Four Doomed Managers

The Setup

The following logic puzzle is titled “Die vier todgeweihten Manager”, “The four doomed Managers”. It’s a version of the hat guessing game, where different people wear hats of different colour. They can see the hats of other people, and need to deduce the colour of their own hat using logic. The story of the four doomed managers goes as follows:

The four managers are burried in sand, neck-deep. Because they appear to lack any ability whatsoever, they are all sentenced to death. Unless they can prove their skill by solving a puzzle.

All they know is that they are placed in a row, which direction each one is facing, and that each of them wears a hat, two of which are white, two of which are black. Further they know that there is a wall between the first and second manager. If one of them correctly identifies their own hat colour in the next ten minutes, they will not get shot. If he or she guesses wrong, all of them die!

There is no second chance. They are also not allowed to move or turn their heads, or to commmunicate in any way. The setup is as follows, where the black dots represents the direction in which each manager is looking.

The first and second manager from the left can only see the wall. The third manager can see the black hat of the second manager. And the fourth manager sees both the white hat of the third, and the black hat of the second manager.

First, everyone stays silent. After a few minutes, surprizingly, one of them correctly guesses their hat colour. Which of the managers was able to guess it with certainty? How did they arrive at the solution?

The solution

The first step is to realize that none of the managers have enough information by themselves to determine their own hat colour. The last manager has the most information at hand, the colour of the two in front of him. Since he sees one black and one white, his own hat could be either of the two colours. If however the setup would be one of the two following two, it would be obvious to him.

All the managers can derive this from the setup: If managers two and three wear the colour white, it is obvious to manager four that he wears a black one (and vice versa), since there are only two white and two black in total. As he wants to live (hopefully), he will not wait to name the colour of his own hat, and save all of them.

Now, because hat color two and three differ in the setup, manager four will not say a a word. Manager three can now take this information to her advantage. The fact that four did not say anything implies that her hat and the hat of the one in front of her must differ. As she sees a black hat, she knows hers must be white.

Therefore, it is manager three who currectly determines white as her hat colour. They are all happy to be alive and continue working their manager jobs, deserving a good pay after all. Only time will show: Will they survive the next math puzzle?

Formal Notation

A more mathematical description is as follows ($M_i$ is the $i$-th manager, $\Rightarrow$ is used for an implication, $\neg$ for negating a statement). Arguing as above, we have these two statements:

$$\neg (M_4 \text{ knows the answer})$$

Using modus tollens (if $P$ implies $Q$, and $Q$ is false, then $P$ cannot be true), the statement $\text{"}M_2 \text{ and } M_3 \text{ wear the same colour}"$ cannot be true. Manager three knows these two statements to be true:

$$\neg(M_2 \text{ and } M_3 \text{ wear the same colour})$$
$$M_2 \text{ wears black}$$

Together with the fact that there are only black and white, $\text{"}M_3 \text{ wears white}"$ follows.