Guilherme Marques

Mais que opiniões, conteúdo embasado.

Written by: on 17 de dezembro de 2015 @ 9:35

Logo PuzzleUp11_fundo

Puzzleup é uma das mais disputadas competições internacionais online de puzzles (desafios de lógica). Anualmente, Emrehan Halici, da Índia, promove o concurso que já está na décima primeira edição. Em 2015 entraram na disputa competidores fortíssimos como Ulrich Voigt, dez vezes campeão mundial do WPC (World Puzzle Championship – Campeonato Mundial de Puzzles).

Veja a lista dos 35 primeiros colocados:

35primeiros

A competição foi realizada entre 29 de julho de 2015 e 9 de dezembro do mesmo ano, sendo composta por 20 desafios semanais de altíssimo grau de dificuldade. A resolução de alguns deles priorizava a lógica, outros exigiam conhecimentos de áreas da matemática como teoria dos grafos e combinatória e muitos exigiam conhecimentos de programação e algoritmos.

Em 2010, outro brasileiro, Carlos Eduardo Rodrigues Alves, que hoje trabalha no Google, na Califórnia, venceu a competição. Carlos, assim como Guilherme, participou do WPC (World Puzzle Championship), em 2005 na Hungria e em 2006 na Bulgária, representando o Brasil na competição. Na edição do Puzzleup de 2010, vencida por Carlos, Guilherme terminou na décima quarta colocação, em sua primeira participação nesse concurso.

Conheça os desafios de 2015, que também estão disponíveis na página da competição http://www.puzzleup.com/

Obs.: Atendendo a pedido do criador dos puzzles, Emrehan Halici, não iremos divulgar as respostas dos problemas nem comentar as resoluções, para não tirar o prazer daqueles que queiram tentar resolver esses belos desafios.

 

1) SAME SUM

You randomly select two different 2-digit numbers and calculate their sum. Your friend does the same operations. What is the probability of you and your friend having the same sum?

If the problem was asked for two 1-digit numbers, the answer would be 29/405.

Enter your answer as a simplified fraction.

2) COLORED CUBES

You have red, blue and green unit cubes. You will create a 2x2x2 cube by using 8 of these unit cubes. What is the maximum number of different cubes you can create?

Note: If a cube can be obtained by rotating another one, these two cubes are not considered to be different.

If the problem was asked for only red and blue unit cubes, the answer would be 23.

3) HALF OF ITS SQUARE

All digits of a number are different and  for each of its digits that has at least two other digits on its right, the half of the square of that digit is greater than the multiplication of the two digits immediately following on its right.

What is the greatest number having this property?

4) SUM OF PALINDROMES

Three different numbers between 10 and 1000 are palindromes and their sum is also a palindrome. What is the maximum possible value of the multiplication of these three numbers?

-The numbers cannot start with 0, for example 010 is not a valid number.

-A palindrome is a number which reads the same backward or forward. Examples: 11, 101, 292.

5) TEN POINTS

You will place 10 points on a paper and you will connect all these points to each other by lines.

What can be the minimum number of intersections of these lines?

Notes:

-There will be a line between every two points.

-Only two lines can intersect at an intersection point.

-Lines are not required to be straight.

6) CHESS COLORS

You have an 8×8 table on a computer screen with all squares colored to white. In each step you will select any square and as a result all the squares on the same row and column -including the selected square itself- will switch their colors (white becomes black, and black becomes white).

What is the minimum number of steps required for obtaining a standard colored chessboard?

7) WORD GUESS

Your friend will pick a 4-letter word and you will make guesses in order to find it.

-A word can contain only the letters A, B, C, and D, and they can be used more than once. (AAAA-DDDD).

-In your guess if at least three letters are in their correct places you will win a prize.

What is the minimum number of guesses in order to guarantee to win the prize?

8) ADDITION

A+B=C. In this operation all of the 10 digits forming these three numbers are different. What can be the maximum value for C?

Example: 324+765=1089.

But 1089 is not the maximum value.

9) CUBE TRAVEL (ANULADO)

p9f1.jpg

You will make a travel through the edges of two cubes connected by their vertices as shown in the figure.

-You will start on A, and finish on B.

-You can pass through the vertices more than once except the finishing vertex B.

-You can’t pass through the edges more than once.

In how many different ways can this travel be done?

Note: Two travels having same edges in different order will be considered as different.

10) 16-DIGIT NUMBER

Using the digits 1, 2, 3, 4, and 5 you will form a 16-digit number such that for every adjacent two digits either both of them will be the same or at least one of them will be 1.

How many different numbers can be formed?

11) WINNING NUMBER

Five numbers will be randomly picked with replacement between 1 and 100 (1,2,…,99,100), and the largest of these five numbers will be named as the winning number. This process will be repeated many times and the average of all the winning numbers will be calculated.

What is the integer nearest to this average?

12) DIGIT TABLE

p12f1

In the table given above in how many different ways can the number “12321” be obtained?

-You can start from any “1”.

-From a square you can move only to a neighboring (adjacent horizontally, vertically, or diagonally) square.

-You can use a square more than once in any “12321”

Example:

p12f2

If the same problem was asked for obtaining “121” in the sample table, the answer would be 208.

13) SET OF CODES

You will produce a set of 7-letter codes using the the letters A, B, C, D, E, F and G.

-Two codes are called similar if they differ by just one letter.

-No two codes will be similar in the set.

-Letters can be used more than once in a code.

What can be the maximum number of codes in this set?

If the problem was asked for a set of 3-letter codes using the letters A, and B then the answer would be 4 (Example: AAA, ABB, BAB, BBA).

14) JUMPING PAWNS

There is a 1×20 board and 10 pawns are placed on the leftmost squares. Your task is to move these 10 pawns to the rightmost squares in minimum steps.

A step consists of either a MOVE or a JUMP

-MOVE: Pawn can move to the empty adjacent square to its right.

-JUMP: Pawn can jump over adjacent right pawn to the next square if it is empty. Jumping must continue till the pawn can’t jump.

What is the minimum number of steps to accomplish this task?

If the problem was asked for 1×6 board and 3 pawns, then the answer would be 5.

15) HANDS OF A CLOCK

How many times at least two of the three hands (hour, minute, second) of a clock exactly overlap between 10:30 and 22:30?

16) SQUARES

What is the minimum number of squares to be drawn on a paper in order to obtain an 8×8 table divided into 64 unit squares.

Notes:

-The squares to be drawn can be of any size.

-There will be no drawings outside the table.

Two examples for a 3×3 table:

The second one with 4 squares is the solution for a 3×3 table.

17) ORDERED CODES

How many 25-letter codes formed by using the letters A, B, C, D and E have their letters in alphabetical order?

If the problem was asked for 4-letter codes with the letters A, B, and C, the answer would be 15:

AAAA, AAAB, AAAC, AABB, AABC, AACC, ABBB, ABBC, ABCC, ACCC, BBBB, BBBC, BBCC, BCCC, CCCC.

18) ADJACENT DIGITS

What is the maximum number having the following properties?

-All of its digits are different.

-When you add any three adjacent digits of this number, the digits of the result will not be found in the number.

19) QUEENS

In how many ways can white and black queens be placed on a 4×4 chess board, such that every row and every column has more white queens than black queens?

Note:

-On each square you can place only one queen (white or black) or you can leave it empty.

If the problem was asked for a 3×3 board, then the answer would be 451.

20) POWERS OF THREE

7 students have selected some numbers which are powers of 3 (1, 3, 9, 27, 81,…). Interestingly, all the students have the same total when they add their numbers. Among all selected numbers if the mostly selected number has been selected X times, what can be the minimum value for X?

Note: A student may select a number more than once.

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