To Win This Numbers Game, Learn to Avoid Making Math Patterns — Quanta Magazine

My latest column for Quanta Magazine starts with a simple game of numbers and ends with some unsolved problems in mathematics.

For example, let’s change the rules to make the loser the first person to complete three in a row of any step size. This means you lose if you make 2-3-4, as in the original game, but also if you make 1-3-5 (three in a row of step size 2) or 1-4-7 (step size 3). These patterns are “arithmetic progressions”: sequences of numbers with a common step size, called the common difference.
Let’s return to our first game board and use the new rules. It’s still your turn. And you’ve lost.

This simple game, where each player tries to avoid completing an arithmetic progression, leads to some complicated math, involving open questions about Salem-Spencer sets and a new result about polynomial sequences.

The full article is available here and includes several exercises to test your game play!