Use a scale.

Good starting point is how would your scheme work for N = 3?

Thereâ€™s a branch of math concerned with â€śfair divisionâ€ť algorithms. You might enjoy this Mudd Math Fun Fact page on the â€ś*Envy-free* Cake Divisionâ€ť problem:

Say you and a friend wish to share a cake. What is a â€śfairâ€ť way to split it? Probably you know this solution: one cuts, the other chooses. This is called a fair division algorithm, because by playing a good strategy, each player can guarantee she gets at least 50 percent of the cake in her own measure. See if you can reason why.

Turns out,

in dividing a desirable object, such as cakes, a 2-person algorithm has been known since antiquity, a 3-person algorithm since at least 1960, but an exact algorithm for achieving n-person envy-free cake divisions was not produced until Brams and Taylorâ€™s solution in 1995.

What Mr. Gardner is asking about is *Proportional fairness*,

proportional fairness is weaker; it only demands each person gets what she feels is at least 1/N of the cake.

Learn more about Fair Division and play around with The Fair Division Calculator

That reminds me of a joke: thereâ€™s a big piece of cake and a little slice of cake. The first guy in line steps up and takes the big piece. The guy behind him says, â€śHey, thatâ€™s rude! If I were you I would have taken the smaller piece.â€ť

The first guy says, â€śWell you got it so what are you complaining about?â€ť

Modern solution? Be the one who cuts the cake. Take what you want. Then convince yourself everyone else is satisfied despite any evidence to the contrary.

There is actually a wonderful board game built entirely around this concept. In Piece oâ€™ Cake, the active player cuts the cake into pieces but is the last person to pick a piece. I use this in classrooms to teach equity - the best way to make sure that something comes back around to you that you want is to give other players something they might want. It is all about satisficing. Brilliant game, and a very important concept in society.

For three people, the first person cuts a piece. The second person gets to choose between taking that piece, or dividing the remaining cake into two pieces. If person 2 takes the first slice, weâ€™re back to the n=2 problem. If person 2 slices, then person 3 gets to choose their slice, then person 1 chooses a slice, and person 2 is left with the last slice.

This should work for all nâ€¦unless Iâ€™m missing something. Thoughts?

I like the idea. What happens if there are four people and four obviously unequal pieces. This might be fair for person one and two, but persons three and four may not feel that they were given a fair slice.

that is exactly what I came up with - if *n* more people agree with us, when can we be said to reasonably approach correctness?

Funny, I only thought to do that with my kids and sweet treats decades after learning it in, um, other business ventures involving sharing substances.

There is a simple procedure by which two people can divide a cake so that each is satisfied he has at least half.

You want *half a cake?* Bleurgh. No wonder thereâ€™s an obesity epidemic. You know what? You can have three-quarters of my half. Satisfied? Me too.

Person 3 may have an argument, but person 4 is the one that cut the cake and so they have nobody to blame but themselves for not creating pieces attractive enough for 1, 2, and 3. It is a hard game (thinking wise, not playing wise) because it requires us to think of satisficing rather than min-maxing or winning for sure.

This was what immediately occurred to me, as well. It lacks the same immediacy as the n=2 version, but it still seems like everyone is incentivized to make as fair a cut as possible.

You could also get much the same effect with a sort of Dutch auction:

- Cutter makes one cut, then rotates the knife until any one person says â€śIâ€™ll take that muchâ€ť â€“ and they get that much. The odds of anyone getting much larger than a 1/n portion is pretty small. Repeat the process until all but cutter and one other person is left. Then they do the you-split-I-pick

(and, @Boundegar, such a system even allows for such self-abnegating cake ascetics as yourself)

Hm, I see what you mean. Imagine this scenario:

Person 1 cuts a 20% piece

Person 2 cuts a 20% piece

Person 3 cuts a 30% piece (and by default, another 30% piece)

Person 4 takes a 30% piece

Person 1 takes a 30% piece

Person 2 takes a 20% piece

Person 3 takes a 20% piece

In this scenario, person 3 is punished, despite having made a fair division. Person 1 is rewarded despite having made an unfair division.

I prefer the belly-flop method of cake distribution

Came looking. Leaving satisfied.

Step 1. Get cupcakes.

Step 2. Donâ€™t eat one if youâ€™re worried about someone not getting one.