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and TST’s highest levels of difficulty contain only unpublished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.
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
Molière,
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 €40. The payment of the fee entitles you to
a complete report with your IQ expressed on the StanfordBinet,
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: sigma_test@sigmasociety.com

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

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, João 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 João
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 João
and an important bachelor businessman, called José, held a large
barbecue. Aside from the businessman José, the mayor João
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 João 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
Antônio 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 João.
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 João, 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. João
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 João
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.
João 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, João 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 João’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 João 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 Udbjorg 
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]
Se você acha que
o Sigma Teste foi fácil, experimente este: Sigma
Teste VI
If you think the Sigma Test was too easy, then try this: Sigma Test VI

