Pebbling a checkerboard. We are given a checkerboard which has 4 rows and n columns

QUESTION:

Pebbling a checkerboard. We are given a checkerboard which has 4 rows and n columns, andhas an integer written in each square. We are also given a set of 2n pebbles, and we want toplace 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 isone constraint: for a placement of pebbles to be legal, no two of them can be on horizontally orvertically adjacent squares (diagonal adjacency is fine).(a) Determine the number of legal patterns that can occur in any column (in isolation, ignoringthe 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 canbe 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 algorithmfor computing an optimal placement.