2016 靜宜大學全國程式競賽題目. Problem1:Largest Square. Description

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1 Problem1:Largest Square 2016 靜宜大學全國程式競賽題目 Given a rectangular grid of characters you have to find out the length of a side of the largest square such that all the characters of the square are same and the center [intersecting point of the two diagonals] of the square is at location (r,c). The height and width of the grid is M and N respectively. Upper left corner and lower right corner of the grid will be denoted by (0, 0) and (M-1,N-1) respectively. Consider the grid of characters given below. Given the location (1, 2) the length of a side of the largest square is 3. The input starts with a line containing a single integer T (< 21). This is followed by T test cases. The first line of each of them will contain three integers M, N and Q (< 21) separated by a space where M, N denotes the dimension of the grid. Next follows M lines each containing N characters. Finally, there will be Q lines each containing two integers r and c. The value of M and N will be at most 100. For each test case in the input produce Q + 1 lines of output. In the first line print the value of M, N and Q in that order separated by single space. In the next Q lines, output the length of a side of the largest square in the corresponding grid for each (r, c) pair in the input. Sample abbbaaaaaa abbbaaaaaa abbbaaaaaa aaaaaaaaaa aaaaaaaaaa aaccaaaaaa aaccaaaaaa

2 5 2 Sample Sample aab aab aaa Sample Hint

3 Problem2:Minesweeper Have you ever played Minesweeper? It's a cute little game which comes within a certain Operating System which name we can't really remember. Well, the goal of the game is to find where are all the mines within a M N field. To help you, the game shows a number in a square which tells you how many mines there are adjacent to that square. For instance, supose the following 4 4 field with 2 mines (which are represented by an `*' character): * *..... If we would represent the same field placing the hint numbers described above, we would end up with: * * As you may have already noticed, each square may have at most 8 adjacent squares. The input will consist of an arbitrary number of fields. The first line of each field contains two integers n and m (0 < n;m 100) which stands for the number of lines and columns of the field respectively. The next n lines contains exactly m characters and represent the eld. Each safe square is represented by an `.' character (without the quotes) and each mine square is represented by an `*' character (also without the quotes). The first field line where n = m = 0

4 represents the end of input and should not be processed. For each field, you must print the following message in a line alone: Field #x: Where x stands for the number of the field (starting from 1). The next n lines should contain the field with the `.' characters replaced by the number of adjacent mines to that square. There must be an empty line between field outputs. Sample * * Sample 1 Field #1: * *

5 Sample ** * Sample 2 Field #1: ** *100

6 Problem3:Simple Addition Lets define a simple recursive function F(n), where %10, if (n%10) > 0 0, if n = 0 /10, Otherwise Lets define another function S(p, q),, In this problem, you have to calculate S(p, q) on given values of p and q. The input file contains several lines of inputs. Each line contains two non-negative integers p and q (p q) separated by a single space. p and q will fit in 32 bit signed integer. is terminated by a line which contains two negative integers. This line should not be processed. For each set of input, print a single line of the value of S(p, q). Sample Sample 46 48

7 Problem4:{sum+=i++} to Reach N All the positive numbers can be expressed as a sum of one, two or more consecutive positive integers. For example, 9 can be expressed in three such ways, 2+3+4, 4+5 or 9. Given an integer less than ( ) or (9E14+1), you will have to determine in how many ways that this number can be expressed as summation of consecutive numbers. The input file contains less than 1100 lines of input. Each line contains a single integer N (0 N 9 14 ). is terminated by end of file. For each line of input produce one line of output. This line contains an integer which tells in how many ways N can be expressed as summation of consecutive integers. Sample 9 11 Sample 3 2

8 Problem5:Matryoshka Matryoshkas are sets of traditional Russian wooden dolls of decreasing size placed one inside the other. A matryoshka doll can be opened to reveal a smaller figure of the same sort inside, which has, in turn, another figure inside, and so on. Pictures from Wikimedia Commons The Russian Matryoshka Museum recently exhibited a collection of similarly designed matryoshka sets, differing only in the number of nested dolls in each set. Unfortunately, some over-zealous (and obviously unsupervised) children separated these sets, placing all the individual dolls in a row. There are n dolls in the row, each with an integer size. You need to reassemble the matryoshka sets, knowing neither the number of sets nor the number of dolls in each set. You know only that every complete set consists of dolls with consecutive sizes from 1 to some number m, which may vary between the different sets. When reassembling the sets, you must follow these rules: 1. You can put a doll or a nested group of dolls only inside a larger doll. 2. You can combine two groups of dolls only if they are adjacent in the row. 3. Once a doll becomes a member of a group, it cannot be transferred to another group or permanently separated from the group. It can be temporarily separated only when combining two groups. Your time is valuable, and you want to do this reassembly process as quickly as possible. The only time-consuming part of this task is opening and subsequently closing a doll, so you want to minimize how often you do this. For example, the minimum number of openings (and subsequent closings) when combining group [1, 2, 6] with the group [4] is two, since you have to open the dolls with sizes 6 and 4. When combining group [1, 2, 5] with the group [3, 4], you need to perform three openings.

9 Write a program to calculate the minimum number of openings required to combine all disassembled matryoshka sets. The input consists of several test cases. A test case consists of two lines. The first line contains one integer n representing the number of individual dolls in the row. The second line contains n positive integers specifying the sizes of the dolls in the order they appear in the row. Each size is between 1 and 500 inclusive. For each test case, display the minimum number of openings required when reassembling the matryoshka sets. If reassembling cannot be done (some of the kids might have been excessively zealous and taken some dolls), display the word `impossible'. Sample Sample 1 impossible Sample Sample 2 7

10 Problem6:n-Queens Problem We place n queens in an n*n chessboard. No two queens are in the same row, column, or diagonal. Write a program to solve the problem. n is the number of queens. (3< n < 15). The number of solutions. Sample 4 Sample 2

11 Problem7:Chained matrix multiplications To multiply a 3*2 matrix times a 2*4 matrix, the result is a 3*4 matrix. We use the simplest matrix multiplication method, we need 3*2*4=24 multiplications. Consider the multiplication of the following three matrices: M1(5*2) *M2(2*8) * M3(8*2) We have two different orders, (M1*M2)*M3 and M1*(M2*M3). We get the same result, but the number of multiplications ae different. (M1*M2)*M3: 5*2*8+5*8*2=160 M1*(M2*M3): 2*8*2+5*2*2=52 Write a program to compute the minimum number of multiplications for n matrices multiplication. n is an integer, the number of matrices. (1 < n < 20). d11 d12 (the dimension of first matrix; the first matrix is d11*d12) d21 d22 dn1 dn2 (the dimension of the nth matrix; the nth matrix is dn1*dn2) The minimum number of multiplications for n matrices multiplication Sample Sample 52