Fourth
International contest **of logical problems**

(organised by the Ludomind society)

The three previous international contests where
organised by Albert Frank and/or Philippe Jacqueroux. This time, the
questions where made by several members of the Ludomind society. It’s
a difficult contest. Send you answers in one single mail __before
June 30 ^{th} 2004 __

by
e-mail to __albert.frank@skynet.be__
(subject: international contest) or
by post to:

Albert Frank

13 Clos du Parnasse / box 45

B 1050 Brussels

BELGIUM

Good luck!

1) 6, 4,
26, 9, 60, ?

2) 4, 7, 11, 12, 14, 18, 20, ?

3) We
draw points on the circumference of a circle.

We have pencils of four different colors.

Every
point is connected to all the others by straight colored lines.

What
is the maximum number of points so that no monochromatic triangle
appear ?

4) From the vertex A of an equilateral triangle A, a laser with thickness zero departs towards the side BC, with angle of 45º measured with the side AB. When it arrives at BC, it is reflected (perfect reflection) towards AC and so on.

What's the minimum number of reflections for the laser to hit a vertex of the triangle? Explain why.

5) 8, 65, 4226, 17859077, ?

6) 4, 4913, 1681, 300763, ?

7) 8, 33, 40, 128, 115, ?

8)
In a building, there is an hexagonal room with one door on each wall.
Each door provides a way to a different room (six rooms in addition
to the hexagonal one). Seen from the interior all of the six rooms
are absolutely identical in content and dimension. They are empty
except for a light bulb on the ceiling. (All bulbs are identical and
have only two states (lit or extinguished). The four walls inside
each room are smooth and white with a door on one of the walls
opening to the central room. The rooms are completely insulated with
nothing leaving the room unless the door is opened. (There is no
keyhole, no sound escapes, etc..) In front of each door, seen from
central room, is a button (a total of six buttons). There is no
interaction between buttons. The hexagonal room is not affected by
the action of the buttons; the hexagonal room is not important to the
problem. A person must discover the function of each button with
regard to its associated room. One does not know beforehand if the
light in the room is on or off. (The rooms may be in different states
at the beginning). Each button can be actuated only one time and
remains blocked thereafter. The person can not actuate the button
after having entered a room (that would be too easy). In each room
there is a sheet of paper and a pencil; the person must write what is
discovered before going out of the room. The doors are marked with a
unique number from 1 to 6 and one must start with door 1. A person
must approach the first button, press it and enter the room. He then
must document the function of that button. He then must leave and
approach the second button, press it, enter the second room, and
document the function of the second button. He must proceed in this
way through the third, the fourth, and the fifth. He must finish with
the sixth to complete the task. Given that the explanation for each
event will be different and that the observations are always correct,
what must be the outcome of the sixth button?

Note that the
man is alone in the building, and that there is no problem with the
electrical power supply in the building.

9) 7, 7, 8, 8, 7, 8, 8, 8, 7, 8, 5, 5, ?, ?, 5, 5

10) Find a way, based on simple probability theory, to get the following finite series:

3, 3, 3, 3, 4, 5, 6, 5, 4

11) Find a way, based on simple probability theory, to get the following finite series:

2, 4, 4, 4, 4, 4, 4, 4, 4, 2

12) 24642, 24976, 28072, ?, ?, 68476, 73372, 73926

13) 1, 1, 1, 2, 3, 1, 1, ?, 2, 4, 1, ?, 2, 1, 3, 1, 1, 1, ?, 2, 1, 1, 2, 3, 1, ?, 1, 3, 3, 4, 1, ?, 3, 2, 1, 1, 1, 3, 1, ?

14) 2, 4, 7, 10, 7, ? (This is not a numerical series).

15) What does the following encrypted word mean and how is it obtained?

LYFKNA

16) 1, 2, 8, 2, 2, 2, 7, 8, 2, ?, ?

17) 52, 72, 11, 23, 31, 31, 15, ?, ?

18) Jacques decides to make an excursion of two days. The first day, he will leave at 7h in 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 go down, leaving at 7h in the morning and arriving back home at 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 ?

19) 5, 6, 7, 8, 8, 8, 8, ?, ?

20) Craig has landed on an island of fun-loving logicians and doesn’t know how to find his way home.

He asks the first person he meets in the street for help, and this native leads him to a secret, mystical place with a large stone engraved with the following drawing:

“I want to go South”, explains Craig. “Is this drawing correct?”

“Judge for yourself”, answers the native. “I can only tell you that one of the arrows points south, but I cannot tell you which one. I cannot tell you how many arrows point in the right direction either, or you would know which way to go.”

Fortunately, Craig was quite bright and worked out which arrow pointed south.

*Can you figure it out too?*