Every week, *The New York Times* challenges its readers to solve a mathematical puzzle in its online Numberplay column. This week’s puzzle was proposed by none other than Dan Bennett, a former editor and author at Key Curriculum Press, and his colleague, Avery Pickford.

Here is their puzzle, as described in Numberplay:

Dan and Avery love playing ping-pong. They love playing ping-pong so much that they devised a new rule to make games last longer. Scoring and play is normal, except that the score is “reduced” whenever possible. In other words, the scores are divided by the greatest common factor. So if Dan is ahead 7-4 and wins a point, instead of going to 8-4 the score becomes 2-1. Like in normal ping-pong, games go to 21. Note: If Avery is leading 20-7 and scores a point, he does not win. The score would go to 3-1.

There are many questions to ask about this game. To get the ball rolling we’ll focus on just one: What are all possible final scores?

Students can explore this puzzle without technology, but I couldn’t resist building the Web Sketchpad model below to accompany the puzzle. This is a great puzzle for implementing the Common Core Standards for Mathematical Practice!