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 co-orbital 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 Formula-1 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 360-degree 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, high-resolution 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 VII-IX 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 real-life events]