manon . michel @math.cnrs.fr
manon . michel @ uca.fr

Laboratoire de mathématiques Blaise Pascal - UMR
6620
Université Clermont-Auvergne
Campus Universitaire des Cézeaux
3 place Vasarely
63178 Aubière Cedex
France

312211. The law of the sequence
is to count sequentially the length of an interrupted
same-digit occurence. '1' = one 1 = 11, '11' = two 1 = 21,
'21' = one 2, one 1 = 1211

Four men in hats

What would happen if prisoners B
and C had the same color hat?

C. The prisoner D sees one black
and one white hat. He could have the black or the
white hat and thus cannot call out. The prisoner C
only sees the white hat of prisoner B but, as prisoner
D remains silent, he can conclude this is because
prisoner D is seeing two different hats in front of
him (if both prisoners B and C had the same hat,
prisoner D would know that he has a hat of the other
color). Thus prisoner C knows that he has a hat of a
different color than prisoner B and then calls out to
say he has a black hat.

The sick monks

Once upon a time there was a monastery built atop a mountain
in which one hundred monks live isolated from the rest of the
world. The rule of the monastery is quite strict: monks cannot
communicate with each other, whether it is through writing, talking or
any body language. No communication at all. They spend the day working
and praying alone but dine together every evening.
One day, the master of the monastery decides to exceptionally break
the rule to inform his fellow monks of one important fact. There is an illness among us. It is fortunately not
contagious nor particularly dangerous, but I would like every
sick person to remain in bed for an entire day. The illness
has only one recognizable symptom: it causes an easily visible
red dot to appear on one's forehead.
The master did not point out the sick monks and
there are no mirrors in the monastery. Seven day pass
without a single monk missing. After the seventh day (on the
eighth day), all sick monks leave the dinner table to go rest
in bed just as they were instructed. How many
monks were ill and how did they find out?

What would happen if there was a
single sick monk?

8. A monk see k monks with a red
dot. If he is healthy, then k = N (N being the total
number of sick monks), k=N-1 otherwise.
A monk will understand he is sick, if, at the k+1-th day, the sick monks are still coming to dinner.
For instance, if there is only one sick monk, he will
see 0 monks with a red dot and will conclude he is then
the sick one. If there are two sick monks, they will see
one monk with a red dot at the first dinner and cannot
conclude they are sick. But, at the second dinner, as
they both return, they can conclude the other monk could
not know he is sick, which means there is two sick monks
and they are one of them. Etc.

The gate keepers

After a long journey, you arrive at last in front of the gate
leading to the Atlantis treasure. But, to your surprise, there is not
only one but two gates and both are protected by giant stone
soldiers. Above the two gates, you can read the following.
One door to Death. One door to Wealth. One soldier of
Lie. One soldier of Truth. On one question you rely.
For yes or no will get you through.
Which question will you ask to which guard to know which gate leads to the treasure?

One guard is always saying the truth and one is always
lying. Find a way to make both of the guards involved in the
question.

Ask to one of the guards picked randomly: If I ask
the other guard whether he is keeping the gate to the
treasure, what will his answer be? If and only
if the guard answers yes , he is keeping the
treasure gate. Depending on his answer, go through
his door or the other one.

The blue coins

At the time, joking about the king sounded like a very good
idea. At the time. Now you are trapped in a room of the
royal castle, in total darkness. You are free to go at any time
though. However you will be able to do so with your head still on your
neck at one condition: the number of blue-top coins outside and inside the
room must be equal when you leave. The room contains indeed a certain
number of coins which have a blue side and a red one. When you entered
the room, you were told that there were exactly 5 coins with the blue
side on top. All the coins are initially in the room, no coin outside
of it. You can do and take whatever you want in the
room. However, you are in total darkness and the coins cannot
be destroyed.
What can you do to succeed this challenge?

What to do if there are only the
5 blue-top coins in the room?

Pick randomly 5 coins, flip them
and walk out of the room with these 5 flipped coins. The number of blue-top coins outside
and inside will be the same.