th]

cicci0salsicci0 apply the same algorithm of course.

[@soufian2733]

Lick them

[@crackedemerald4930]

cicci0salsicci0 they are just noise

[@Halosty45]

Crumbs are a fight to the death. It's the only way.

[@KarinaVeiga16]

what about the smaller crumbs that come out of that

[@picksalot1]

I suggest cutting the cake into quarters, which is simple and easily accurate, and distribute 3 of those pieces. Then divide the last piece into quarters again. Once you do that a few times the pieces are so small, there is nothing to argue about.

[@cern1999sb]

These are mathematicians you're sharing with. You will never be able to cut the cake enough times for them to be satisfied

[@picksalot1]

@@cern1999sb Let them eat cake. 😁

[@Joe_Payne]

Actually there's an even better way than this. I just commented. Lemme find it to copy paste.

[@DaveWhoa]

just cut it into sixths and give everyone two pieces.

[@jtris01]

@@DaveWhoa Sixths requires you to cut into thirds.

[@pencrows]

"Aw James, it's my birthday. Stop doing math"

[@sillysausage4549]

Maths.

[@4ltrz555]

Meth

[@SgtAbramovich]

Different trios of professionals splitting a round cake: *Mathematicians* See above *Physicists* Search for the cake's fundamental frequency, make a sound exactly three times that frequency and cut it precisely at the nodes. *Engineers* Divide by three? Three is like so close to five, isn't it? And five is so close to ten, might as well cut it roughly in ten pieces, give each one three and give the remaining slice to the heftier one. *Philosophers* But really, what is a cake? *Lawyers* Discuss why you should get the whole cake instead, because you deserve it and others wanting it is a violation of your rights. *Chemists* Any way of dividing the cake fairly would be integrally destructive. *Bankers* What cake?

[@pedromaneiro123]

"But really, what is a cake?" Vsauce would say that.

[@4skullman]

But really, what is frequency?

[@TheMudDragon]

*Bakers* Bake two more cakes, everyone's happy.

[@SgtAbramovich]

@William Rush That would either produce 8 slices if you cut it like a pizza or a maximum of 4 slices if you cut each slice precisely once. I don't see how this would generate 6 slices. :/

[@someoneelse325]

What cake?

[@PlayTheMind]

Cake?! Should've been *π*

[@whitherwhence]

τ

[@DesmondAltairEzio]

but π will only get you half way.

[@dramawind]

This π and pie thing is getting really old...

[@homensdeti]

Boa

[@eddylebreton9544]

Dramawind no it's not

[@p.stahli4652]

4:13 „In general, the number of cuts that it takes to divide a cake between n people fairly is n^n^n^n^n^n „ But that is not true for the presented example of n=3?! Am i right?

[@yinan88]

It's the "maximum" number of cuts. So it's like an upper bound, not necessary the exact count. The question wants to find a bounded solution, so here is the bound, thus the solution.

[@EebstertheGreat]

All that is currently known about this problem for general n is that for sufficiently large n, the optimal envy-free cake cutting procedure requires at least cn² cuts for some constant c and at most n^n^n^n^n^n cuts. Obviously there is a lot of work to do to improve those bounds. But in the special cases of n = 1, 2, 3, provably optimal algorithms exist. In particular, for n = 1, 0 cuts are needed (you eat the whole cake yourself), for n = 2, exactly 1 cut is needed (called the "divide and choose" algorithm), and for n = 3, exactly 5 cuts are needed (the algorithm in the video), assuming the first two cuts don't immediately divide the cake into 3 pieces everyone is happy with. No optimal solution is known for any particular value of n > 3. It has also been proved that no general solution for connected components exists that can always be completed in a finite (even if unbounded) number of steps. That means if you want to ever finish cutting the cake, you have to accept either that some people may not get whole pieces or that some people may envy other people's pieces.

[@TheBouli]

@@EebstertheGreat What do you mean by "connected components"? Is the cake represented as a circle in the general solution you described in the beginning and the case for the cake being any connected 2D shape doesnt have a general solution?

[@EebstertheGreat]

@@TheBouli The simplest case is to use the unit interval, but it doesn't actually matter. It applies equally well to the circle or disk or indeed any simply-connected compact region.

[@brandonhenley3597]

@@EebstertheGreat It takes 6 cuts in the video. 3 to cut the cake into 3rds 1 to approximate 2 equal slices 2 to cut the remainder into 3rds In general you are right that it could take 5 if the cake is cut into 3 pieces disregarding original shape or desired shape. Just not what was shown here.

[@Eidako]

Place cake in blender. Blend at high speed. Pour the contents of the blender into three glasses, alternating between glasses as you do so, so each glass has an equivalent volume of cake.

[@__teles__]

It's not equal sharing, its poltical bargaining meaning everyone feels cheated but can't complain about it.

[@neilwilson5785]

That's the best kind of politics. The 2017 one, where you get a 'strong' middle-aged leader who tells you how to feel is vastly inferior.

[@rafaelfaria1960]

Jhb

[@Silverwind87]

This is why communism fails.

[@orionmartoridouriet6834]

@@Silverwind87 This is why authority fails

[@LeWildSister]

Lol true

[@jettycomlego]

Hannah fry is amazing

[@sleepysteev2735]

If Dr. Hannah Fry was my math teacher, I'd never be late to class.

[@farrel_ra]

me too!

[@nfineon]

Yes but you would spend each day being envious of mr fry, as you dont get any slice of that cake 😉

[@jugodats9990]

@kha0snv7 I think he means "cake", as in, Hannah's "cake", if you know what I mean

[@MJW238]

Call down simp

[@libbyd1001]

“There’s more to a cake than just size” I’ve been telling myself that for years. 😂

[@classicaf]

Adult humour 101

[@Thundernoob88]

This sounds like something a mathematician thinks up when he is drunk AF

[@nikkehautapelto1323]

Yeah rather high

[@unounk9415]

What if instead of Numberphile, it was called Number Pedo phile

[@liam12317]

Thundernoob88 I

[@nathanberrigan9839]

These problems often have business applications. e.g. A business dissolves and the three investors need to divide up the assets. Or three children dividing up their parents' estate.

[@geordonworley5618]

Nathan Berrigan: wow, very interesting point...ill need to remember that in the future

[@jackiejikariti8718]

Jigsaw: I want to play a game. There are three of you. Bobby, you must equally divide the cake amongst the three of you. You have 1 minute.

[@JonathanCook5000]

This, in reality, doesn't work. No-one is happy with the way they cut the cake to begin with!

[@m__6930]

exactly

[@missionpupa]

Wrong, this totally works. This has nothing to do with the cake being equal, of course no one can cut the cake perfectly. What this has to do with is allocating blame. If you are the agent that made a decision to take a piece or cut the piece, then you cant blame anyone else for getting a bigger piece.

[@0xCAFEF00D]

@@missionpupa Yeah I think people would get it better if you work through this in the negative example. You can really only blame yourself.

[@Vide0Browser]

This cake cutting experiment I think expects each person to be completely selfish and biased... Because if you are not and cut a tiny piece and 2 equally big pieces.... Too bad.....

[@Michael-ul2dg]

@@Vide0Browser usually kids are selfish and the rule for two cuts works very well

[@Kurzes_Spiel]

Mathematicians back then: Pythagorean Theorem Mathematicians now: Cake

[@rohangeorge712]

mathematicians in the future: how long will it take until we all die?!

[@Kadath84]

1:21 Alice, put the knife down, you frigging psycho!

[@thecatofficial3562]

Ahahahaha! You made my day :D

[@purple.cube.]

🤣🤣

[@Trifoon]

Bob is bart simsons

[@kevinscorner4883]

No offence, but if any of my friends tried to do this, they wouldn't be friends for much longer, thus only 2 people would be left and the problem would be solved.

[@pranishkhadgi2723]

an accurate representation of how siblings share their food

[@Adam-zf8hn]

Hannah is so perfect honestly

[@thickfingersw.1730]

Simp

[@borscht7743]

c00mer

[@schadenfreudebuddha]

Why are Alice and Bob constantly getting themselves into these situations?

[@schadenfreudebuddha]

holy sh! and Charlie! why didn't I see that?

[@mr.gentlezombie8709]

Yep.

[@Acid31337]

schadenfreudebuddha so how you explain Eve? )

[@mr.gentlezombie8709]

Alice, Bob, Charlie, David, Eve

[@jacobseymour823]

In cryptography and computer science, EVE represents an EAVEsdropper; someone who is observing an exchange between Alice and Bob.

[@maxhaibara8828]

maybe this is why those useless cutting knife gadget is invented

[@OntheFloorLaughing1]

now are they really useless then?

[@wellesradio]

And yet you've just proven their usefulness.

[@TheTororist]

Cake katting knife gajiks are awesome. Safety is number 1 priority

[@chongjunxiang3002]

Me: instead of cutting the cake, I cut my friends.

[@vedantneema]

or you could eat them whole

[@particleonazock2246]

I was drinking while reading this comment. xD

[@caribbeanman3379]

Here's another simple solution if you don't have a protractor: Use a tailor's measuring tape. Mark the center of the circle of the cake. Then run the tape around the outside of the cake to measure its circumference. Divide the circumference by 3 which will give you the arc size of the back of each slice. Use that arc measurement to mark off the three points along the outside edge of the cake and then just use your knife to cut from each outer point to the center.

[@jimallysonnevado3973]

That only works if the cake is uniform. But cakes are typically not, so each person will value each piece differently based on their preference. The goal of the method is to ensure that each individual is satisfied with their choices (ie, each one think they got a fair share based on their preference). And second, each person will think that any other person's share is not greater than what they have gotten.

[@caolanlennon3684]

Why does bob (a child presumably) have the hairline of a 78 year old pensioner. Even more questionable why is this in recommended to me two years later.

[@deivisony]

Lol

[@theworldoverheavan560]

Lol

[@alexandruvaratic2276]

loled so hard, I was drinking while reading this btw xD

[@chrishansen456]

It's Rooney.

[@andrewsparkes8829]

Why presume he is a child when he has a receding hairline and no other information has been given about his age at all?

[@kingpopaul]

Why cut the cake when you can just grind it and divide it by weight?

[@sakanagakyoko]

You are the best

[@bbgun061]

Who would like a scoop of birthday cake?

[@selfcentered3406]

Now then it wouldn't be cake would it?

[@kingpopaul]

Definition of cake 1 a :a breadlike food made from a dough or batter that is usually fried or baked in small flat shapes and is often unleavened b :a sweet baked food made from a dough or thick batter usually containing flour and sugar and often shortening, eggs, and a raising agent (such as baking powder) c :a flattened usually round mass of food that is baked or fried a fish cake Source: Merriam-Webster Still would be a cake.

[@nathanwilson7929]

Love it. I have a feeling each piece sans the drop piece is really close to actually being 30% than any price in the first round being 33%

[@maxcai3795]

"it's not about size" "Cut until equally preferable"

[@HawkOfGP]

THIS TAKES TOO LONG. I JUST WANT MY CAKE.

[@boerharms2209]

I'd just blend it, and divide by weight....

[@LePoseidon]

what if the weight is not divisible by 3 :)

[@LePoseidon]

@@anthonythomas For example when the weight is 200 gram...how will you divide the cake equally then?

[@LePoseidon]

@@anthonythomas yes so I mean the original comment just missed the point of this video. The main goal is to divide the cake envy-freely so that everyone is happy with their choices, not to divide it physically equally.

[@driziiD]

No one wants a blended cake 🙄

[@tomr6955]

Or just cut it and divide by weight?

[@caribbeanman3379]

Another solution with no protractors or measuring tapes required: Lightly mark the outer edge of the cake with your knife at regular intervals of one blade width, counting each mark as you go. When you've completed the full circle of the cake, your count is a measure of the circumference of the cake in blade widths. Just divide that number by 3 then use the resulting number and the marks on the outer edge to know how many blade widths wide at the outer edge, each slice should be. Better yet, when baking your cake have some kind of regular pattern at the outer edge that you can use for measuring the cake's circumference and know the measure in advance so when it's time to cut the cake you just divide by the number of eaters and measure the result on the pattern to know where to cut.

[@tomdekler9280]

This still leaves room for guesswork. For a tiny cake and a big knife, you may get 5 marks. That leaves you with no way to estimate the center of the circle, nor does it leave you with a number divisible by 3. Plus this assumes we're dealing with a perfectly circularly symmetrical cake. Who gets the chocolate lettering on the top? The part of the rainbow with purple skittles? The slightly burnt part? The marzepan giraffe? The head of the poorly-thought-out eerily realistic baby shaped cake at the baby shower?

[@zsoltsz2323]

You can reduce the number of cuts required for 3 people from 5, as shown in this video, to 4. "A" cuts twice, leaving 3 parts: X, Y and Z. B chooses 2 favorites (Y and Z) and trims the bigger one (Y), as in the video. C cuts the trim into 2 pieces and distributes them onto Y and Z, trying to make them equal. So far 4 cuts. Now C can decide: He either takes X, or declines in favor of Y or Z. If C takes X, then: --- B gets Y, and the appendix ox both Y and Z. --- A gets Z without appendix, which he considered fair in the beginning. If C does not take X, then: --- B gets to choose Y or Z. He has first choice and is happy. --- C gets the remainder of Y and Z. He distributed, so he is happy. --- A gets X, so he is happy. This is effectively a recursive algorithm. Consider 1 person and the rest, make one cut, the repeat with the remaining pieces and remaining persons.

[@lulabel64]

Are we in agreement that Charlie got the best trimming because damn, icing

[@LJM1566]

Wow this was a piece of cake.

[@Szobiz]

Bearkabear Lunabug not at all

[@calebwright9539]

me too

[@Legendaryan_]

I laughed out loud 😂😂

[@clickpause8732]

please stop

[@vansf3433]

The road edge of each of the 3 equal pieces of the cake is approximately corresponding to an arc which subtends an angle of 120 degree

[@Scara433]

My solution: find friends that are ok with a slightly smaller or bigger piece 😂

[@rohangeorge712]

u will have 0 friends then :)

[@ContinualImprovement]

I'll share the cake with 3 people - they're all my split personalities, so technically I get the cake all to myself.

[@Meocross]

This man is a genius.

[@Darkskytornado]

This man is three geniuses.

[@Darkskytornado]

*+Répeateàrs Répeateàrs* Autism has nothing to do with MPD.

[@lokukis8007]

If you share the cake with 3 other persons there won´t be a problem... just quarter it

[@verioffkin]

and it's called schizophrenia

[@plop010]

half this video is an ad but I still stayed for hannah fry.

[@adshille8987]

1 cake = 2π 1/3 cake = 2π/3 cos2π/3 = -0.5 Draw a line x=-0.5 Cut the cake from the point where the line crosses the circumference to the center. Then cut from centre down the ordinate. Voila!

[@TheOriginalItchyman]

I could listen to Hannah talk about anything and it would be lovely.

[@puupipo]

Probably the best Audible ad I've ever heard. And I've heard plenty.

[@pocket83]

1) Add four scoops ice cream to cake 2) Liquefy mixture in blender until homogenous 3) Distribute evenly between three similar glasses by apparent volume 4) Measure mass of each glass' contents 5) Use pipette to closer approximate equality by weight 6) Freeze liquid 7) Break glass, remove, discard 8) Cut each frozen cakecream cylinder into nine even portions by thickness 9) Distribute disks between group, cycling (as would a poker dealer) to mitigate random error through even distribution of slice discrepancy 10) Disregard the fact that step #8 poses inherently the same problem as does the original 11) Add cakeshake slices to new glass 12) Thaw, serve.

[@deathstorm1190]

Can any of you stop to think how elegant and fascinating this idea. Think about the real world applications, this is just a demonstration.

[@raghumahi8913]

Cut the cake horizontally 😂😂

[@GodzillaFreak]

All the cuts ruins the cake

[@GodzillaFreak]

D-MMA LEL

[@111vincento]

they only cut the trimmings. meaning 90% of the cake wont be cut after the first 2 cuts.

[@GodzillaFreak]

vincent schuurhof the trimmings are what makes it fair, so ruining those kind of removes the purpose of even doing this in the first place

[@vt9205]

But it'll be equally ruined for everybody.

[@IllidanS4]

Another method - cut a cake into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and cut the remaining piece into 4 pieces, give 3 to the people, and at this point, everyone will be already so bored they will not care what you do with the remaining atom of cake.

[@cmuller1441]

IllidanS4 This reminds me of a friend. Some cake was let alone in the kitchen so it is shared. Every time he came to the kitchen he was taking half of what was left. So if he is spotted he can pretend that he took only half of it. Of course after a little walk around, he was coming back to take half of what he left before... and again... A nice way to eat all the cake...

[@okay_then8472]

W

[@main135s]

I don't know if it was intentional or what, but you just created a nigh-perfect line of spaces down the majority of your comment. I applaud this.

[@IllidanS4]

I know, of course. Just a hyperbole.

[@georgelionon9050]

Splitting cake atoms is dangerous. BTW: by the amounts of splitting you did, you are nowhere near atom level tough...

[@kovid07]

I have four boys in group but I’m watching cause this video features Hannah Fry

[@duckmeister5385]

This is a way to ensure satisfaction for all the eaters, but I don't see how it guarantees all the pieces are equal.

[@le_science4all]

Great video! Cake-cutting is such a nice problem! Note though that the method presented is NOT the envy-free Selfridge Conway method. In the Selfridge-Conway method, it is the trimmer Bob that should cut the trim and pick last. In particular, the method presented here seems proportionally fair but not envy-free. Indeed, while "Alice got an extra bit", so did Bob, especially if Bob first got an untrimmed piece. This means that, in the end, Alice still might envy Bob's share. More details in the Quanta Magazine article "How to Cut Cake Fairly and Finally Eat It Too".

[@randompastahandle]

that is what I was thinking thanks fore pointing that out but alredy with 12 likes (: . although 1. omly if bob got the untrimed peace oatherwize he got less that a far cuut whill alice got more 2. who pike the cuting first and 2nd?

[@Stormhatt]

Yeah if I were Alice I would be pissed all day seeing Bob munching away on his two top pics.

[@suprguy]

Alice should cut the cake more equally next time instead of envying Bob.

[@Liam-qr7zn]

I don't understand. Shouldn't it be the person who receives an untrimmed piece and did not make the first cut who divides up the remainder and picks last, rather than the trimmer, as you say? If, as you say, it is the trimmer (Bob, in this case) who divides up the remainder, there could still be envy between Alice and Charlie, as the person who picks first will have an advantage. If the person who receives an untrimmed piece and did not make the first cut divides up the remainder, however, the person who receives the trimmed piece can be made to pick first among the pieces of the remainder. It wouldn't matter to Alice which he picks, since his share can never exceed hers, and after he's made his choice what happens between the other two is equivalent to the two-person situation, 'I cut, you choose'.

[@Dimpl739]

I think the caveat the video didn't explain is that Bob *must* take the trimmed slice if it's not taken by Charlie.

[@TheSempleGardener]

Cut like normal. Into 6 pieces Each person gets gets 2 pieces.

[@mhm3z]

Shane Semple IQ 600

[@TheSempleGardener]

@@mhm3z I just like cake

[@TheSempleGardener]

@@jessicu511 because you can just make 3 full length cuts across the cake. Not have to partially cut through, then guess the spot for the next partial cut. Then guess again, then trim, then trim, then trim.........

[@donna30044]

Bonus: that way everybody gets twice as much cake. 😎

[@TheSempleGardener]

@@donna30044 well...... Not really, but I like the enthusiasm!

[@lostindixie]

This is actually of significant importance, not because of cake but more important things. What if three siblings are to divide the household possessions of their deceased parents? Or nations are negotiating a treaty? An equitable division does not necessarily involve dividing something into quantitative equal or measurable amounts. Nicknacks, land, resources can also be divided in this way.

[@user49917]

This algorithm takes me back to my childhood days when we were three very quarrelsome brothers. We did it like that. You cut, we choose.

[@macron905]

I'm pretty sure that this problem was solved in one of Martin Gardner's books (at least 20+ years ago), but I can't recall which one. As I recall, it started with one person moving the knife across the cake. Anyone can say "cut" when they would be satisfied with the piece cut off and take that piece. The remainder of the cake would be divided by the remaining two people using the one cuts the other selects solution. If the first person moved too far before the cut, the others would feel cheated and therefore would call cut before that happened. According to Gardner, the method would work for N people wanting to share a cake, just repeat the procedure with the N-1 people that haven't gotten a piece of cake yet.

[@cartman19892]

Yay Hannah is back

[@arnavrawat9864]

That's amazing... I thought it was way too complicated, but this envy free stuff is mind-blowing

[@Choujeen]

You can just have Alice cutting, and Bob looking away. Then Alice points to a piece and asks who Bob wants to give it to. Works for any number of people.

[@prezadent1]

fair ≠ equal

[@stijnvandrongelen5625]

Either find a way to type Unicode codepoint U+2260, or the Unicode codepoints U+003D and U+0338 in sequence.

[@DuringDark]

Stijn van Drongelen thanks a bunch, man. Now I'll just insert that into my phone, and... Agghh C04r7Pt1oNO~π&rj[¢β51!:*\πnsye{¥™L~®.A#,l

[@chattyw87]

fair > equal. If someone wished to have more icing/fruit/crust they wouldn't prefer equal.

[@prezadent1]

Title says equal. Video shows fair. I don't know what you're on about.

[@Crazy_Diamond_75]

Marnige What I do is I google "not equals symbol" and then copy/paste. "≠" Some symbols also have corresponding alt codes. ≠, unfortunately, does not.

[@poutouellet]

I came for Dr. Fry.

[@armin0815]

Cut it into six pieces. Very easy to gauge (so you start out with very similar pieces) and everyone can take turns picking a piece. That's my practical advice.

[@richboomparkland]

You might feel it is easy to gauge but what if the other two thinks one piece is bigger? And also this method works for any shape, any topping distribution in cake as well xD

[@armin0815]

@@richboomparkland I cut, then the others each pick a sixth, then me and everyone else again and the last one for me. That makes it simple and because everyone gets to pick two pieces it evens out any inconsistency. Just talking from practical experience between three kids. :)

[@richboomparkland]

@@armin0815 XD that's pretty cool. Very practical too. The problem arises if both people choosing think the same one slice is better than the other though. Not saying this cutting method is practical or realistic, but it's just mathematics :P needs everyone to be perfectly satisfied

[@armin0815]

@@richboomparkland I think the only practical, simple, mathematically most precise, but least satisfying method is to use a mallet, turn it into crumbs and weigh the shares. :D

[@richboomparkland]

@@armin0815 Pefect. If my kids were fighting about which slice is bigger, all they'll get is weighed 'used-to-be' cake crumbs

[@5Detective]

This doesn't work, since the cutter's original "happy" cake slices possibly get altered before they get theirs, which means they may no longer be happy with "any" slice.

[@bdur5637]

Use a protractor and dive into 120 degree each everyone is happy 😂😂

[@bruce9421]

The best solution

[@ashantraveler]

assuming that the cake is a perfect circle

[@JamilKhan-hk1wl]

@@ashantraveler find the center of mass and cut from there

[@beningram1811]

@@JamilKhan-hk1wl also assumes the cake is perfectly even all the way through.

[@moth4256]

@@beningram1811 cutting from the center of mass solves that, does it not?

[@bluefire2755]

So, let's say Charlie takes the "Dud" piece, then, Bob, seeing the trimmed and untrimmed slices as being equal, takes the untrimmed piece, would that not make Alice unhappy due to getting a piece with some cake trimmed off? Or is that in the rules, Bob must take the piece he trimmed if it is available?

[@rigelestbit]

He has to take either one of the pieces he chose before trimming

[@bencheevers6693]

I could listen to Hannah for hours

[@petergustafsson1670]

Former bass chorister here - I noticed the bit starting at 6:10 Green apples are used to improve voices before a performance, and both milk and chocolate are to be avoided.

[@louis058]

This very obviously has more useful applications outside of cake cutting.

[@Itwasalwaysme_Noone]

You know what? You maybe thinking that you made a funny/ sarcastic comment (I certainly laughed when I first read it) but after checking the other comments here I propose your comment to be pinned.

[@rosepierce9382]

like?

[@robertholder]

Missed the chance to use Knuth's double arrow notation! n^n^n^n^n^n is just n^^6. Or, as Knuth originally wrote it, n↑↑6.

[@CatfoodChronicles6737]

Everyone gangsta until the birthday boy asks to cut the cake all by himself

[@PiriyapanBEGo]

Bold of you to assume they're all happy

[@ForViewingOnly]

Get real! Here is a normal, practical person's solution: - Cut the cake into quarters (in half, then half again). Only a person with no sense of proportion could mess this up. - The three people get a quarter each, and the final quarter is laid on its side and cut into three pieces. Any errors on cutting the final quarter into three will be so minimal that nobody will mind. Simple. That's the difference between an engineer and a mathematician!

[@GojiraPrime]

ForViewingOnly Alternatively, cut it in half, in half again and in half again for 6 equal pieces and give everyone 2. Like a pizza.

[@ChristopherKing288]

That's only if the cake is consistent. What if half hour chocolate and half is peppermint?

[@capnskiddies]

Christopher King you make the first diametrical cut at the equator between flavours. I'm a problem solver. You may now have cake

[@ChristopherKing288]

capnskiddies I never said the regions where disjoint.

[@oobkyleomars6663]

but what if it had icing on the top? then the bottom two pieces of the last cut wouldn't have any of the toppings on top. that's not fair.

[@superscatboy]

Aka "how to divvy up weed without scales".

[@MrKydaman]

Ah yes, the old I split / you pick or you split / I pick.

[@walow6549]

superscatboy

[@walow6549]

Deus Vult.

[@IAMmrhardcore]

Yeah this is pretty close to how my friends always did it

[@atteindresiempredad]

I thought I was the first human to invent the "one person cuts, the other person choses" rule when I was 8 years old

[@kaczynskis5721]

Is there a mathematical formula for cake crumbliness?

[@alessandroceloria4573]

Mass of crumbs = 0.99 x mass of cake

[@alanjones3104]

@@alessandroceloria4573 please don't tell me the other 1% in the cake is 'love'...

[@patricioansaldi8021]

Yeah and turns out it's a fractal pattern

[@alessandroceloria4573]

@@alanjones3104 Nah, it's obiouvsly magical unicorn powder

[@Eidako]

All cake cutting takes place in a frictionless vacuum.

[@walterh2113]

Gotta love the sparkly bits.

[@21.nivruttt14]

I'd cut once from any point on the circumference to the middle , run to get a protractor measure 120˚ cut , and then another 120˚ and cut and then done

[@TheStarDreamer]

"Ahh, we gotta cake, we gonna cut it" Sounds so legit...

[@bordeterre5234]

The best way to share a cake fairly is to make a cake big enough so everyone get as much cake as they want, and there is still trimmings left

[@solsystem1342]

You just keep on trying until you run out of cake!

[@warpman345]

you Never run out of cake

[@jesseward6332]

And the science gets done. And you make a neat gun for the people who are still alive.

[@ZoggFromBetelgeuse]

There is also the democratic method: - A divides the cake into five equal parts. - B choses two of them and is more than happy - A choses two and is also happy - C gets the remaining part and is not happy - but as A and B have the majority, this doesn't really matter.

[@jamesmarker3956]

The method I would advise is to divide the cake into 16 slices, and then give each person 5 slices, following which the final slice is awarded through three-way rock, paper, scissors (winning is determined by the first person to chose something that defeats both of the other participants).

Cut in three, let the other two pick two slices and do RPS over those two, and the third piece goes to the cutter of course.