Ping Pong Redux: A Visual Representation

Before reading this post, you might want to read my previous post on Ping Pong Redux since below is a solution of sorts. Another option, though, is to not read the previous post and instead attempt to backwards engineer the problem. I'd be curious what people came up with (and if the fact that you know the name is too constricting). I'll leave out a key for now.

Ping Pong Redux

Heard this problem from a colleague at lunch earlier this week and loved it.



Dan and a childhood friend loved playing ping pong. They loved 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, if the score is 7-4 and I win a point, instead of going to 8-4 the score becomes 2-1. Like in normal ping pong, games go to 21. Note: If you are leading 20-7 and score a point, you do not win. The score would go to 3-1.

There are some great mathematical questions to explore here. I'd recommend thinking of your own before reading the below list of mathematical questions (some of which are Dan's).

• What are all the possible final scores?
• What scores are impossible to get?
• In ordinary scoring rules, the winner has to win by 2. Is that rule necessary with these reducing rules? Explain.
• What strategies might a player use to avoid (or cause) a score being reduced?
• If two players were not evenly matched, do you think these rules would favor the weaker player or the stronger one? Explain.
• Describe as generally as possible games in which no reductions occur. How likely do you think such a game would be?
• How long would you expect a game to last if players were evenly matched (each player had a 50% chance of winning the next point)?