Apple Cutting Puzzle

When I cut an apple for my kids, I normally cut it into 6 pieces. I give the biggest piece to my younger son, since he insists on only eating one piece, but I want him to eat more apple.

I take the three smallest pieces for myself, and my elder son takes the middle two pieces.


Of course, this means I get the most apple, but it doesn’t have to be that way. I could make sure we all got the same amount by cutting three pieces equal to 1/9 of the apple each, two equal to 1/6 each, and the last piece would be 1/3 of the apple.

I’ve noticed, by the way, that my local pizza shop never cuts the pizza into equal slices. Do they know that some people want a bigger slice?

But what if I just cut that apple randomly?

Today’s puzzle is this. Suppose I divide an apple (or pie, or pizza) into six pieces, by making six straight cuts from the middle in random directions. Then I distribute the pieces as I described above. What’s the chance that the three smallest pieces are less than the biggest piece? Or less than the other two pieces?

You might want to start with an easier puzzle – if I cut the apple into just three randomly sized pieces, what’s the chance that the largest piece is more than the other two combined?

Think about this next time you’re cutting an apple…