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 30th 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 1 door on each wall. Each door gives a way to a different room. (6 rooms in addition to the hexagonal one). Seen from interior all the rooms are absolutely identical in contents and dimension. They are empty except for a light bulb on the ceiling. (all bulbs are identical and have only two states (lit or extinct). The 4 walls inside each room are smooth and white and a door on one of the walls open a path to the central room. Rooms are completely insulated and nothing leaves from if the door is not opened. (no keyhole, no sound etc...). In front of each door, seen from central room, is a switch. (6 switches). There is no interaction between the switches. the hexagonal room is not concerned with the action of the switches and is not significant. A person must discover what each switch produces in each associated room. It does not know before if the light is lit in the room or not. (the rooms could be in a different state at the beginning) . The switch can be actuated only one time and remains blocked. the person can not actuate the switch after having entered a room .(too easy; -) In each room ,there is a sheet of paper and a pencil and the person must write what it discovered before going out from the room. The doors are marked with a number from 1 to 6 and it must start with door 1. The person approaches the first switch, actuate it and enters the room. He then gives an explanation of the function of the switch. It approaches the second switch then actuate it , enters the room, and gives an explanation of the function of the second switch. It makes in the same way for the third the fourth and the fifth. Then finishes by the sixth and is victorious. Knowing that the person have to give a different interpretation to each event and that it is victorious, which event produced the sixth switch?
Note that the man is alone in the building, and that there is no problem with the electricity 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?
