Maths lets them eat cake
Mathematicians have devised a fairer way to share a cake, according to new research.
It allows you to choose a slice with more chocolate icing. But the trade-off is that you get a smaller piece.
The research, which is based on the principle of surplus procedure, is published in the Notices of the American Mathematical Society.
The traditional way of dividing a cake is the 'cut and choose' approach, where one person cuts the cake in two and the other chooses a half, write Professor Michael Jones and colleagues from Montclair State University in New Jersey.
While this ensures both people get the same sized portion, it doesn't necessarily mean they are equally happy.
For example, one may like chocolate icing, sprinkles or cream more than the other.
To even this out, the researchers propose people tell a referee how much they value certain qualities of a cake.
The referee then uses these weighted values to calculate where to cut the cake.
In an ideal world, both parties would get 100% of what they want. But in reality there's a compromise.
Jones says his team's algorithms ensure that both parties get about 65% of what they want, based on the principle of giving each person at least 50% of the cake plus the surplus as they value it.
The catch is that the system depends on honesty and requires a referee and a calculator.
The surplus procedure system can be used for two or three people, Jones says, but doesn't work quite as well for three or more.
He says the method can also be applied to other heterogeneous and divisible items, like dividing land or deciding how much rent each flatmate pays.
The wedge system
Dr Burkard Polster, an Australian mathematician from Monash University, isn't convinced the proposed method is best.
"How do you associate a number with something like happiness?" he says.
"If you really wanted to sell this to a 10 year old, they wouldn't go for it, They'd still think the other piece is better."
He says a square cake, for example, can be cut into five fairly by using the equation that says the area of a triangle equals the length of its base times its height, divided by two.
The cake can't be cut into equal squares because there will only be four pieces. And it can't be cut into parallel pieces because the pieces on the edge will have more icing (if the cake is iced around the edge and not just on top).
The answer is to cut the cake in equally sized wedges.
"If the cake has a perimeter of 50 centimetres, you start at one corner and measure 10 centimetres and make a mark, then you go around until you're back to the beginning and you have five marks.
"From those marks you cut through the middle and those wedges will be exactly one fifth."
Eggs and sandwiches
Polster says applying surplus procedure to cakes isn't the first time maths has been used to share food.
The 'egg yolk' theory describes how best to divide a fried egg into pieces with an equal portion of yolk and white.
And he says the 'ham sandwich' theory makes sure you can cut a sandwich and get equal amounts of ham, lettuce and bread.