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)?