Roland Dürre
Monday April 3rd, 2017

A Very Special Task!

The Solution will be Supplied Later!

A short time ago, a good friend of mine came up with a brainteaser. He did not know the source, otherwise I would gladly have cited it. My friend was not able to solve the problem, neither was I. But it is a truly exciting scenario. And it has a surprisingly simple solution, including a beautiful mathematical reasoning. It also gives us a nice metaphor for our lives.

Among other things, it shows that mathematics can also, once in a while, be quite useful. Here is the story:

Here is a female criminal. On her card, the number 1 is written. But she does not know this. After all, she only knows the nine other images with their numbers.

A – not dislikeable – gang of 10 persons constantly violates the prevailing moral concepts in an outrageous manner. The gang members are creative and wise – this is how, with great finesse, they remain unmolested by the arm of the law for their abominable activities. That is lucky for them, because the legal penalty for their crime is death by strangulation.

In the public perception, the gang soon has a legendary reputation, and is idolized by quite a few simple people. For the authorities, this development is totally unacceptable. Consequently, the increased manhunt of the authorities, along with a growing arrogance and flippancy among the gang members led to the capture of the group.

All 10 gang members are quickly sentenced to death due to their abominable behaviour in a show trial. However, there is a way for the ten comrades in crime to save their lives – through an appeal for clemency. The head of state who decides upon said appeal is a very prudent and well-meaning woman. She is very wise; there are even some rumours insinuating that she may to some extent sympathise with the gang.

Actually, she works hard to come to a fair decision. She hands down a conditional amnesty (a little like a “Judgement of God”):
Before the verdict is executed, the ten members are permitted to see each other once more. There is a farewell meeting, the ten gangsters can spend the afternoon before their execution together and without supervision.

Here is a male criminal. On his card, the number 2 is written. But he does not know that. After all, he only knows the nine other images with their numbers.

As the meeting starts, the gangsters are told how the amnesty will work. A picture of each of the members is taken (two of them can be seen here). On each of those pictures, a number from the set 0 – 9 {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} is drawn. Each number can be used several times. Consequently, it is possible that the same number is written on all the pictures. Or that only some numbers are used, for instance {1, 2, 3)}. Or maybe only the even or uneven numbers. Whatever. But perhaps all numbers have been used. Nothing is impossible.

After the meeting, each of them is taken into solitary confinement until the time of execution. Each of the ten gang members is shown the nine pictures of the other nine members – but not his own one. And then they ask him the number on his own photo. And if even one of the gangsters gives a correct answer for the number on his card – all of them will get the amnesty.

Initially, you will think that the gangsters have quite a good chance to avoid their punishment and enjoy clemency over justice. And there is no doubt that their situation will have improved. After all, chances are not too bad that one of the ten will guess correctly and thus free them all.

But it is nowhere near as easy as that. Matters may turn out poorly. And there is one thing the wise regent forgot (or perhaps not): by applying a simple agreement, the ten gangsters can make sure that one of them will inevitably say the right number, as written on his picture. And this is how he can guarantee that he and his comrades will enjoy the amnesty.

It is a small problem: what agreement makes it possible for the gang to use the meeting that was meant as a farewell to make sure that “their heads” are out of the sling with a 100% chance?

I will publish the solution in a few weeks – and until then, I look forward to having many email solutions sent to me!

RMD
(Translated by EG)

3 Kommentare zu “A Very Special Task!”

  1. rd (Friday April 7th, 2017)

    Wow – eine richtige Lösung mit Beweis (!) kam schon rein! Großes Kompliment an Jörg!

  2. Chris Wood (Tuesday April 18th, 2017)

    There is a mistake here.
    I assumed the pictures of cards were just for people who could not understand the problem description.
    But a friend points out that it says that “two of the cards are shown”. This extra information makes the problem easier, but less elegant. It also contradicts the statement that all cards can have the same number, (since we see a “1” and a “2”).

  3. rd (Tuesday April 18th, 2017)

    No – the pictures are only for illustration and just two examples of young criminels. Of course both can have the same number. Perhaps I should use not #1 an #2 but #x and #y out of {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} 🙂

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