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Sharing some cakes

By David, 15 December 2016 Activity

three cakes, ready for sharing

How can we make sure everyone gets a fair piece?

Picture this: you’ve just bought a nice little cake, just for you. Then your friends show up, wanting a piece. Lucky for you, mathematicians have developed some pretty cool ways of sharing.

You will need

  • Three cakes (include a fancy one with icing, filling and sweet decorations on it, and at least one rectangular one)
  • Knife
  • Friends to share with
  • A plate for each person
  • Measuring cups, spoons, scales

Method 1: I cut, you choose

Someone is cutting a cake into three slices.

One person cuts the cake into pieces.

  1. Select one person to be the cutter.
  2. The cutter cuts the cake into even-sized slices, with one slice for each person.
  3. Once the cake is cut, everybody takes turns to select a slice each. They can choose any slice that hasn’t already been taken.
  4. The cutter gets the last slice.

Method 2: Moving knife

A knife is movine along a rectangular cake. There is a speech bubble saying 'stop'

When someone yells ‘stop’, cut them a slice.

This sharing method is easier with a rectangular cake.

  1. Choose one person to be the (temporary) cutter.
  2. The cutter slowly moves the knife down the length of the cake. At any point, any person can yell “stop!”
  3. When someone yells “stop!” the cutter immediately cuts the cake at that point and gives that person the slice. The person with a slice of cake is not allowed to yell “stop!” again.
  4. After the first round, the cutter should give the knife to the person who has a slice of cake – now that they have their cake, they have no reason to cheat.
  5. Repeat the process until only one person has no cake. That person gets everything that remains.

Method 3: Measuring exactly

Someone is scooping out the contents of a tart.

Separate the cake into different materials.

Use a fancy cake with icing, filling and sweet decorations for this method. Warning: this can get very messy!

  1. Look at your cake, and identify the different materials in it. For example, a cake may have filling, icing, fruit on top, and so on.
  2. Break the cake up and put each material into its own pile.
  3. Use measuring spoons to divide each pile evenly between all the people.

Wrapping it all up

Several small piles of filling, crumbs and pieces of fruit.

Measure out an even share using scales and measuring spoons.

After you have shared all three cakes, talk with your friends about the different ways of sharing a cake. Which one is fairest?

What’s happening?

There is no single way to share all things. Instead there are many different ways, and each has good and bad points.

‘I cut, you choose’ might sound fair, but if you’re cutting several slices, there’s a good chance one of them will end up a bit smaller than the rest. This means ‘I cut you choose’ isn’t fair for the person cutting.

The ‘Moving knife’ method is reasonably fair as long as no one tries to get more than they deserve. However, if all players try to get more than their fair share, some people might end up with very little.

‘Measuring exactly’ might be the fairest way in this activity, but it’s not necessarily the best way. For many people, a pile of icing and a pile of cake crumbs isn’t as nice as a complete slice of cake.

It’s useful to consider different methods for sharing things. If you’re just sharing a muesli bar, then a simple method, like ‘I cut, you choose’, is quick and gives a pretty good result. If you’re trying to share out chores around the house, a more complicated method might be useful.

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