Professor's three daughters. Solution explained.

La Habitación de Fermat (2007)

In the Spanish film La Habitación de Fermat (English title: Fermat’s Room), four mathematicians are locked in a room by an unknown host and are forced to solve a series of math puzzles. The film does a pretty good job of explaining most of the puzzles with a certain amount of detail—sometimes to an extent that it is frustrating to watch the mathematicians waste time going over the problem and explaining it to one another when all they have to do is write down the answer. The only puzzle that the movie deliberately leaves unexplained is the following number puzzle.

Un alumno le pregunta a un profesor: “¿Qué edad tienen tus tres hijas?”, y el profesor contesta: “Si multiplicas sus edades, da 36; y si las sumas da el número de tu casa”. “Me falta un date”, protesta el alumno. Y el profesor le responde: “Es verdad. La mayor toca el piano”. ¿Qué edad tienen los tres hijas?

Or for those who are more familiar with English:

A student asks a professor, “How old are your three daughters?” The professor replies, “If you multiply their ages, you get 36, and if you add them, you get the number of your house.” “I need more information,” protests the student. The professor says to him, “You’re right. The oldest one plays the piano.” How old are the three daughters?

In the movie, one of the mathematicians happens to know the problem from somewhere else and blurts out the answer without discussing it. The movie attributes such rush to the limited time that the characters have, but I think the real reason may have been that although this problem is cool to present to an audience, it’s not easy to verbally explain how to solve it. Nevertheless, for those who were curious to know, here is the 

solution.

Take a look at the first piece of information given to us. The product of the daughters’ ages is 36. Fortunately for us, there are not many combinations of three numbers whose product is 36. Below is the complete list of such combinations.

1 × 1 × 36

1 × 2 × 18

1 × 3 × 12

1 × 4 × 9

1 × 6 × 6

2 × 2 × 9

2 × 3 × 6

3 × 3 × 4

The second part of the problem states that the sum of the three numbers is the same as the number of the student's house. It would be reasonable to assume that the student knows the number of his house. What we need to focus on is that this bit of information was not sufficient for him to figure out the answer. We can logically reason that there are more than one group of numbers whose sum add up to the number of his house. If we add the numbers above we get:

1 + 1 + 36 = 38

1 + 2 + 18 = 21

1 + 3 + 12 = 16

1 + 4 + 9 = 14

1 + 6 + 6 = 13

2 + 2 + 9 = 13

2 + 3 + 6 = 11

3 + 3 + 4 = 10

Note that there are only two combinations that add up to the same number: [1, 6, 6] and [2, 2, 9]. The answer has to be one of these; otherwise there would have been no reason for the student to still need more information.

The last bit of information that the professor provides is that his oldest daughter plays the piano. All we need to do is pick the combination in which “the oldest daughter” exists. In other words, [1, 6, 6] is out because there is no single oldest daughter; the two oldest ones are twins.

The answer: two-year-old twin sisters and a nine year old.