## Friday, March 8, 2013

### 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)?

1. This sounds like a fun exploration. I haven't thought it through, but it seems like you might be able to eventually link up with Farey sequences (http://en.wikipedia.org/wiki/Farey_sequence).

And while this new scoring system is a bit contrived, there are definitely some interesting strategic consequences. A more targeted follow-up to #4 on your list: under what conditions would you be best served by conceding a point?

2. Am I correct that 11-0 or 11-1 are required for a game to be completed? This is weird stuff.

Jonathan

3. I simulated 1000 games, and observed these final scores:

1 - 21: 502 games
21 - 1: 484 games
21 - 11: 7 games
11 - 21: 5 games
13 - 21: 1 game

The average number of points scored per game was 1,356,352. One game was not completed even after 10,000,000 points were scored. I think they must play this version of Ping Pong in Hell.