Sharing a pizza
Scalena's makes the best (perfectly circular) pizza in town. They are also notoriously bad at cutting the pizza into equal slices, although they do make cuts that are straight lines.
Alice and Bob decide to share a pizza. When the pizza arrives, Alice is dismayed to see that it looks like this:
Bob, who was distracted, grabs the largest piece, AGF, and takes a bite. And then he says, "Oh no, that was so rude of me! Tell you what, I'll try to make it fair. You can pick any cut, and I will take the side of the cut that contains my piece, and you can have the other side."
Alice agrees, and picks chord AD (which happens to also be a diameter) and takes the three slices AGB, BGC and CGD, while Bob gets the other three. Notice that the friends each got half the pizza.
Alice and Bob are so pleased with the pizza and the outcome of their game, that they decide to always share pizza from Scalena's using this protocol:
1) Bob selects any piece
2) Alice selects one of the cuts
3) Bob gets the side of Alice's cut that contains his piece, while Alice gets the other side.
Which of the following is true?
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1 solution
Bob can always guarantee himself half the pizza by picking a slice that contains the center of the pizza. If there is a unique such slice then Bob will always get strictly more than half.