Every day, we use basic arithmetic operations-addition and multiplication-to solve a variety of problems, from simple calculations to complex equations. These operations are fundamental to our understanding of mathematics and are essential tools in numerous fields. The development of the binary system introduced a new way to represent numbers and perform calculations, revolutionizing computation and digital technology. However, the concept of carries in arithmetic operations can become cumbersome in certain contexts.
Dismal Arithmetic, discovered by Martin Gardner, offers an intriguing alternative by eliminating carries altogether. In Dismal Addition, the sum of two digits is simply the maximum of the two (that is, a b = max{a, b}), while Dismal Multiplication takes the minimum value (that is, a b = max(a, b)), This carry-less system provides a novel approach to arithmetic, with distinct properties that set it apart from the traditional base- 10 operations.
This study aimed to explore the properties of Dismal Arithmetic in comparison to conventional arithmetic using base-10 digits (0, 1, 2, ..., 9}. The research examined the analogues of fundamental number-theoretic functions, such as prime numbers, sum of divisors, and number of divisors, within the context of Dismal Arithmetic.
Using an expository method, the research described and analyzed the construction and unique characteristics of Dismal Arithmetic, highlighting its applications in number theory. Additionally, the study introduced a strategic board game designed to engage players in mathematical reasoning and logical deduction. The game simulates a tactical duel, challenging players to use the principles of Dismal Arithmetic to overpower their opponents. By employing a \"fog of war\" mechanic, the game enhances players\' skills in mathematics, logic, memory, and spatial awareness.