
Some notes about the test:
Changes:
MPT SHORT FORM (2002 Version): A smaller test, with the first 40 questions (numerical and grids sequences) is available. Be aware that this short form is not considered valid for admission at Sigma Society. Click HERE.
"Every man ought to be inquisitive through every hour of his great adventure down to the day when he shall no longer cast a shadow in the sun. For if he dies without a question in his heart, what excuse is there for his continuance?"
Frank Moore Colby, The Colby Essays
PART I – NUMERICAL SEQUENCES
Replace the question marks with the numbers that best complete the sequence
PART II  GRIDS
Fill in the empty grids
Suggestion: Send your answers as (10010 00100 00011 11000 11010 ) where each group of five digits corresponds to one row of the grid, starting from TOP, i.e. first 5 digits=first (top) row. A "1" will stand for a filled square in the grid, a "0" for an empty square.
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PART III  LOGIC
41. You are locked in a room with two strangers, Joe and Jim. One of them has the key that opens the only door for the way out. You do not know who has the key and you are allowed to ask both of them "Which of you has the key ?". Jim answers "Everything Joe says is true and I have the key."; Joe, in its turn, answers "Whatever Jim says is false, but he has, in fact, the key." Which of them has the key ?
42. Several statements, numbered from 1 to 100, are written in a sheet of paper. Statement n says "Exactly n statements in this sheet are false". How many false statements are there ?
43. Assume now that in the previous question the expression "Exactly n..." is replaced by "At least n...". In this case, how many false statements are written in the sheet of paper ?
44. You want to send a valuable object to a friend. This object is a valuable unique old piece of jewelry, desired by several individuals, some of them wellknown thieves. You have a box which is more than large enough to contain the object and offering several ways of putting a lock in it, rendering it virtually inviolable. You have several locks with keys, but your friend, obviously, does not have the key to any lock that you have. How do you do it? Note that you cannot send a key in an unlocked box, since it might be copied and used later for opening the locked box while in transit...
45. Hard job you are going to face: You have eight bags, each of them containing 48 coins. Five of these bags contain only true coins, the rest of them contain fake coins. Fake coins weigh 1 gram less than the real coins. You do not know what bags have fake coins and what bags have real coins, You do not know also, besides that it is an integer value, the weight of the real coins. You can use a scale, a dynamometer type one, with precision up to 1 gram. Making only one weighing and using the minimum number of coins, how can you find the bags containing the fake coins ?
PART V  PROBLEMS
46. The five images below represent five views (from five of its six sides) of a solid object. This object has been assembled by gluing together several identical small cubes so that at least one face of each small cube is totally adherent to a face of another contiguous small cube. Each black line shown represents a side of the object that is perpendicular to the plane of this page. Draw the sixth view of the object and calculate smallest number of cubes needed to assemble the object, as well as its respective colours.
47. A PingPong ball maker packs the balls in card boxes in such a way that the balls end up perfectly packed, not suffering any damage, in a box of minimum size. Trying to optimize this operation, he has decided to put more balls in each box, going from 6 balls per box to 9 balls per box. Knowing that each ball has a perimeter of 9 centimeters and that each square centimeter of card used in the box making costs 0.01 US$, calculate how much it will cost more than the old box, in percentage, the new box that will be used to pack the 9 balls.
48. What should be the minimum area of a chessboard so that it would be possible to place 19 coins of 2.5 centimeters in diameter, following this set of rules: (1) Any coin should touch at least one of the contiguous coins; (2) None of the coins is allowed to overlap, partially or totally, any other coin; (3) None of the coins is allowed to pass over any of the chessboard edges (4) The remaining free area, not occupied by coins, of the chessboard, should be the smallest possible.
49. A bar waiter uses a circular salver with a diameter of 24 centimeters, having an elevated border of about 1 centimeter height. Knowing that all the glasses in the bar are identical, with a shape of a cylinder having a diameter of 6 centimeters, what will be the maximum number of totally filled glasses the waiter is able to safely carry (not putting, for instance, a glass on top of another), and without spilling, using this salver. Consider that the referred diameter of the salver is the "usable" inner diameter.
50. What is the minimum number of ellipses needed so that the maximum number of distinct and not further subdivided areas resulting from its intersection would be equal to the maximum number of distinct and not further subdivided areas resulting from the intersection of 120 circles ?
51. Regarding the image below, check if it is possible to draw a path that, starting from the vertex marked red, passes through all the vertices, do not pass by any vertex more than once and finishes, again, in the red vertex. If you do think so, draw such a path.
52. Consider the image shown below: In how many different ways can you place only positive integer numbers in the vertices of this solid so that: (1) the sum of all external vertices (the eight vertices more distant from the solid's "center") is 6; (2) the sum of all inner vertices (the eight vertices in the solid's "hole") is 7; the sum off all sixteen vertices is 5. Configurations attained by rotations or symmetry are not considered different and it is allowed to repeat the numbers to be placed at the vertices.
53. A bag contains 10 balls of different colors. Randomly select a pair, repaint the first to match the second, and replace the pair in the bag. What is the expected number of times you will have to repeat this procedure until, undoubtedly, the balls are all the same color? Another question: suppose that after 10 repetitions, you decide to stop. You randomly pick a ball, check its colour (assume it is blue) and put the ball back in the bag. You repeat this procedure 10 times and in every repetition, the ball has the same colour (blue). What are the probabilities that all the 10 balls in the bag are blue ?
54. In a special dice game the "house" rolls two 20sided dice and the "player" rolls one 20sided die. If the player rolls a number on his die between the two numbers the house rolled, then the player wins. Otherwise, the house wins (including ties). What are the probabilities, in the beginning of the game, that the player wins?
55. Two spheres are the same size and weight, but one is hollow. Each is made of an uniform material, obviously not the same material, apparently equal to sight and touch. With a minimum of apparatus and not causing any damage to the spheres (you can not just make an hole in one of the spheres) how can you find out which is hollow?
56. Take a good look at the map, from a particular area of New York City, shown below. You are standing at point "1" and your mission is to reach point "2":
(cont. 56) We want to know how many different ways are there of reaching "2" starting from "1", if you travel always (and only) in the (approximate) North or East direction.
57. The two images shown below depict two opposite views of a Rubik's cube after an undetermined number of moves. Your job is to find out the minimum number of moves needed to, starting from the depicted position, completely solve the cube, so that each side of it has a single colour. You may consider that the position shown was reached after a undetermined number of moves having the objective of completely solve the cube.
58. Using nothing more than a ruler without marks, a pencil and a square sheet of paper, your task is to divide the angle "alpha" shown below in three perfectly equal parts.
59. In a conventional chessboard, a knight has been placed in the position shown in image 59I, being that square numbered as "1". Now, making only legal movements (by the chess rules), numbering any square visited by the knight with a sequential number (i.e. 2,3,4...) and not visiting any square more than once, devise a way of visiting all the squares in the chessboard in such a way that the sum of each row and each column would be equal to maximum number of distinct and not further subdivided regions of space resulting from the intersection of ten spheres (as you can see in image 59II, a maximum of 8 distinct and not further subdivided regions of space are the result of the intersection of at most three spheres).


59I  59II 
60. The image below shows an icosahedron, a regular polyhedron (also called platonic solid), having 20 equivalent equilateral triangular faces, 12 vertices and 30 edges. If we consider that no two faces with a common edge, may share the same colour, how many ways are there of painting an icosahedron using only (a) five colours? (b) three colours? Since you are holding the icosahedron in your hands, find out if there is a way of drawing a path starting from any vertex that visits all the vertices and no vertex more than once, ending in the starting vertex. Yes ? Then, please, draw it.