Pebbling a checkerboard. we are given a checkerboard which has 4 rows and n columns, and has an integer written in each square. we are also given a set of 2n pebbles, and we want to place some or all of these on the checkerboard (each pebble can be placed on exactly one square) so as to maximize the sum of the integers in the squares that are covered by pebbles. there is one constraint: for a placement of pebbles to be legal, no two of them can be on horizontally or vertically adjacent squares (diagonal adjacency is fine). (a) determine the number of legal patterns that can occur in any column (in isolation, ignoring the pebbles in adjacent columns) and describe these patterns. call two patterns compatible if they can be placed on adjacent columns to form a legal placement. let us consider subproblems consisting of the first k columns 1 ? k ? n. each subproblem can be assigned a type, which is the pattern occurring in the last column. (b) using the notions of compatibility and type, give an o(n)-time dynamic programming algorithm for computing an optimal placement

Consider this data sequence: "3 11 5 5 5 2 4 6 6 7 3 -8". any value that is the same as the immediately preceding value is considered a consecutive duplicate. in this example, there are three such consecutive duplicates: the 2nd and 3rd 5s and the second 6. note that the last 3 is not a consecutive duplicate because it was preceded by a 7. write some code that uses a loop to read such a sequence of non-negative integers , terminated by a negative number. when the code finishes executing, the number of consecutive duplicates encountered is printed. in this case, 3 would be printed. assume the availability of a variable, stdin, that references a scanner object associated with standard input. that is, stdin = new scanner(system.in); is given.

Technician a says that if a valve is open when a piston rises to the very top of a cylinder, the piston may actually strike the valve head and cause serious engine damage. technician b says if the camshaft is located in the engine block, then the engine is called an overhead valve engine, ohv engine, or an in-block camshaft. who is right? a. b only b. both a and b c. a only d. neither a nor b

1. the program must provide following functions to extract some statistics. note that the data_list parameter specified in these functions may be the same for all functions or different for different functions—that is your choice. a skeleton file is provided on mirmir. a) open_file()prompts the user to enter a year number for the data file. the program will check whether the year is between 1990 and 2015 (both inclusive). if year number is valid, the program will try to open data file with file name ‘year.txt’, where is the year. appropriate error message should be shown if the data file cannot be opened or if the year number is invalid. this function will loop until it receives proper input and successfully opens the file. it returns a file pointer and year. i. hint: use string concatenation to construct the file name b) read_file(fp)has one parameter, a file pointer read. this function returns a list of your choosing containing data you need for other parts of this project. c) find_average(data_list) takes a list of data (of some organization of your choosing) and returns the average salary. the function does not print anything. hints: i. this is not the average of the last column of data. it is not mathematically valid to find an average by finding the average of averages—for example, in this case there are many more in the lowest category than in the highest category. ii. how many wage earners are considered in finding the average (denominator)