**PUZZLES.**

Albert FRANK

1. Four miners are at the same place, inside a mine. The
mine is going to explode in precisely one hour. They don't move at the same
speed, and are situated respectively at 5, 10, 20 and 25 minutes of the unique
exit. They cannot move without light and have only one lamp. Besides, they can
move maximum simultaneously by two (at the speed of the slowest)

How are they going all to run away?

2. We have a board 8 x 8 (like a chessboard). Let's call
domino an element measuring 2 squares on one square. With the help of 32
dominoes, one can cover the board.

Now remove the squares situated below on the left and
right on the top. We now have 62 squares.
Can one cover this picture with the help of 31 dominoes ? ATTENTION: It
is necessary that the solution can be
understood by a child of 8 years !

3.Jacques, that live in Malines, has two girlfriends,
one in Antwerp, the other in Brussels. Having noted that the frequency of the
trains for Antwerp and for Brussels is the same, he decides to go randomly at
the station and to take the first train that presents itself. At the end of one
month, he notes that he visited his girlfriend of Brussels more often than the
one of Antwerp! How to explain this?

4.

10 11 12 13 14 15 16 17 20 22 24 31 100 ? 10000

5. We will admit (for this problem) that all report
presents a danger of contamination by the virus of AIDS.

Jacques wants to honour three girls, but he only has two
condoms. How is he going to do?

6. In the statement following, only the weights weighing
a integer number of kilograms are taken in consideration.

One has a flat-bed balance, and four weights of
reference. What must be these 4 weights in order to be able to weigh all
weights from 1 to 40 kilograms?

7. Find a number of 9 different digits (all digits
except zero) so that the number formed by the k first digits from the left is
divisible by k.

8. Mr. and Mrs Smith have habits: Every day, Mr. Smith
arrives at the station of his city at the same hour. At the same moment, Mrs.
Smith arrives with the car, embarks Mr. Smith and they go back home. They
arrive at home every day at the same hour. We will admit all along the problem
that the speed of the car is constant.

One day, Mr. Smith arrives at the station one hour
early.

Mrs. Smith, who did not know that, is not there. Mr.
Smith decides to make the beginning of the road on foot, until he meets his
wife and go back at home. That day, they arrive at home 20 minutes earlier than
usual.

How many minutes did Mr. Dupont walk ?

9. Ten bags, numbered from 1 to 10, contain each 1000
pieces of currency. The pieces of nine of the bags weigh each 1 gram, while the
pieces of one of the bags weigh each two grams. One has a plain balance with
graduating by gram. How, in one only weighed, to find the heaviest bag?

10. A scientific calculating machine has some problems:
all numeric keys are out of use, and one cannot even use the numbers e or pi.
Only are usable the functions (of which ln, square root, factorial,... but not
exponential). In a memory, we have a number (unknown) strictly positive.

How, using three times this number , to get the number
6?

11. A known problem is , with the help of a balance to
determine in three weights what is the
“guilty ball” among a set of 12 balls of which one has a weight
different of the others, and to determine if it is heavier or lighter than the
others.

Is it necessary to generalize the problem now:
considering n balls (any n > 2), of which one has a weight different of the
others, in how many weights can we identify the different ball and determine if it is heavier or lighter than
the others?

.

12. You have the choice between three doors. Behind one
among them, a treasure; behind the two other, nothing.

1°. You choose one of the three doors.

2°. An individual, showing you one of the two doors that
you didn't choose, tells you, while opening it: “you were right not to choose
this one - watches - the treasure is not here."

3°. Do you maintain your first choice (made at 1°), or
do you choose the other remaining door? What are your probabilities according
to the choice?

13. One makes a list of the number of inhabitants of
every city (or village) of U.S.A. What
is approximately the percentage of the
elements of this list of which the first number will be " 1" or
" 2" ?

Same question if one is interested in the surface,
expressed in square kilometres, of all lakes of the world.

14. Two cubic dice have, on each of their faces, a
natural number from one to twelve. (all faces must be marked with one of this
numbers, and nothing forbids that two or several faces carry the same number).
When one throws the two dice, a sum between two and nineteen always appear.
Besides, all this sums have the same probability. Describe the two dice.

15. As in an old easy problem, you are at a bifurcation
of two roads (one leads to Paris -
where you want to go - , the other elsewhere.) At the bifurcation, there is a
house, in which lives two brothers, among which you know that there are AT
LEAST a liar (which always lies.) You speak to one of the brothers (you don't know which). You have the right
to ask him only one (short) question. What question will you ask?

16. You have the following material : a lighter and two
ropes. Each of these two ropes burns in precisely one hour, but in a nonlinear
way (some parts burn more quickly than others) . Besides, they don’t have
the same length. How do you make to measure a time of 45 minutes ?

17. Bernard and Claude play the following game : Bernard
aligns, hidden faces, five cards of which 1 Ace. Claude chooses two adjacent
cards then. If he finds the ace, he wins. They play a lot of games. What is
Bernard's optimal strategy (for the placement of the Ace) ? And the one of
Claude (for the choice of the two adjacent cards ) ? If they play all two
in an optimal way, what will be the probability of win of each ?

18. Find a number of 10 digits (this number cannot start
with zero), so as the first digit is equal to the number of 0 in the number,
the second digit is equal to the number of 1 in the number, and so on, the last
digit being equal to the number of 9 in the number.

19. You have 10 white balls and 10 black balls. You must
place these 20 balls in two urns. When you will have placed the balls in the
urns, a man will choose one of the urns at random, then, always at random, one
of the balls in the urn that he will have chosen. If he pulls a white ball, he
will have "won ." How do you arrange the balls in the urns to
maximize the probability of win of this man ?

20. Dogs, cats, mouse. Knowing that a dog costs 15
francs, a cat 1 franc and a mouse 0,25 franc (it is not expensive, but this is
not the problem !), and than one must spend exactly 100 francs to buy exactly 100 animals precisely, and
that besides one must buy at least one animal of every sort, how many animals
of each sort will one buy? It is necessary to solve the problem by a simple
reasoning, not with a system of equations !

21. Jacques decides to make an excursion of two days.
The first day, he will leave at 7h on the morning to climb a mountain and to
arrive on top at 7h in the evening. There is only one path that goes to the
mountain. He will sleep on the mountain, and the following day will retort to
7h the morning to have come back to 7h
in the evening. To go as to
return, he is not in a hurry, sometimes walks, sometimes races, stop
several times to eat, at any hours. What is the probability that he passes, the
two days , at a same point precisely at the same hour ?

22.

5,6,7,8,8,8,8, ?, ?

23. How to get the number 24 while using, each once, the
numbers 1, 3, 4 and 6, as well as the 4 elementary operations and the
brackets ?

24. Antoine, Bernard and Claude are going to fight in a
duel : They are placed at the summits of an equilateral triangle, of
which they don't move. They draw by lot the order of shooting, and each should
shoot on its turn (on one the two others or in the air). The duel will end when
it will only remain one survivor. The three gunmen have different
abilities : Antoine always succeeds, Bernard succeeds in 80% of the cases,
and Claude only once on two. What is the strategy that will give to Claude the
best chances of survive ? What are then the probabilities of survival of
each of the three men ? (Martin Gardner mentions this problem in 1938).

25. On the 25 squares of a chessboard 5x5, how to place
five white queens and three black queens, so that no queen can be taken in one
move ?

__PART II__

* = “easy”

** = medium

*** = difficult

**** = VERY difficult

1 **. Two cubic dice have, on each of their faces, a
natural number from one to twelve. (all faces must be marked with one of this
numbers, and nothing forbids that two or several faces carry the same number).
When one throws the two dice, a sum between two and nineteen always appear.
Besides, all this sums have the same probability. Describe the two dice.

2 ***. As in an old easy problem, you are at a
bifurcation of two roads (one leads to
Paris - where you want to go - , the other elsewhere.) At the bifurcation,
there is a house, in which lives two brothers, among which you know that there
are AT LEAST a liar (which always lies.) You speak to one of the brothers (you don't know which). You have the right
to ask him only one (SHORT) question. What question will you ask?

3 *. Jacques decides to make an excursion of two days.
The first day, he will leave at 7h on the morning to climb a mountain and to
arrive on top at 7h in the evening. There is only one path that goes to the
mountain. He will sleep on the mountain, and the following day will retort to
7h the morning to have come back to 7h
in the evening. To go as to
return, he is not in a hurry, sometimes walks, sometimes races, stop
several times to eat, at any hours. What is the probability that he passes, the
two days , at a same point precisely at the same hour ?

4 **.

5,6,7,8,8,8,8, ?, ?

5 ****

Some clues :

In the sequence, the
followings have something to do :

- Bases (the given numbers are in base 10)

- Probabilities

- Extrema of functions

- Absolute values

- Aesthetic (!)

- Prime numbers

- A very known (easy) "paradox"

Of course, the solution has to be explained.

2,2,3,3,4,5,5,5,6,7,7,7,8,8,8,9,9,9,10,10,10,11,11,11,12,12,12,12, 13,13,14,14,14,14,15,15,15,15,? |