11/08/2021

In Folha, the curious relationship between marriage and logarithms.

Casamento da princesa Diana com príncipe Charles, em 1981. Foto: Flickr

Reproduction of Marcelo Viana's column in Folha de S.Paulo

When the king announces he is seeking a groom for his daughter, numerous suitors come forward. The choice will be made by the princess herself, naturally, based on conversations with the candidates. To avoid discrimination, the order of the interviews is drawn by lot. After each conversation, the princess is able to rank the candidate among all those she has spoken to, but she still knows nothing about the others.

Ideally, we would talk to everyone and only then choose, of course. The problem is that the noble knights have sensitive egos: they get offended and leave if they aren't chosen right at the end of their interviews.

What to do? Choose one of the first suitors who seems reasonable, giving up on finding Prince Charming? Or wait until near the end to choose, when you've already met most of them, risking letting the love of your life slip away with a broken heart? Every strategy has risks; is one better than the others?

This problem was formulated by Merrill Flood in 1949, but similar questions had been proposed earlier by Arthur Cayley and even Johannes Kepler. It became widely known from 1960 onwards, when Martin Gardner publicized it in his famous column in Scientific American magazine.

The complete solution was given by Thomas Bruss in 1984, based on the work of R. Palermo: to maximize the chance of getting a groom to her liking, the princess should reject the first N/e suitors, however good they may seem to her, and from then on should accept the first one who is better than his predecessors.

This is yet another example of what I wrote last week about the fascinating connections between distinct themes that mathematics can reveal: who would have thought that Euler's number e=2.718281828459…, so important in the theory of logarithms, would be useful in the search for our soulmate?

There are many other "when to stop evaluating and make a decision?" type of problems that frequently arise in everyday life. For example, when refueling the car, we need to decide whether it's worth continuing to look for a gas station with a more favorable price or whether it's more advantageous to stop right away and fill the tank.

To read the full text, visit the newspaper's website.