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3 people are dividing a cake. Each person wants as much cake as possible for himself, but each person thinks that the other two might be colluding. Lets call the three A, B, and C.

A suggests that he will cut the cake into 3 portions, and then C will pick his piece from the 3, and B then would pick his choice from the 2 remaining. and A takes the last piece. B objects to this scheme, saying that A can cut the cake into 1 large piece, and 2 smaller equal pieces. C then would have the largest piece, and B would have to take the smaller piece. In fact, B claims, If A and C were really colluding, this method would allowing them to get the entire cake to share among them two.

Given this distrusting atmosphere, is there a way to divide the cake so that each person is satisfied? Any proposed method would have to convince each person that he would get his fair share even if the other two were colluding.

I know of two solutions to this, but I wouldn't be surprised if the den can come up with a couple extra solutions.

Edited by bushindo
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3 people are dividing a cake. Each person wants as much cake as possible for himself, but each person thinks that the other two might be colluding. Lets call the three A, B, and C.

A suggests that he will cut the cake into 3 portions, and then C will pick his piece from the 3, and B then would pick his choice from the 2 remaining. and A takes the last piece. B objects to this scheme, saying that A can cut the cake into 1 large piece, and 2 smaller equal pieces. C then would have the largest piece, and B would have to take the smaller piece. In fact, B claims, If A and C were really colluding, this method would allowing them to get the entire cake to share among them two.

Given this distrusting atmosphere, is there a way to divide the cake so that each person is satisfied? Any proposed method would have to convince each person that he would get his fair share even if the other two were colluding.

I know of two solutions to this, but I wouldn't be surprised if the den can come up with a couple extra solutions.

Have one person cut the cake into thirds, the other two play ro sham bo to determine who gets to pick first and second. The one cutting the cake gets the remainder.

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do there have to be only 3 pieces and do they have to be straight cuts? if the 3rd guys feels hes getting **** can he just cut it into as many pieces as necessary to make them all the smallest?

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do there have to be only 3 pieces and do they have to be straight cuts? if the 3rd guys feels hes getting **** can he just cut it into as many pieces as necessary to make them all the smallest?

The cuts don't have to be straight cuts, and there don't need to be only 3 cuts. If the first decide to cut the cake into many pieces, how would the division process go?

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why would C be conspiring to get a smaller piece?

I assume you meant why would A conspire with C in order to get the smaller piece. The assumption here is that if these two were really colluding, they would share their collective pieces after B goes away.

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Everybody gets 9/27. The cake is cut into thirds by A, then B cuts those slices in thirds, then C cuts those in thirds.

That way, everybody did equal cutting.

How would they divide those 27 pieces? Who gets to pick first, and then how would it go from there?

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Have one person cut the cake into thirds, the other two play ro sham bo to determine who gets to pick first and second. The one cutting the cake gets the remainder.

That would work. There is also a way to divide without resorting to random events.

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how could 2 people colluding get the whole cake if third party is not a moron?

If A, B, and C agrees to a schemes where A cuts, C picks first, B second, and A last. It is possible for A and C to collude to get the entire cake.

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Cut the height of the cake into equal thirds (making three cakes of equal volume with 1/3 the height of the original cake). Then let A cut the cake into 3 pieces (really 9 considering the original cuts). Then pick as follows (ABCBCACAB). This way, there will be 9 pieces and each will have an opportunity to pic one big piece, one medium piece and one small piece assuming the pieces were cut to different sizes.

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That would work. There is also a way to divide without resorting to random events.

Have A cut the cake into thirds. B chooses a piece and cuts it in half for he and C to share. C gets what he feels is the larger of B's cut. Next C cuts one of the two remaining pieces, of which A gets first pick of half. Finally, A bisects the remaining piece and gives B first pick of its half, keeping the remainder.

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Well,

First make one cut from the center to any side. Then have one person rotate the knife around. When one person is happy with how much there is in the cut, they will say their name. Then, the person will cut and whoever said their name would get that piece. This would continue for the other two people. Then, everyone is happy.

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I assume you meant why would A conspire with C in order to get the smaller piece. The assumption here is that if these two were really colluding, they would share their collective pieces after B goes away.

Have A cut the cake by slowly moving the knife over the cake until either B or C is satisfied that the piece is 1/3 of the cake. Then that person would get that slice. A would then start moving the knife over the cake until person C is satisfied that it is 1/2 of the remainder of the cake. Then he would get that slice and A would get what is left.

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this problem can be solved if none of them dare to take bigger piece because doing that might cause them heavily..... let a 4th person say D uses a knife and start moving the knife from the rightmost side of the cake towards left very slowly.....all three participant have the right to say STOP at any point of time......as soon as someone says STOP D will stop moving the knife and will cut the cake at that point only....whoever says STOP will get the right side of the cake.....this way the person who has said stop couldn't complain about the size as it was his decision.....and the other two also can't complain as they were waiting for a bigger piece....

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this problem can be solved if none of them dare to take bigger piece because doing that might cause them heavily..... let a 4th person say D uses a knife and start moving the knife from the rightmost side of the cake towards left very slowly.....all three participant have the right to say STOP at any point of time......as soon as someone says STOP D will stop moving the knife and will cut the cake at that point only....whoever says STOP will get the right side of the cake.....this way the person who has said stop couldn't complain about the size as it was his decision.....and the other two also can't complain as they were waiting for a bigger piece....

Phatfinger got two solutions, both of which are good. Good job to kidsrange, TerriDawn21, and Nitin Agrawal, who posted about the moving knife method, that's a very slick way.

Edited by bushindo
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1. Have A cut the cake into three equal pieces.

2. Have B and C consider A's cuts.

-If all three pieces are deemed larger by either B or C then that means that B thinks a piece is larger that C does not think is the case (neither B or C can think all three pieces are larger!). Give that to B then let C choose from the remaining two.

-If there are 1 or 2 pieces B or C deem larger then give one of the remaining pieces to A. A is happy since A made the cut. So now B and C are left with they consider to be greater than or equal to 2/3 of the cake.

3. Now this problem has reduced to the two person splitting problem.

Ah. looks like i just came up with a variation of an earlier solution. However...this method only requires 4 cuts!

Edited by birleis
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Have the group agree that person A cuts the cake, B gets the biggest piece, C gets the first pick after that, and A gets whatever C doesn't pick. Person A cuts the cake in half. Person B does not get any cake due to the pieces being the same size.

Edited by DodgeStealthTT
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(Lets assume that we have a round cake, and not a sheet cake for this method)

What about having A, make the first cut into the cake. Then B makes a second cut and that piece is given to C. Then C, makes a third cut of the supposedly 2/3 piece remaining and gives that piece to B. The last piece then obviously goes to A. This teaser is a "peanut butter" problem. Two kids make one sandwich but both want a bigger piece, so one cuts and the other chooses. Therefore making the person who's cutting it be honest and try to make them as equal as possible so the second person doesn't get the bigger half.

-Ibelin

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A wud cut a piece for himself and can't claim unless other 2 have taken theirs

similarly, B wud cut a piece for himself and can't claim unless other two have taken theirs

now ...in the order C,B,A.. each can claims their share.

:)

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