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Programming logic is the foundational framework for creating computer software and applications. Like a blueprint for a construction project, it outlines the steps and procedures that a computer must follow to execute a specific task. In the context of generating and displaying a random number, programming logic comes into play within the definition and execution of the pseudocode statement Display random(1,20).

Initially, the computer needs to understand the task, which here is to show a randomly chosen number to the user. To achieve this, the logic is broken down into sequential steps. The first part of the logic involves invoking a random number generator with specific boundaries, in this case, 1 through 20. This step necessitates an understanding of how to work with functions — which are predefined pieces of code that perform a particular task. Once the random number is generated, the second part of the logic is to display this number to the user. The separate tasks of generating and displaying are combined into a coherent sequence that satisfies the requirement of the exercise.

Control structures are the decision-making aspects of programming languages that enable the flow of a program to be controlled and manipulated. While the initial exercise does not overtly present a control structure, understanding them is essential for enhancing the functionality of our pseudocode.

Imagine if you were asked to generate a random number only if a certain condition is met. In such cases, we would use constructs like if, else, or while loops. An improved pseudocode could be:
if user_input equals 'yes' then
Display random(1,20)
end if
Here, the 'if' control structure dictates that the random number should only be displayed if the user wants to see it. The ability to incorporate such logic is what turns simple instructions into dynamic and responsive programs.

Algorithmic thinking is a methodical approach to problem solving that involves breaking down problems into smaller, more manageable parts and then developing a step-by-step solution. In the case of our exercise, algorithmic thinking is reflected in the systematic process of generating a random number and then displaying it. To delve deeper into this concept, consider the steps you would take if you wanted to select a random student from a list to answer a question in class.

You would first ensure the selection is fair by assigning each student a unique number. Then, you'd use a method—akin to our random number function—to pick one of those numbers. Afterward, you'd identify and announce the student associated with that number. In a program, such an algorithm involves representing each student by a numerical value and using the random() function to select a value within that range. The chosen number corresponds to the student, mirroring the logic in our pseudocode example.