Equally sharing a cake between three people - 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.

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

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

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

that is the saddest cake i've ever seen

No friends = No problems

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

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

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.

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

Moral: never share cake with a mathematician

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

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

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

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

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.

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-

What about the crumbs?

Cake?! Should've been *π*

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.

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

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

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

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?

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!

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.

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

Hannah fry is amazing

Why would someone even consider sharing a cake?!

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.

maybe this is why those useless cutting knife gadget is invented

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.

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.

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

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

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

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?

Wow this was a piece of cake.

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

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

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

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

All the cuts ruins the cake

fair ≠ equal

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.

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.

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

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".

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

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.

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

THIS TAKES TOO LONG. I JUST WANT MY CAKE.

Hannah is so perfect honestly

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

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

I came for Dr. Fry.

Mathematicians back then: Pythagorean Theorem Mathematicians now: Cake

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

Is there a mathematical formula for cake crumbliness?

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.

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.

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?

Actually, listening to Hannah _is_ like having honey poured into your ears.

Aka "how to divvy up weed without scales".

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

lol just order a cake for everyone

Hannah Fry in a Numberphile video... Perfection!

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

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

A great video, but the ad was even better. Craving warm mucousy milk now...

Gotta love the sparkly bits.

Yay Hannah is back

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

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.

I would love to have honey poured in my ears by Dr Fry.

Parker cake

an accurate representation of how siblings share their food

There's no way that n^n^n^n^n^n (at 4:34) is correct - the number of cuts would exceed the number of atoms in the universe when n is greater than 2

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

Seing as we're dealing with cake, I think using "m" is better than "n". So it's m^m^m^m^m^m. Mmmmmm, cake.

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!

Hannah is always a delight. I really enjoyed the Mathematics of Love, it's a great book (and listen!)

Berlin is the trimming that was left when Germany was divided efter WWII

Easy. One half for me, the other half for you guys. I cut it in two, you may take one half of your choice.

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

well but if you put in for n 3, n^n^n^n^n^n^n != 7

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%

You had me at Hannah Fry. Delightful.

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

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

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.

Seems more psychological than mathematical.

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.

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.

I would cut in half, then into quarters. Each person takes a quarter, and then you cut the remaining quarter into 3 slices.

They should all be depressed because their cake is mangled.

Other's: "Lets discuss some theorems" Dr. Hannah: "CUTTING THINGS" I love it!

Or you know, could just be less fussy after all you guys are friends

That formula makes no sense. You already provided a counter example: n=3. ???

this was a triumph