Sigma
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INTRODUCTION
The Sigma Test (TST) intends to be innovative in many aspects. The
main objective when TST was created was to put together a test of
high intrinsic difficulty without resorting excessively to questions
requiring specific knowledge of mathematics. TST bears no resemblance
to traditional test models based on matrix reasoning or number series,
and its level of difficulty has not been artificially increased
by using exhaustion techniques in question analysis. The test questions,
36 in total, fall to ten levels of difficulty, and TST’s highest
levels of difficulty contain only unpublished questions. See
this article on Hierarchical Factor Analysis of Sigma Test items:
Análise
Fatorial Hierárquica do Sigma Test
(NEW,
jun 2005)
The weighted scoring system, used in conjunction with the raw score,
improves the accuracy of the results as testees’ scores do
not suffer unfairly due to a momentarily lapse or two in concentration
when working on easy questions. Moreover, we believe that the fact
that some of the harder questions have more than one acceptable
answer is an improvement over other tests.
Apart from the varying level of difficulty of the questions, also
the type of reasoning needed to come up with the right answers varies
between questions. Convergent thinking can solve most of the questions
111, while the questions 1220 require more complex convergent
thinking as well as some elemental divergent thinking. In going
from question 21 to question 28, the proportion of divergent thinking
needed increases progressively, and from question 29 onwards, powerful
convergent and divergent thinking is necessary. Only highly inventive
people with strong logical reasoning power may score high on TST.
As for the preliminary norms, we estimate that a person of normal
intelligence would get 4 or 5 questions right. An average academic
with a Bachelor’s degree would be able to answer 9 or 10 questions
correctly. An academic with a Master’s degree would get 13
or 14 right and could become a subscriber to Sigma III. Members
of Mensa would, on average, get 16 or 17 right and would meet the
admission criteria for members of the Sigma Society. An average
Doctor of some of the Exact Sciences would be expected to get 18
or 19 right. Based on the work of Dr. Catherine Cox, we may estimate
that:
Men of Noteworthy Talent:
Napoleon or George Washington would achieve a raw score of about
20
Rousseau or Lincoln would get 23 right (and would qualify for membership
in Sigma III)
Geniuses:
Swift, Rembrandt, La Fontaine, Cervantes or Balzac would get 25
right
Moličre, Lamartine, Benjamin Franklin or Copernicus would get 26
or 27 right
Beethoven, Darwin, Montaigne, Mendelssohn, Watt or Diderot would
get 28 or 29 right
(Sigma IV)
Luther, Lavoisier, Raphael or Alexander Dumas would get 30 right
Great
Geniuses:
Kant, Kepler or Spinoza would get 31 or 32 right
Descartes, Michelangelo, Victor Hugo, Dickens, Musset or Byron would
get 33 right
(and would qualify for Sigma V)
Newton, Voltaire or Galileo would get 34 right
Universal
Geniuses:
Da Vinci, Pascal or Leibniz could get a raw score of 35. (Note:
Da Vinci’s IQ was estimated by Cox at 180, but it was surely
higher than that, possibly close to 200)
Instructions
The
test fee is €
500. The payment of the fee entitles you to a complete
report with your IQ expressed on the StanfordBinet scale,
Wechsler and Cattell scales and statistical data on your
standing relative to the world’s population. The document
will be issued in the name of the Sigma Society Directorate
and is recognized by the founder. Payments can be made in
cash, by bank deposit or by international postal order.
In the case of international postal order, it should be
sent together with the answer sheet. In the case of bank
deposit, a copy of the receipt should be sent together with
the answer sheet.
For information about the postal addresses
to which the answers are to be sent, write to:click
here.
To
receive your certificate shortly, for kindness, send your
answers summarized in this Excel
spreadsheet. Thank you.

Try to answer all the questions even if you’re not sure your
answers are correct, and send in the answer sheet with all the questions
answered.
There is no time limit and there are no restrictions as to the use
of books, calculators, softwares, hammers, pliers or any other tools.
The test can be done over various sessions.
If you want your score to be correct, you should not consult other
people on the questions.
The answer sheet must be printed or typed, and should contain your
full name and address, scores obtained on other tests (including
the names of the tests), and current and former memberships in other
societies.
Provide explanations for your answers only when requested (from
question 26 onwards).
Partially correct answers will also be considered.
From question 26 onwards the following criteria will be applied
in the correction of the answers: functionality (the method must
work in practice), accuracy (the result obtained must be close to
the correct one) and economy (of time, money, material etc.). Most
importantly, the method must work, but the functionality of the
method does not yield the most points. On the other hand, if the
method does not work, points are not awarded at all. Another criterion
is that the method must afford the right result with a small margin
of error. Finally, the method has to be fast and consume little
material. Most points are awarded to answers best meeting these
criteria. It is allowed to consult books whn solving the problems,
but the people occuring in the questions only have the described
material at their disposal or they may acquire material within the
specified budjet.
In some of the questions you may be requested to supply certain
pertinent details or comment on some fenomena affecting the answers.
Failure to do this will result in lower scores for those questions.
Good luck!
To know the scoring method, see
the New Norm
 since 2004
New
norm  Sep 2004
Please, don't discuss the solutions
for these problems in open nor private forums. This would invalidate
this test.
Level
I
1)
In 1976 Marcelo was 11 years old. How old will he be in 1999?
2)
If 13 bullets cost $3.90, how much do 31 bullets cost?
3)
A box measures 60 cm x 50 cm x 30 cm. What is the maximum number
of smaller boxes measuring 10 cm x 10 cm x 10 cm that fit in it?
4)
12 people do a job in 12 days. How many people are needed to do
the same job in 1 day?
5)
A collection consists of 12 volumes. There are 300 pages in each
volume, 50 lines on each page, and 100 letters on each line. What
is the total number of letters in the collection?
Level
II
6)
A company has enough stock to supply its clientele of 2,500 people
for 12 months. How long would this stock last if the clientele grew
to 6,000 people?
7)
If one horse can pull 600 kg, how many horses are needed to pull
6,150 kg?
8)
Fernanda’s and Andreia’s ages total 18 years. What are
their individual ages, given that Andreia is twice as old as Fernanda?
Level
III
9)
Ricardo weighs 30% more than José. If Ricardo were to lose 10% and
José gain 20% of weight, which one of them would weigh more after
that. Explain.
10)
A planetary system has, in addition to the main star, 9 planets.
Each planet has 7 primary satellites. One out of every 21 primary
satellites has 3 coorbital satellites. How many celestial bodies
are there all together?
11)
On a staircase with 1,000 steps there was 1 gram of gold on the
1st step, 2 grams on the 2nd, 3 grams on the 3rd, 4 grams on the
4th, 5 grams on the 5th and so on, so that there was 1 kg of gold
on the last step. Given that 1 gram of gold is worth 11 dollars,
calculate the total value of the gold on the staircase (in dollars).
Level
IV
12)
99% of the people in a room are men. How many men would have to
leave the room in order for this percentage to decrease to 98%?
It is known that the number of women in the room is 3.
13)
On a chessboard with 64 squares (8 x 8), two kings can occupy 3,612
different positions. How many different positions can two kings
occupy on a chessboard with 117 squares (13 x 9). Two kings may
not be on the same square at the same time or occupy adjacent squares.
14)
Marcelo had apples, half of which he gave to his brother. The latter
gave 75% of the apples that he had gotten to be equally shared between
his three cousins: Anderson, Joćo and Mané. Anderson bought 7 apples
more and gave half of all his apples to his brother Mané. Mané’s
apples then totaled 17. How many apples did Joćo get?
15)
Maria went to a farm to buy eggs. Returning home, she gave half
of them to her sister who, in turn, gave a third of those she had
gotten to her boyfriend. The latter, after eating one third of the
eggs that he had gotten, gave the rest to his cousin. Given that
each egg weighs 70 grams, that Maria cannot carry more than 2.5
kg, and that the eggs were raw, calculate how many eggs the cousin
of Maria’s sister’s boyfriend received.
16)
The mayor Joćo and an important bachelor businessman, called José,
held a large barbecue. Aside from the businessman José, the mayor
Joćo and his wife, the number of people present equaled the number
of 100 dollar notes that the mayor spent multiplied by the the number
of 100 dollar notes that the businessman spent. Given that, on average,
every person consumed the equivalent of US$6.40 and that the mayor
invested US$1,700, calculate how much the businessman José invested.
(Note: the businessman José, the mayor Joćo and his wife took part
in the consumption)
Level
V
17)
A Formula1 race car is racing around a circular track, completing
the first lap in 3 minutes with an average speed of 144 km/h. In
what time must a second lap be driven in order for the average speed
of the two laps to increase to 300 km/h?
18)
When Antōnio looked at his watch, he noticed that the hour hand
was lying exactly over the minute hand. What time will it take for
this to happen again? (both hands move at constant rates)
19)
A train with 2 cars is traveling at a speed of 80 km/h from town
X to town Y, located 800 km from each other. At the same moment
that the train departed, a passenger started to walk back and forth
from one end of car B to the other at a speed of 100 cm/s. Arriving
in town Y, the passenger had already gone and returned 720 times.
The length of car A is that of car B plus one fourth of the length
of the locomotive, and the length of the locomotive equals the length
of car A plus one fifth of the length of car B. What is the total
length of the train?
Level
VI
20)
Several faucets were used to fill up six tanks. For one hour, all
the faucets discharged water in a reservoir, which distributed it
between four of these tanks: A, B, C and D. After that, for one
hour, the faucets discharged water in a double funnel which directed
half of the water to tanks E and F and the other half to the reservoir
which, in turn, continued to distribute its water between tanks
A, B, C and D. With this, tanks A, B, C and D were full. To fill
tanks E and F up, it was necessary to use one faucet, which, for
two hours, distributed its water between tanks E and F. After this
all the six tanks were full. What was the number of faucets initially
used? (Note: all the faucets had the same water flow rate and all
the tanks had the same volume).
21)
Several rectangles are drawn on a plane surface in such a way that
their intersecting lines form 18,769 areas not further subdivided.
What is the minimum number of rectangles that must be drawn to form
the described pattern?
22)
Several straight line segments are drawn on a plane surface in such
a way that their intersecting lines form 1,597 areas that are not
further subdivided. What is the minimum number of line segments
that must be drawn to form the described pattern?
23)
1 + 10^1,234,567,890 triangles are drawn on a plane surface. What
is the maximum number of areas, not further subdivided, that can
be formed as these triangles intersect each other? (Contributed
by Rodrigo de Almeida Rodrigues)
24)
According to Fermat’s Last Theorem, a^n + b^n = c^n has no solutions for n > 2 (a,
b, c
and n must be positive integers). In 1992, I proved
this in a simple, yet incorrect manner. This was my reasoning: Fermat’s
Theorem is a generalization of Pythagoras’ Theorem, which
asserts that the sum of areas of the squares drawn on the legs (short
sides) of a right triangle equals the area of a square drawn on
the hypotenuse of the same right triangle (a^2
+ b^2 = c^2). If we try to generalize that theorem, going
from 2 to 3 dimensions (a^3 + b^3 = c^3), we have a triangular prism formed
by displacement of a right triangle along an axis perpendicular
to its face, as illustrated by the figure below.
We
can construct a cube on one of the three quadrangular faces of that
prism. Two of those faces correspond to the legs of the right triangle
(ADFB, BFEC) while the larger face corresponds to the hypotenuse
(ADEC). It is possible to construct a cube on one of the faces,
implying that the 4 sides of that face have the same length. This
affects the whole prism, causing the cube constructed on the other
face to have the same size than that constructed on the first, for
if AB=BF and BF=BC, then AB=BC. In that way, no cube can be constructed
on the third face, for if AC represents the hypotenuse, then AC
cannot be equal to to AB. Therefore, a^n
+ b^n = c^n has no solution for n=3. Following the same
line of reasoning, we can show that it has no solution for any number
of dimensions larger than 2. What is the error in this proof?
Level
VII
25)
A certain gear system consists of 5 concentric, superposed discs:
A, B, C, D and E, which are mounted on a solid platform, taken as
a stationary reference. The discs have different sizes and spin
at different speeds. All the discs spin at constant rates, some
clockwise, some anticlockwise. Each disc has a red dot on its surface,
and initially all these red dots are not lined up. At a given moment,
all the discs start to spin simultaneously, each at its own speed,
without any contact between them. It takes 7 minutes for disc A,
13 minutes for disc B, 17 minutes for disc C, 19 minutes for disc
D and 23 minutes for disc E to complete a full 360degree spin.
After a certain time, all the red dots were aligned, disc A being
in the same position that it was 2 minutes after the discs started
to spin, disc B being in the same position that it was 3 minutes
after the discs started to spin, disc C being in the same position
that it was 4 minutes after the discs started to spin, disc D being
in the same position that it was 7 minutes after the discs started
to spin, and disc E being the same position that it was 9 minutes
after the discs started to spin. How much time elapsed from the
moment the discs started to spin until the discs reached that configuration
for the first time?
26)
Pedrinho entered Dona Maria’s Stationer“s Shop and asked her
to sell him a geometric ruler for drawing a spiral with a small
concentric circle. Dona Maria, a Sigma Society member, told the
boy that there were no rulers for drawing spirals. But after thinking
the problem over, she found a way to make a drawing like that, and
described the method to the boy. She sold the boy the material needed
right away, which he paid with a US$ 10.00 note, receiving some
change. He went home and made the drawing without any problems.
Describe a method to perform Pedrinho’s task having the same
US$ 10.00 at your disposal for buying the material needed. The drawing
must show a satisfactory agreement with the described pattern (a
spiral with a small concentric circle), without large irregularities
in the spiral. (Modified in 31 Aug 2001 at the suggestion of our
friends Petri Widsten and Nikos Lygeros, as the earlier question
with the 9 cubes was similar to one of the questions of the Eureka
Test).
27)
A man takes a deep breath, filling his lungs completely with air.
Then he holds his breath and a tape measure is used to measure his
chest circumference, which turns out to be 106 cm. After that, the
man exhales so that all the air is expelled from his lungs. His
chest circumference is measured again, and is now 84 cm. Having
$10 at your disposal for buying material, find out the volume of
air that his lungs are able to hold.
28)
The speed of a person’s reflexes can be determined based on
the time elapsed between a stimulus and the response of that stimulus.
For example: A lamp remains unlit while we observe it. On receiving
the stimulus “the lamp was lit”, the reaction is to
be “close the eyes”. The shorter the time between “the
lamp was lit” and “close the eyes”, the faster
the reflexes. Describe a method to determine the speed of a person’s
reflexes, without using a chronometer or any other equipment allowing
the measurement of time intervals shorter than 1 second. It is possible
to devise a rough method on a US$ 1 budjet for equipment, and a
sophisticated method with good precision having US$ 1,000 at one’s
disposal. Describe a method for both budjets.
29)
In 1993, in an essay about Science and Religion, I described a project
regarding the possibility to build an “invisibility machine”.
On describing the details, I realized that some problems were insolvable,
not only because of technological limitations but also for physical
reasons imposing theoretical and possibly insurmountable limits.
The project starts from the central idea that in order to make an
object invisible, it is necessary for an external observer looking
in its direction to visually stop noticing its presence. This can
be done in the following way: A sphere is constructed, and its whole
external surface is covered with minute, highresolution TV cameras
and monitors. Millions or even billions of cameras and monitors
are to cover the whole sphere in such a way that each monitor transmits
the image captured by a camera located in the point diametrically
opposite to that monitor. The result will be as shown in the figure
below.
The
image of the object (blue square) is captured by a camera located
in point A, which transmits the image to a monitor in point M. As
a result, an observer in point O will see the blue square as if
there were nothing in front of him. In that way, everything inside
the sphere will be invisible to the external observer. But this
scheme presents two problems. One of them can be solved in theory
while the other one is insolvable. Indicate those two problems and
explain why one of them can be solved but the other one cannot.
Level
VIII
30)
The porous and gray “lead” inside a pencil consists
of a mixture of graphite and clay. The ratio of graphite to clay
is not known. On writing on a sheet of paper, a fine layer of “lead”
remains on the surface of the sheet. Describe a method for calculating
the mass of “lead” in the dot of the letter “i”.
You may use only US$10 to buy the material needed for the experiment.
31)
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.
32)
A sophisticated aircraft is hovering like a hummingbird over a terrain
located on the equatorial line of a planet, at an altitude of 1,000
m. The planet is completely spherical and homogeneous, and has a
small satellite on a circular orbit on a plane parallel to its equator.
At 15:58:30h a man parachutes down from the aircraft, descending
perpendicularly to the ground. At the moment that he jumps off the
aircraft, he notices a satellite starting to rise on the eastern
horizon. He lands and, without leaving the landing site, continues
to observe the satellite, which at 17:40:00 h reaches the zenith.
He remains in the same place, observing…and at 19:20:00h sees
the satellite disappear on the western horizon. Still in the same
place, at 22:40:00h, he sees the satellite rising again in the east.
What is the approximate diameter of that planet? Explain how you
arrived at your answer and the usefulness of all the pieces of information
given. Explain also why the result cannot be exact.
(If you have doubts as to the meaning of zenith, horizon, equator,
orbit etc., you may consult dictionaries or encyclopaedias).
Level
IX
33)
Describe a practical and fast method that can be used with good
precision to determine the number of words in a person’s vocabulary.
34)
There was a brillant anthropologists, member of Sigma V, named Joćo.
During an expedition to Africa he was captured by a tribe of cannibals
and sentenced to serve as a meal. However, the “legislation”
of the tribe offered the prisoners a chance to be freed should they
be able to overcome a challenge. In the case of Joćo, the challenge
was as follows: he would be presented with two chicken eggs, one
of them raw and the other one boiled. There would be two boxes.
The raw egg would be placed in one box and the boiled egg in the
other one. Joćo doesn’t know what the dimensions of the boxes
are until he faces the challenge. The walls of the boxes are rigid
and opaque. The have the shape of a parallelepiped. One of the boxes
has a window in one of its walls. The window is covered by a wire
screen whose mesh size Joćo does not know until he faces the challenge.
Through the window it it is possible to observe the egg inside the
box.
The challenge is to find out within 2 minutes which one of the two
eggs is raw. It is not allowed to break the eggs, take the eggs
out of the boxes or open the boxes. Nothing solid, liquid or gaseous
may be put inside the boxes.
Joćo is informed that the challenge would be presented to him after
90 days. Before that, he may count on the help of the villagers
to help him work out a solution to the problem. Aside from that,
he has the use of all “sophisticated” instruments and
everything else that he can find in the village and its surroundings.
When the time came for him to face the challenge, at dawn, Joćo
was blindfolded and his hands were tied. A few minutes after that,
an old villager took an egg, boiled it, dried it, and placed it
in a box which he closed. He then took a raw egg and put it in another
box, closing the box immediately. The two boxes were placed on a
table where they stayed until nightfall. Then Joćo’s hands
were untied, the blindfold was removed, he was supplied with the
equipment he had requested earlier, and he was taken to the table
on which the boxes containing the eggs were lying. He examined them
carefully and succeeded in finding out which box contained the raw
egg. The challenge was repeated daily for a period of 20 days, each
time with different eggs, and every time he was able to identify
the raw egg. The admiring cannibals then proceeded to set him free
and even gave him lots of jewelry as a gift. How did Joćo
manage to save himself?
We
are recommending those who are taking the "Sigma " test ,not
to try out the quest in real life !It can bring you into very
dangerous situation.
We don't take any kind of responsibility for possible physical
or other problems caused by trying out the questions in real
life.
We would like to tell you about the following true story,
which has made a deep impression on us, the story tells what
might happen if you try to carry out the questions in reality.
Our friend, David Udbjorg, from Denmark, risked his life by
trying to solve the problem. He traveled to Africa. He found
a local tribe of cannibals, in order to try out question no.34.
But the Cannibals, didn't know about the Sigma test and consequently
haven't read the agreements. So they decided that David should
be the next meal. Fortunately, on the same day as David was
going to be served, there would be a solar eclipse at 12 o“clock.
OF course David knew this, and threatened to take the sun
away forever. The cannibals didn't believe David, but as the
sun started be shaded by the moon, they let him loose. David
told them that he would forgive what they had done and bring
the sun back. And the sun returned ! Our hero was celebrated
by the cannibals because he saved the town. David sent a photo
as proof.
Photo:
curtesy of David Udbjor

35)
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.
Level
X  EXTRA
(it is required to get at least 3 questions
of levels VIIIX right to try to answer this question)
36) The great
poet Joao spent the last days of his life lodged in the cellar
of his friend Jose·s house. Jose, a petty merchant, was
a man with meager possessions but very generous. Before dying,
Joao entrusted his friend with an unpublished poem. The title
of the poem, published posthumously, is irrelevant to the problem
at hand.
Joao called that humble and generous friend just ·Amphibian·.
Once his friend asked him why he always called him by that name,
and Joao explained.
Take into consideration that Joao held his friend in high
esteem and, within the context, find a logical explanation for
the meaning of ·Amphibian·.
[This text
is based on reallife events]
