Can You Master Competitive Programming Without Mastering Theory and Proofs?

Competitive programming is a field that requires both algorithmic knowledge and problem-solving skills. However, is it possible to excel in competitive programming without a deep understanding of mathematical theories and proofs?

Let's explore the relationship between mathematical theory and competitive programming by examining the nature of problem setting for programming contests. It's essential to understand how problem setters increasingly rely on mathematical concepts to create engaging and challenging problems.

Mathematics and Problem Setting in Programming Contests

The utility of mathematics in software development today is undeniably significant. When it comes to programming contests, the fundamental question is whether a deep understanding of mathematical theories and proofs is necessary for success. In short, the answer is increasingly yes.

Experimenting with AIME Problems

To illustrate this point, let us consider the 2020 AIME I (American Invitational Mathematics Examination) problems and attempt to convert them into programming contest problems.
Here is a quick thought experiment:

Problem 1: Can be modified to include a ratio between AB and AC. However, it's still tricky to convert to a programming problem due to multiple variables. Problem 2: Can replace 2, 4, and 8 with numbers, but it remains a constant-sized input problem. Problem 3: Can replace base-eleven with base-b in input, which is still a constant-sized input problem. Problem 4: Can be turned into a general n-digit number problem, more akin to a typical programming contest problem. Problem 5: Can replace 6 with n, keeping it a constant-sized input problem. Problem 6: Can convert it into a constant-sized geometry problem by specifying the radii of the holes or sphere. Problem 7: Can be turned into a general input problem by allowing multiple groups and different sizes. Problem 8: Can be transformed into a problem by specifying a general trajectory. Problem 9: Can be turned into a general programming problem by giving an arbitrary large integer w or exponents instead of 20^9. Problem 10: Can replace 210 with other numbers but might be hard to optimize beyond brute force. Problem 11: Can turn into a constant input size problem by replacing 10, 2, and 4 with other numbers. Problem 12: Input constraints feel specialized, so it might be difficult to generalize. Problem 13: Can be transformed into a constant input size problem by replacing 4, 5, and 6 with arbitrary numbers. Problem 14: Can be transformed into a constant input size problem by replacing 3 and 4 with arbitrary inputs. The general higher degree version may not be particularly interesting. Problem 15: Can be turned into a constant-sized geometry problem similar to Problem 13.

Based on this experiment, only 3 out of 15 problems (1, 10, and 12) are particularly difficult to convert into programming problems due to specialized configurations. The majority of the problems (2, 3, 5, 6, 11, 13, 14, and 15) can be transformed into simple input/output problems by requiring the numbers to be input. At least 4 problems (4, 7, 8, and 9) can be converted into typical programming contest problems that involve reading multiple numbers and outputting results.

Conclusion: Is Mathematical Theory and Proof Important?

These observations lead to an inevitable conclusion: problem setters in programming contests are increasingly drawing on mathematical theories and proofs to create engaging and challenging problems. This raises the question: does this preference for mathematical theory and proof-based problems actually favor certain contestants? As a result, it appears that contestants who excel in math competitions often have an advantage in programming contests.

The intrinsic reach of math competitions allows them to capture a broader audience. Therefore, in times of a shortage of programming contest problems, the standard approach is to borrow ideas and tricks from math contests.

While it is possible to excel in competitive programming with a strong foundation in problem-solving skills alone, a deep understanding of mathematical theory and proofs can significantly enhance your ability to tackle complex and innovative problems.