Kissing a Frog, A Math Problem for the Princess

Editor's Note: We found this 2008 maths essay again when trying to find something unique for Halloween ... we had purchased a witch doll kissing a frog*, if you must know. In fact, in a post centering on the Scout Report, we ran their recommendation of +Plus Magazine.

Kissing the frog: A mathematician's guide to mating by John Billingham, a Professor of Theoretical Mechanics in the School of Mathematical Sciences at the University of Nottingham:

After introducing the original fairytale, Professor Billingham posits another way to play the selection game. The paragraphs and link that follow are from a 2008 +Plus Magazine:

How attractive is my frog? Kissed Too Soon?

I'm told that when men meet women, they sometimes rate each other on a scale of 1 to 10. Of course, mathematicians are far too intelligent and sophisticated do this. We rate people on a scale of 0 to 1. In our original mathematical model, all we could do was compare one possible frog with another. The numbers didn't mean anything in themselves; they just told the princess whether one frog was "better" than another. Let's change the fairytale slightly so that the 100 frogs are now labelled with numbers drawn randomly from those that lie between 0 and 1, with the handsome prince having the highest number. What's the princess's best strategy now?

The Frog Prince by Paul Meyerheim, 1889; illustration for the fairy tale The Frog Prince. Wikipedia

Well, the princess now has much more information to use. There is a highest and a lowest number (0 and 1), and the frogs' numbers are uniformly, but randomly, distributed between the two. If the first frog to hop out is numbered, for example, 0.99, then she knows it's a top quality frog, and could well be worth a peck on the cheek. What if the first frog is numbered 0.8? Is that good enough to kiss? It turns out that the best strategy is, as anyone aged over 25 knows, to start with high standards, and then lower them as the frogs keep on coming. We're meant to be doing some maths here, so by "standards" I mean that for each frog there is a number, called a decision number, below which the princess shouldn't kiss it (here's how to calculate the decision number). If the frog is numbered above the appropriate decision number, and is the best frog so far, she should kiss it. This strategy nets her the handsome prince a whopping 58% of the time. In fact, if the first frog is numbered 0.99, she shouldn't kiss it, because the first decision number for 100 frogs is about 0.992. She's more likely to find Mr. Right by holding out for a more attractive frog.

The best-looking frog ... but he doesn't fancy me .. and I don't know why not!

There are lots of other things that we could add to our mathematical model to make it more realistic. For example, in real life, if you kiss the second best frog, you don't have to stay in the enchanted forest. Unless you're an incurable romantic who thinks that there's just one perfect person out there for you, you can be very happy with frog number 2. Maybe you're more interested in avoiding a very bad frog. What's more important, making sure you bag frog number 1 or avoiding frogs 51 to 100? The strategy you should choose depends upon what you're trying to achieve.

Read the rest of Prof. Billingham's math essay at the +Plus Magazine site

*A catalog appeared at our house during this Halloween season that might supply another frog kissing witch doll: Olive & Cocoa

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Kissing a Frog, A Math Problem for the Princess

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