Sigma Society

Each question will receive a raw score ranging from 0 to 1. The raw scores will be multiplied by the weight factors of the questions. The sum of the products thus obtained constitutes the weighted score on the test.

Question 1 = weight 2
Question 2 = weight 6
Question 3 = weight 2
Question 4 = weight 15
Question 5 = weight 20
Question 6 = weight 15
Question 7 = weight 20
Question 8 = weight 20
Question 9 = weight 25
Question 10 = weight 25
Extra question = weight 20

                 All answers must be submitted at the same time. After you present your answers, it won't be allowed to modify them.

The questions have been arranged in order of increasing difficulty, the first two forming part of the Sigma Test. The last two questions are probably the most difficult and most important because they involve fundamental concepts such as time, identity, and limits of empirical and theoretical knowledge. These topics are essential in a test that aims to discriminate with some degree of confidence at a level above 190.

There is no time limit. It’s allowed to use computers, consult books etc. If you want your score to be correct, you shoudn’t consult other people.

                 This is a hard test. Based on the degree of difficulty of the first two “easiest” (or least difficult) questions and the norms of the Sigma Test we can estimate that 9.999 (or maybe 10.000) out of every 10.000 people will not be able to solve any of the questions.

                 At first the test was only available to members of Sigma V. However, after several people expressed their interest in the test it was decided to make it available to everybody.

                 A suggestion to optimize your score: believe that the questions are difficult and they are approximately in order of difficulty.

1) We have a cylinder with a radius of 50 cm and a tape measure 0.01 cm thick. The height of the cylinder equals the width of the tape measure. The thickness of the tape measure is invariable and one of its wider sides is inextensible. What is the minimum length of tape necessary to wind it around the cylinder 9 times, all rounds overlapping, as in a roll of scotch tape. The top and base of the cylinder may not be covered with tape. The solution must be given with 14 significative digits and it is not allowed to cut the tape or cut or deform de cylinder.

2) An Arab man and an Israeli woman are abducted by extraterrestrials. The E.T.s promise to return them to Earth unharmed, provided that they succeed in the following task: three rooms are designated A, B and C. Each room is square and measures approximately 25 m2. The rooms are connected in such a way that each room has two doors, and each door provides access to one of the other two rooms. The three rooms are acustically isolated and have no furniture or windows. The walls, doors, ceiling and floor of the rooms are solid and opaque, and contain no cracks, holes, hidden passages or the like. The man is placed in room A and the woman in room B. They both receive the following instructions:

1- They both have 1 hour to traverse the three rooms and return to the room where they started, always walking in the direction A - B - C - A.
2- The both have to remain seated, on the floor, in their respective rooms, until a signal would be emitted, indicating that the time count had started. The signal was as follows: on each door there are two lamps (one on each side of the door), and the nearly simultaneous lighting of the all the lamps constitutes the signal. Each lamp is bright enough for a person to notice easily even when he is not paying attention to it.
3- The moment that the woman touches the doorknob of a room, the man cannot be in that room any more.
4- The moment that the man touches the doorknob of a room, the woman cannot be in that room any more.
5- The woman has to get up from the floor after the man.
6- The man and woman are not permitted to communicate between each other in any way, or obtain from others any information allowing them to figure out where the other one is. They may not beat the walls or the doors, or try to generate any kind of shock wave. On leaving a room and entering another one, it is required to close the corresponding door. Initially all the doors are closed. Two or more doors may not be open at the same time.
7- None of them has a clock or any other instrument that can be used to measure time.
8- 1 minute before the 1 hour period is up the light signal will be given again, indicating that the time is running out.
9- When the 1 hour period is up the man has to be sitting in the center of room A and the woman in the center of room B.
10- The woman may only sit down after the man.
11- The man is told that the woman is exceptionally intelligent.
12- The woman is told that the man is exceptionally intelligent.

The man and the woman did not know each other and had never been in any contact with each other before. They did not communicate with each other during the whole process (to clarify the matter, it can be told that they both were mute and deaf). The experiment is carried out and they manage to perform the task. The experiment is repeated 10 times and each time they complete the task successfully, making it clear that the first time was not due to mere good luck. Afterwards they are returned to Earth where they convert to Zoroastrianism, get married and live happily everafter! Describe the method they used and the way of thinking of both of them.

3) In an orthogonal hyperspace of 256 dimensions there is a hypersphere of 6 dimensions that we want to divide into as many pieces as possible with no more than 12 cuts. The pieces cannot be moved from their original positions. Each cut is an orthogonal 5-dimensional Euclidian hyperspace (hyperplane). What is the maximum number of pieces into which the 6-dimensional hypersphere can be cut in this way?

4) There was an island inhabited just by seagulls. Some of them caught a lethal but non-contagious disease. The only symptom of the disease is a dark spot on the neck. The neck shows no protuberance or increased sensibility so that it is not possible for a given seagull to know whether it has one. A few months after contracting the disease, all infected seagulls die in a terrible way. Therefore, to minimize its suffering, when a seagull is sure of being infected, it commits suicide exactly at 11 pm on the day that it learns of its illness. The seagulls are highly intelligent, but they are unable to communicate with each other. They know how to count and they know the total number of seagulls on the island. Once a day, exactly at noon, all of them meet so that some of them see the spots on the other seagulls' necks, but they never see the spot on their own neck or find out from other seagulls whether they have one or not. A seagull with a spot on its neck always carries the disease. During the first 39 meetings, none of the seagulls commit suicide. After 39 days and as many meetings, all of the seagulls with a spot on their neck commit suicide at 11 pm. The seagull population is constant from the first meeting until the day the suicides occur. How many seagulls commit suicide and how do they find out about their spots?

5) At the starting position of a chess match, the whites have 20 different ways to make the first move. Whatever the first move of the whites, the blacks also have the same 20 possible first moves. Based on that, we can calculate that after the whites and blacks have made their first moves, 400 different positions are possible on the board. It is believed that there are nearly 100 billions of galaxies in the universe and that, on average, each galaxy has some 100 billions of stars. The average mass of a main sequency star is about 0.7 times the mass of the Sun, i.e., about 1,4x10^33g. As Avogadro’s number is approximately 6x10^23mol^-1 and 90% of the atoms of the universe are hydrogen (atomic mass ~1) and 9% are helium (atomic mass ~4), we can calculate that the number of atoms in the universe is about 6x10^78. Is the maximum number of different positions on a chessboard after a match of 29 moves (29 for whites and 29 for blacks) greater or smaller than than the number of atoms in the universe? Explain how you arrived at your answer and tell the total number of different positions that can be produced in all possible chess matches.
For practical reasons, considering that some solutions may be too lengthy, you have to get one of the first 4 questions right in order for your remaining answers to be scored. Otherwise your test will end here and your final score will be that obtained on the first 4 questions.
Click here to see clues for this problem. (since September 9 2002)

It's required to get at least one (full) of the first five questions right for the remaining answers to be scored.

6) An alien with limbs resembling arms and legs, coming from a distant planet, arrived on Earth. As he landed on a field, he was attacked by a farmer who cut one of his legs off at knee height. Not happy with the way he had been welcomed, he set off on a return journey to his planet. The journey would last several millions of years. As exobiologists examined the leg that the farmer had cut off, they concluded that it could only belong to an alien being. The farmer told them how he had cut the leg off and described the alien as a large-headed, bald, long-limbed and slow-moving hominid. The scientists then started sending off messages to different directions to inform the inhabitants of the alien’s home planet that their emissary had lost a leg. Only two kinds of signals may be transmitted, represented by two different wave lengths. Even though the scientists did not know the language of the aliens, they managed to find a way to express the desired information. Describe a way to to transmit the information concerning the alien’s amputated leg. Bear in mind that the aliens receiving the message are highly intelligent.

7) If it were possible to travel to a friendly planet situated millions of lightyears from the Earth, and you had to describe “an orange” to an intelligent alien who doesn’t know anything about our culture, what kind of description would you give?

8) Create a functional model of a fictional universe. The universe should be as different as possible from ours and governed by coherent laws. You need not know physics. It’s enough for your universe to be coherent.
You have to get one of the first 3 of the first 7 questions right in order for your remaining answers to be scored, otherwise your test will end here and your final score will be that obtained on the first 7 questions.

Your answers 9 and 10 will only be corrected if you get right at least 3 of the first 8, otherwise, your test will finish for here and your final score will be what you have obtained in the first 8 questions.

9) Which of the sentences below cannot be wrong? Explain.
a) The Sun cannot assume the shape of a cube.
b) The ratio pi/2 cannot be smaller than the square root of 2.
c) A person cannot have 3 arms.
d) An animal cannot observe with any other organs than eyes.
e) The choices ‘a’ and ‘b’ are correct.
f) The choices ‘a’ and ‘c’ are correct.
g) The choices ‘a’ and ‘d’ are correct.
h) The choices ‘b’ and ‘c’ are correct.
i) The choices ‘b’ and ‘d’ are correct.
j) The choices ‘c’ and ‘d’ are correct.
k) The choices ‘a’, ‘b’ and ‘c’ are correct.
l) The choices ‘a’, ‘b’ and ‘d’ are correct.
m) The choices ‘a’, ‘c’ and ‘d’ are correct.
n) The choices ‘b’, ‘c’ and ‘d’ are correct.
o) All the above are correct.
p) None of the above is correct.

10) Which of the sentences below cannot be wrong? Explain.
a) The Earth orbits the Sun.
b) Cats cannot lay eggs.
c) You cannot be me.
d) If João is Pedro’s son, João was born after Pedro.
e) The choices ‘a’ and ‘b’ are correct.
f) The choices ‘a’ and ‘c’ are correct.
g) The choices ‘a’ and ‘d’ are correct.
h) The choices ‘b’ and ‘c’ are correct.
i) The choices ‘b’ and ‘d’ are correct.
j) The choices ‘c’ and ‘d’ are correct.
k) The choices ‘a’, ‘b’ and ‘c’ are correct.
l) The choices ‘a’, ‘b’ and ‘d’ are correct.
m) The choices ‘a’, ‘c’ and ‘d’ are correct.
n) The choices ‘b’, ‘c’ and ‘d’ are correct.
o) All the above are correct.
p) None of the above is correct.