How do number games use arithmetic and logical reasoning?

Number games and mathematical recreations utilize arithmetic and logical reasoning to challenge the mind through various puzzles and problems.12

Arithmetic Recreations

Arithmetic-based games often involve manipulating numbers to reveal patterns or reach specific values. For instance, digital problems—such as the "four 4s" challenge—require using a specific digit and various algebraic operations (like addition, division, or factorials) to express a sequence of integers. Other recreations explore the intrinsic properties of numbers, such as:3
* Figurate numbers: Visualizing numbers as geometric shapes (e.g., triangular or square numbers) to understand their relationships.4
* Number curiosities: Identifying patterns like narcissistic numbers, where an integer equals the sum of the $n$th powers of its digits, or multigrades, which involve identities between sums of powers.5

Logical Reasoning

Logical reasoning is central to puzzles that do not rely on calculation. For example, sudoku is based entirely on logic, requiring players to fill a grid by ensuring numbers appear exactly once in each row, column, and subgrid. Similarly, general logical inference puzzles are solved by testing hypotheses, rejecting inconsistencies, and using substitution and elimination to reach a conclusion.67