Julian D. Allagan

Problem-solving is essential in various fields, including business, technology, and everyday life. It often involves a combination of experience, knowledge, intuition, and rational analysis. Furthermore, it requires integrating disciplines such as logic and reasoning, mathematics, algorithms, Python, and generative AI in today's complex world. We will provide detailed descriptions of how to solve problems based on integrating previously mentioned disciplines. Later, we will discuss how to guide students through making intelligent investment decisions.

This paper investigates the dynamic interplay between Artificial Intelligence (AI) and human logic in the domain of mathematical problem-solving. By critically examining a series of case studies, we compare the efficacy of AI-generated solutions, particularly those offered by ChatGPT, against traditional human problem-solving methods. The study employs various mathematical challenges, ranging from abstract logical puzzles to applied numerical problems, to evaluate AI's problem-solving approach and alignment with human cognitive processes. Our analysis highlights instances where AI's computational strategies complement or diverge from human reasoning, shedding light on AI's potential and limitations in deciphering mathematical problems. Furthermore, we explore the implications of integrating AI tools in educational contexts, specifically their role in enhancing students' mathematical problem-solving skills. The paper aims to contribute to the ongoing discourse on the optimal utilization of AI in education, proposing a balanced approach that leverages AI's computational power while fostering the depth and creativity of human logic. Through this comparative study, we advocate for a collaborative model where AI and human reasoning merge to enrich the educational landscape, particularly in the teaching and learning of mathematics.

Even though the task of multiplying matrices appears to be rather straightforward, it can be quite challenging in practice. Many researchers have focused on how to effectively multiply two 2 by 2 matrices by applying Strassen Algorithm in the past 50 years. They worked on the complexity from both the mathematical and algorithmic points of view. In our paper, we will discuss the comparison of processing time in Python IDLE, Jupyter Notebook, and Colab from a practical point of view. Several open problems are then presented to challenge our readers.

Multiplying matrices can be very challenging although it seems straightforward. Many researchers have studied the multiplication of two 3 by 3 matrices by using Strassen Algorithm in the past 50 years. They focused on the complexity from the mathematical and algorithmic points of view. We will discuss the processing time comparison from a practical point of view in Python IDLE, Jupyter Notebook, and Colab. Several open problems are posted to challenge our readers.

The domination (number) of a graph G=(V,E), denoted by γ(G), is the size of the minimum dominating sets of V(G), also known as γ-sets. As such, the dominion of G, denoted by ζ(G), counts all its γ-sets. We proved a conjecture from one of the authors on the dominion of cycles C3k−1 and C3k−2, k≥2. Further, we found the formulae and recurrence relations for the dominions of several grids, Gm,n, with 2≤m≤4 and other results when m≤9 and n≤20. In general, domination and dominion play important roles in assessing certain vulnerabilities of any given network system.