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Share some cakes (or pies)!

By David, 14 March 2013 Activity

It’s a nice thing to share a cake with your friends and family, but what’s the best way to do it? Don’t worry, mathematicians are on the case!

food safety hazard iconSafety: When dealing with food, use clean hands and clean equipment!

You will need

Three cakes on a plate with knife and two measuring spoons.

What you will need.

  • Three cakes or pies (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

  1. Select one person to be the cutter.
  2. The cutter carefully cuts the cake into even-sized slices, with one slice for each person.

    Custard tart cut into three pieces.

    When cutting for ‘I cut, you choose’, the cutter should try and make the slices as even as possible.

  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

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 (but carefully) 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.

    Slice of cake being cut on a plate.

    When someone calls “stop!”, carefully cut a slice.

  4. After the first round, the cutter should carefully 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 it exactly

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.

    Hand touching a tart.

    Break the cake down into different materials.

  2. Break the cake up and put each material into its own pile.
  3. Use measuring spoons to divide each pile of stuff evenly between all the people.

Wrapping it all up

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

‘Measuring it 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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