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; please propose your own in the comments. To get the ball rolling we’ll focus on just one: What are all possible final scores?
The big thing here is to figure out which scores can lose (ie which numbers can’t be reduced), essentially we need to figure out all the prime numbers less than or equal to 21 that are ALSO not factors of any higher number.
Prime numbers less than 21: 2, 3, 5, 7, 11, 13, 17, 19. The numbers 2, 3, 5, and 7 are all factors of bigger numbers that are less than or equal to 21. So the list of possible winning scores are 21-11, 21-13, 21-17, 21-19. But WAIT! We forgot two scores…21-0 and 21-1. Even though 0 and 1 aren’t prime, they can’t be reduced.
The tougher part would be figuring out how the scores happen. 21-0 is pretty easy, and the 1 point in 21-1 could happen at any time, but with 11, 13, 15, 17, and 19, you have to be really careful that the scores don’t hit reducable scores. For example, on the way to 21-11, the score could potentially be 11-11 and go back to 1-1.
This sounds like a great way to practice fractions and factors in a fun way that includes table tennis, but also a way to play an insanely long and complicated match. We can’t even fathom how serving would work if you were following this scoring. The big takeaway is that table tennis is full of all sorts of weird math possibilities.