This is the solution to the Birthday problem.
The fun
Three friends, Javi, Juan and Manolo, all mathematicians, want to celebrate the third anniversary of the presentation of the IMUS Blog by a mutual friend, Antonio, also a mathematician, but neither Juan nor Manolo can remember the exact date of the event with any certainty. Javi remembers it well, but instead of giving it to them, he suggests that they use their mathematical knowledge to deduce it from a clue he is going to give them and from the comments they can make when they receive it.
To do this, he gives them a list of 10 possible dates:
|
2 September |
3 September |
6 September |
|
4 October |
5 October |
|
|
1 November |
3 November |
|
|
1 December |
2 December |
4 December |
and tells one of them the day and the other the month of that presentation, individually and separately, without each of them being aware of their conversation with the other. He then allows each of them to make some comments about that conversation.
These comments were as follows:
Juan: I don’t remember when that presentation was, but now I know that Manolo doesn’t remember either.
Manolo: Initially I didn’t remember when the celebration was, but now I do.
Juan: Then I know that date too.
The question is: On what day and month did the presentation of the IMUS Blog take place?
Solution
It follows from Juan and Manolo’s comments that Javi told Juan necessarily the month of the presentation, because if he had told him the day, then there would be no possibility of Manolo knowing the full date (day and month). Therefore, Juan knows the month.
From the statement of the problem we know that Juan is sure that Manolo does not know the date of the celebration, so we must rule out the months of September and October, since the 5th only appears in October and the 6th only in September (if Juan knew that it is in September or October, then he cannot be sure that Manolo does not know it, since Manolo could not know whether it is a 5th or a 6th).
This allows Manolo to already know what month that presentation was in, by the following reasoning. It could be the 3rd of November, the 2nd of December or the 4th of December (not on the 1st of December, since he cannot know it). Juan, on the other hand, can subsequently be sure of the date: he knows that it is in November, because if it were in December, he could not be sure whether it was the 2nd or the 4th of that month.
Therefore, the day and month of the presentation was November 3.
¿Por qué la siguiente solución no es válida?
Si a Juan le hubiesen dicho el día, al decir que no recuerda la fecha, podríamos descartar el 5 de octubre y el 6 de septiembre. Luego dice que Manolo seguro tampoco lo sabe de primeras. Entonces Manolo sabe que debe descartar esos días mencionados, y si ya lo sabe, es que tiene que ser el 4 de octubre. Juan lo entiende, y también sabe que es el 4 de octubre.
¿Falla algo?
Al razonar debemos asumir que antes del diálogo entre Juan y Manolo ambos saben que Javi le dijo el día a uno y el mes al otro. Si Juan hubiera sabido el día, no tiene sentido que diga “pero ahora sé que Manolo tampoco la recuerda” (la palabra “ahora” es la importante), porque mientras entre las opciones del día hay algunas con un único mes asociado, no ocurre lo mismo al revés. Es decir, si Juan hubiera sabido el día, debería haber dicho “…pero sé que…”.
Ese matiz da la clave para deducir que Juan sabía el mes y no el día. Es cierto que en la solución propuesta no está del todo bien explicado.