Judgment of Solomon

Logic Level 4

What is the minimum number of cuts required to cut a cake such that it is the fairest way to split among three people?

Details and Assumptions :

  • The cake is not necessarily circular. And no one is allowed to measure the cake. You are not allowed to communicate with each other in any form while cutting the cake.

  • You may want to look up Fair Division first.

Image Credit: Flickr Risa .
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Pi Han Goh by Pi Han Goh , 30, Malaysia

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1 solution

Pi Han Goh
Jul 6, 2015

Credits: Hongyu Chen .
2 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Why can't we do better than 4?

Ho do we tell that this is a fair distribution?

I get a solution of 2 cuts.

The first person places the knife in such a way that the piece they intend to cut is to their minds, 1/3 of the cake. They ask the other people if, in their minds, that piece is more than 1/3 of the cake. If the other two people say no, they make the cut and take that piece. If the second person says no and the third person says yes, the third person positions the knife to cut off less cake, what in their mind is 1/3 of the cake, and makes that cut. If the second person says yes and third person says no, the second person adjusts the knife to cut off less cake, what in their mind is 1/3 of the cake, and makes the cut.

If both the second and third person say yes, the second person adjusts the position of the knife to cut less cake, what in their mind is 1/3 of the cake, and asks the third person if the piece they intend to cut is more than 1/3 of the cake. if the third person says no, the second person makes the cut. If the third person says yes, they adjust the position of the knife to cut less cake, what they feel is 1/3 of the cake, and make the cut.

In any of the above cases, we have 1 person cutting a piece of cake they feel is 1/3 of the cake, with 1 cut, and the other 2 people feeling that the remainder of the cake is at least 2/3.

Call the two remaining people A and B. With the second cut, A cuts the remaining cake into what s/he feels are equal sized pieces, and B selects which piece s/he wants.

Everyone gets a piece of cake they feel is at least 1/3 of the original cake, and only 2 actual cuts were required.

Note that a similar procedure can divide a cake fairly among n people with n-1 cuts.

Denton Young - 5 years, 11 months ago

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You got the wrong idea for the word fairest . There's no point in asking this question if everyone gets to vote on whether one person should cut the cake the cake in a certain way or not.

Pi Han Goh - 5 years, 11 months ago

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I did look up fair division. I still get 2, and you don't even need to hover the knife. You can use 2 mobile markers which can be adjusted.

Denton Young - 5 years, 11 months ago

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@Denton Young – That's the thing, there is no "2 mobile markers which can be adjusted".

Pi Han Goh - 5 years, 11 months ago

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@Pi Han Goh – Then you can hover the knife. You're surely not going to tell me there is no knife.

I think the issue here is that the problem is using some very strange, technical, obscure definition of the word "fair" that isn't making sense to me. Perhaps an admin can weigh in on this? I really don't want to file a report when I'm sure the issue is one of definition: there's nothing wrong with the problem itself.

Denton Young - 5 years, 11 months ago

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@Denton Young – The point I'm trying to make is that you can't have people vote on whether you should cut the cake in a certain manner or not. The "adjust the position of the knife" is equivalent of measuring the cake. In other words, you already failed to meet the criteria that "no one is allowed to measure the cake".

Pi Han Goh - 5 years, 11 months ago

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@Pi Han Goh – I knew it was a matter of definition. I don't consider someone judging by eye "measuring". No measuring instruments are in fact used.

The word "measuring" threw me off. If you had specified that the people aren't allowed to communicate with each other in any form during this process, then you have to use your more convoluted way of doing it.

Thanks for the clarification.

Denton Young - 5 years, 11 months ago

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@Denton Young – I find that unnecessary to specify that no communication is allowed, but there's no harm in adding it in.

No problem ;)

Pi Han Goh - 5 years, 11 months ago

Zero: I eat the cake. That's fair in my book :D

Marc Brothers - 5 years, 9 months ago

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