Unlocking the Art of Creating Compelling Math and Computing Contest Problems

When we often talk about solving complex math problems, it's easy to underestimate the challenge of generating interesting and engaging questions, especially for exams and Olympiad-style challenges. As someone who has created contest problems in the past, I can attest that crafting a good, fresh problem is typically more challenging than solving it. This article delves into the nuances of problem creation in the context of math and computing contests, providing insights and references to aid problem posers.

The Complexity of Problem Creation

While the initial idea for a problem might seem straightforward, the act of creating a compelling contest problem is an intricate process. Problem posers accumulate a diverse set of ideas over time, and at some point, they select one to develop further. The problem in development often has various parameters that need to be defined, resulting in a family of potential problems. The key challenge lies in determining which parameter combinations can be solved and which cannot. Unlike problem solvers, who know in advance that the problem has a well-defined elegant solution, creators must focus on developing a precise and convincing formulation. Afterward, they need to consider how to evaluate solution attempts, often requiring a deep understanding of diverse solutions beyond just one approach.

A Deeper Dive into Problem Creation

Here are three references that explore problem creation in detail:

Guidelines for Producing a Programming-Contest Problem Set by Tom Verhoeff. Personal Note, October 1988, expanded July 1990. This resource provides a comprehensive guide to problem set creation, detailing the nuances of problem formulation and solution evaluation.

The Lost Group Chart and Related Problems by Tom Verhoeff. Part of the Liber Amicorum for prof. dr. F. E. J. Kruseman Aretz, Faculty of Mathematics and Computing Science Eindhoven University of Technology, December 1995. This article delves into specific problem families and highlights the subtleties in problem-solving techniques.

Finding the Median under IOI Conditions by Gyula Horvath and Tom Verhoeff. Published in Informatics in Education, Vol. 1 Number 1 pp.73-92, 2002. This paper provides a detailed exploration of problem-solving methodologies and the evaluation of solutions.

An Example Problem Family: Color-Based Permutations

To illustrate the complexity of problem creation, let's consider a specific problem family:

Assume you have a sequence of n colored objects. You are only allowed to swap adjacent objects having a different color. The key questions that arise are:

Is it possible to present each permutation of this sequence exactly once using only such adjacent swaps?

If so, how can it be done?

If not, what is the minimum number of permutations to show twice or more often?

Alternatively, what is the minimum number of permutations not to show? How can you achieve this?

If we use digits to represent colors, this problem can be quite diverse, with various instances serving as potential contest problems, such as:

012

01234

0011

000111

00001111

001122

011222

The creator has considerable freedom in defining the problem instance, including the number of objects, the number of colors, and the frequency of each color. When all objects have distinct colors, the problem is relatively straightforward and well-known, being treated as a pure permutation problem. However, when there are two colors, the problem is only solvable under specific conditions, such as when each color appears an odd number of times, or when the number of objects is trivial.

For three or more colors, the problem is more complex. It was settled in 1992 that for cases where at least two colors occur an odd number of times, the permutation is possible. However, when neither color occurs an odd number of times, the problem was solved in 2015, with a recursive algorithm that improves upon the 1985 solution for two objects and two colors.

Conclusion

Problem creation, especially in the context of math and computing contests, is akin to research. It requires a deep understanding of the problem at hand and the ability to formulate a compelling and solvable problem. By exploring and contributing to existing problem families, problem posers can create engaging and challenging problems that inspire and challenge contestants.