It’s envy, and the problem is to avoid it.

Stefan Banach and Bromslaw Knaster extended Steinhaus’s method of fair cake division to any number of people, and simplified it for three people. Their work pretty much summed up  the whole area until a subtle flaw emerged: the procedure may be fair, but it takes no account of envy. A method is envy-free if no one thinks that anyone else has got a bigger share than they have. Every envy-free method is fair, but a fair method need not be envy free. And, neither Steinhaus’s method, nor that of Banach and Knaster is envy-free.

For example, Belinda may think that Arthur’s division is fair. Then, Steinhaus’s method stops after step 3, and both Arthur and Belinda consider all three pieces to be of size $1/3$. Charlie must think that his own piece is at least $1/3$, so the allocation is proportional. But, if Charlie sees Arthur’s piece as $1/6$ and Belinda’s as $1/2$, then he will envy Belinda, because Belinda got first crack at a piece that Charlie thinks is bigger than his.

Can you find an envy-free method for dividing a cake among three people?

Ref: Professor Ian Stewart’s Cabinet of Mathematical Curiosities

We will see from treasures from Professor Stewart’s cabinet soon,

-Nalin Pithwa

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