Discrete Mathematics Propositional Logic Combinatory Logic P

Discrete Mathematics, Propositional Logic, Combinatory Logic Problem

Must Provide Answer and reasoning (and confidence level of both).

I will rate and pick best answer within one day.

Problem 1. The picture above shows a puzzle based on a four-by-four array of nested squares (each of the 16 locations has a \"center\" and a \"border\" square). The puzzle needs to be filled in with four colors: black, blue, yellow, and red.

The puzzle rules are as follows:
Rule a. Each square is two-colored (the center and border are different colors).

Rule b. In each row of four squares (horizontal, vertical, or diagonal), all the centers have different colors and all the borders have different colors.

In the grid above, five black regions, three red, and two blue have been filled in. Anything shown in white needs to be filled with one of the four colors. Rows in the grid are numbered 1-4 from the top, and columns are numbered 1-4 from the left.

1a) Solve the puzzle by filling in the remaining regions, and briefly explain your reasoning.

Now, suppose you have propositional logic statements of the form:

SQ11CENTRED(\"the center of the square in row 1, column 1 is red\")

SQ32BORDBLACK (\"the border of the square in row 3, column 2 is black\")

As it happens, these propositions are both true in the given grid. Answer the following questions:

1b) How many distinct propositions of the form given in the previous two examples do we need to cover all combinations of square regions and colors?

1c) Show all the propositions you need to represent the idea that there is exactly one color in the center of the square in row 1, column 1.

1d) Show all the propositions you need to represent the idea that all the borders in column four have distinct colors. (There are multiple ways of doing this, but regardless you\'ll need well more than a few propositions.)

Solution

The above table shows border square colors.

The below table shows the centre colors