182 // The Sixth Deadly Sin

The Sixth Deadly Sin It's envy, and the problem is to avoid it.

Stefan Banach and Bronislaw 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?

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Answer on page 290

Weird Arithmetic `No, Henry, you can't do that,' said the teacher, pointing to Henry's exercise book, where he had written

1 8 18

6 ¼

4 5 45 `Sorry, sir,' said Henry. `What's wrong? I checked it on my calculator and it seemed to work.'

`Well, Henry, the answer is right, I guess,' the teacher admitted. `Though you should probably cancel the 9's to get 2,

5 which is simpler. What's wrong is--'

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