Board Game Tales: Aaron’s Story

Mathematics Meets Board Gaming

Teaching with a Purpose

” For the past few years, I have been teaching college classes for math majors to make students aware of how the mathematical topics they are learning can be found in “every day” activities, like tabletop games. I am writing a book that reflects on my experiences with this material, and Project L will be included because it is a popular and enjoyable game that connects to several mathematical topics. A common pursuit in mathematics is counting objects of a particular type, and various mathematicians have worked on counting polyominoes (the tiles and shapes used in Project L).”

Counting Polyominoes

“The main question is: how many polyominoes of a particular size are possible? Asking students to determine a systematic way to count polyominoes of size up to 5 cells (along with proof that they have succeeded in finding all possibilities) can be assigned as an in-class activity or as homework. Students will find that Project L does give the player access to all polyominoes of size 1 through 4 in the base game and 3 of the 12 polyominoes of size 5 with the Ghost expansion. It has been proven that the number of polyominoes with n cells grows exponentially as n increases, which makes computation difficult for large values of n. It wasn’t until this year (2023) that anyone had counted the number of polyominoes with 50 cells when John Mason calculated this as 2150182610161041739167164220 and noted that to compute the number of polyominoes with 52 cells would require at least 4.5 CPU years.”

The Question of Equivalence

In Project L, they are equivalent, but different games have different rules about whether a polyomino can be rotated or flipped. This introduces the idea of symmetry, and mathematicians often study this using a field of algebra called Group Theory. Although since Project L allows rotations and flips of polyominoes, this is typically explored more deeply when discussing games that disallow such actions.

The central puzzle in Project L (filling one shape with polyomino tiles) is a packing problem that is also of interest to mathematicians. A packing problem asks which shapes can be tiled with copies of a particular polyomino. For example, an 8 by 8 grid missing one square can be covered with right triominoes

Polyomino Puzzle: Problem-Solving & Rewards in Project L

” Project L’s puzzle solving is similar but limits you to the polyomino tiles in your supply rather than requiring all polyominoes to be the same. So, students can be asked if they can find conditions that allow or disallow a collection of polyominoes to tile a particular shape. At first, students are given a shape and collection of polynomials and asked to either cover the shape or prove that such a covering is impossible. Once familiar with completing this task for a particular shape, students can be asked to generalize the solution by attempting to find a way to easily identify all shapes that can be (or cannot be) covered with a collection of polyominoes.

Finally, players of Project L are rewarded for completing puzzle shapes in different ways. Some shapes provide players with new polyominoes to add to their supply, others award victory points, and others reward players with a combination of both.”

Mathematical Exploration

” Some rewards can be directly compared. For example, a reward of 2 victory points and a domino is strictly better than 2 victory points and a monomial. However, some rewards will not be easily compared since there is no way to convert between polyominoes and victory points. For example, whether you would rather receive a new monomial or 1 victory point depends on where you are in the game. This provides an example of a partial order, where there is not necessarily a comparison between every pair of rewards.

Given this partial order, students can be asked to determine relationships between the puzzle’s shape and the reward it provides. In addition to exploring these mathematical topics raised by the game, I always take time to play the game with my students in a relaxed atmosphere. I have enjoyed introducing Project L to my students as it is a very enjoyable game that touches on many mathematical topics.”

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