## Half a pizza and half a balloon

This is a puzzle

If you’ve ever actually tried to divide a pizza into 15 equal pieces – and I mean the kind of dividing that uses a knife or a cutter or piano wire – you’ll find that knowing fractions like $\frac{1}{15}$ isn’t necessarily a whole lot of help.  My favorite way, by far, of cutting pizza in 15 equal pieces is to cut the pizza into 16 equal pieces and abscond with one of the pieces before doing the actual distribution.

But! – you may object – That wasn’t fair!  Division is supposed to be fair.

Sure, it is important to be fair.  But fairness isn’t always a function of cutting into equal pieces with a sharp enough knife.  Solomon demonstrated this a long time ago.  And if I divide up the remaining odd balloon between two kids by cutting it carefully in half, I’ll get the enthusiastic thanks of neither kid.

The puzzle: can you think of some ways to “divide” the balloon in a way that is fair?  Whatever method you come up with, we’ll insist that the same method work to “divide” the slice of pizza I absconded with too.  How do you know that your method is fair?

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### 4 Responses to Half a pizza and half a balloon

1. Carol Friends says:

Bert,
I’d give the children the choice of either being first or getting an extra balloon. That way the they can have either the color they want or an extra balloon.
Then I’d have the kids negotiate.
Is this a logical solution, a maternal solution or both?
Carol

2. Bert Speelpenning says:

Carol,
I like how you turned the situation in a game with moves and choices. The question then becomes: is it a fair game? Would the children relate to it as a fair game?
It occurs to me that even though you and me may regard the extra balloon as fair compensation for the other child choosing the coolest colors (and vice versa), the children may not agree. What if both kids want the extra balloon, or both kids want to choose the colors?

Are there any ways to set up the game so that both children would agree up front that the outcome will be fair?

3. Carol Friends says:

Bert,

This is fun.

So I’ve given the children a possible solution, an example of how to solve the problem. When they negotiate they will find their own solution.

Perhaps one chooses first and gets two balloons. The second child gets three.

Maybe they let the air partially out of one, tie two knots with string in the middle, cut between the two knots and each get a half of a balloon.

Or they have a little brother. They give him the extra balloon to get him to go away.

Or they have an extra balloon and they give it to mom for two cookies.

Then they get really smart and give all the balloons to mom for 10 cookies.

Carol

4. Bert Speelpenning says:

Carol,
I like how many different approaches you found for coming up with a fair system for dealing with five balloons among two kids. Very young kids have no problem with that, but typically stop playing like that when they get older. Somehow, the concern for the right answer (which often means: the answer the teacher likes) takes over.
Each of your answers could be tested with a bunch of real-life kids to see if they would consider the scheme to be fair. It would be important to have them judges the fairness of the scheme before they end up with their balloons. A scheme may be judged fair by both kids and yet one of the kids may be disappointed afterwards. E.g. one kid may have an exaggerated notion of his strength and may agree to settle who gets the balloons by arm wrestling for it with the other kid.

One thing that strikes me about each of your answers is that I don’t see fractions in any of them. When we talk about dividing pizzas we always talk about fractions – when we talk about balloons, what happened to the fractions?