TITLE: Splitting a sandwich and not feeling deceived
QUESTION [679 upvotes]: This is a problem that has haunted me for more than a decade. Not all the time - but from time to time, and always on windy or rainy days, it suddenly reappears in my mind, stares at me for half an hour to an hour, and then just grins at me, and whispers whole day: "You will never solve me..."
Please save me from this torturer.
Here it is:
Let's say there are two people and a sandwich. They want to share the sandwich, but they don't trust each other. However, they found the way how both of them will have a lunch without feeling deceived: One of them will cut the sandwich in two halves, and another will choose which half will be his. Fair, right?

The problem is:
Is there such mechanism for three people and a sandwich?

EDIT: This was roller-coaster for me. Now, it turns out that there are at least two books devoted exclusively on this problem and its variations:
Fair Division
Cake Cutting Algorithms


Yesterday, I was in a coffee shop in a small company. We ordered coffee and some chocolate cakes. As I was cutting my cake for my first bite, I felt sweat on my forehead. I thought, 'What if some of my buddies just interrupt me and say: Stop! You are not cutting the cake in a fair manner!' My hands started shaking in fear of that. But, no, nothing happened, fortunately.

REPLY [16 votes]: My professor for Game Theory asked me this question,and I gave him an answer that he said that is the simplest he had ever heard (also Steaven Brhams said it's nice):
P1 divides the cake to 3.
P2 notes the smallest,in his opinion; let it be A.
If P3 wants A, let him have it.
Else P1 takes it, and we have only 2 which we know how to solve.
If P3 does not want piece A, then P1 will take it so then we have P2 and P3 and piece B and piece C give piece B to P2 and put piece C on the side and then have P2 divide piece B into two pieces (B1 and B2) and let P3 pick The one he thinks is bigger. Next give C to P3 and have him cut it (into C1 and C2) and let P2 pick the one he thinks is bigger and give the other to P3