Inspired by the Fiddler on the Proof (formerly The Riddler), X’s Puzzle Corner aims to produce a weekly puzzle for readers that enjoy math, probability, and algorithms. Please submit your solution! Solutions will be accepted until 11 pm the following Sunday after the puzzle is posted (in this case 5/25/25). While it isn’t required, I encourage you to opt to have your solution shared so that we all get the chance to see how others thought about and attempted the problem! The solution and submitted responses will be posted around Wednesday at 10 am.

I make no guarantees my solutions are correct! You are all smart people so please comment if you think I made a mistake!

The next few weeks will feature puzzles intended to set the scene and build into a fascinating result from the recent paper A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode. I was tempted not to mention the paper as the influence, but it’s such an interesting result that I couldn’t resist. Just know that reading the paper will result in puzzle spoilers :)

As an analyst at the New York Times games division, you’ve been tasked with understanding how users play their new game Lines’n’Gons. The game proceeds as follows:

Your board is […] a regular octagon. You take turns drawing lines between two non-neighboring vertices with the following rules:

1. You can’t cross an existing line.

2. If your line creates a triangular face, you get to claim that face and score +1. If you manage to create two triangular faces, you get both of those faces for +2.

3. You and your opponent proceed until no more lines can be drawn.

At the end, whoever has more points wins.

Below is a simple animation to illustrate:

You want to analyze how users play the game, but your initial analysis shows that many people don’t complete their games. To aid your analysis, you consider grouping and counting every final board configuration—finished and un-finished—so that all your subsequent analysis is speedy. Before you begin, you decide you want to determine how many distinct final board configurations there are. Every board has a predetermined ‘bottom’ so we can consider a board distinct if it has the same configuration once the bottom is properly positioned. For example, these two hexagonal game boards would be considered distinct.

Final board configurations can include any number of player moves (including 0). By final board configuration, we just mean the configuration the board was in when the players stopped, so we don’t need to concern ourselves with the order of moves or anything like that.

The question this week is, how many distinct final board configurations are possible with an octagon board? How about an n-gon board?

For extra credit, find a way to compute the number of final boards that are separated into exactly d₃, d₄, … , d_n polygons where d_i is the number of i-gons (polygons with i sides). For example, the figure above would be (d₃=2, d₄=1, d_n-1=0, d_n=0).

Please submit your answers here. Please ask any questions in the comments.