Martin Gardner talk, Oct. 18, 2018
Dennis Mancl, Princeton Chapter of ACM / IEEE-CS
The math quiz
Your math teacher, who has a reputation for always telling the truth, announces the next quiz:“There will be a quiz next week, and it will be on day when you aren’t expecting it. You won’t know what day the quiz will be until Quiz Day.”
Everyone is worried about the quiz, because they know from experience that they must prepare in advance for the quiz if they want to do well. Can we guess which day will be Quiz Day??
Mon Tues Wed Thurs Fri ---- ---- --- ----- ---- Quiz Quiz Quiz Quiz Quiz Day? Day? Day? Day? Day?
One student thinks for a minute and tells the others: “The quiz won’t be on Friday. If we get to Thursday and we haven’t had the quiz yet, then a quiz on Friday would not be a surprise.”
In other words, four school days with no quiz is a sure sign that Quiz Day will be Friday:
Mon Tues Wed Thurs Fri ---- ---- --- ----- ---- No No No No Quiz Quiz Quiz Quiz Quiz Day!
A second student is also thinking about this: “The quiz won’t be on Thursday either. We already ruled out Friday, and if we get to Wednesday and we haven’t had the quiz yet, we would be expecting the quiz on Thursday.”
Mon Tues Wed Thurs Fri ---- ---- --- ----- ---- No No No Quiz Imposs- Quiz Quiz Quiz Day! ible
Three more students figure out the pattern. One explains that the quiz won’t be on Wednesday (we already ruled out Thursday and Friday... if we get to Tuesday without a quiz, we will expect Wednesday to be Quiz Day). The next eliminates Tuesday with the same argument. And the last student rules out Monday.
Mon Tues Wed Thurs Fri ---- ---- --- ----- ---- Imposs- Imposs- Imposs- Imposs- Imposs- ible ible ible ible ible
Everyone comes to the same conclusion: There can’t be a quiz next week, because our logic rules it out.
On Wednesday, the teacher comes in and she announces to the class: “Today is Quiz Day! I’ll bet you weren’t expecting it today...”
What happened??
There have been many attempts to explain this logical paradox.
- It might be OK to rule out Friday as Quiz Day, but you are getting into trouble once you start using deductions instead of facts to exclude more days....
- Math teachers are sneaky....
- No one expects the Spanish Inquisition!!
But my favorite explanation is to split the teacher’s statement into two parts:
-
part 1: Quiz Day will be next week (a choice of five possible days)
part 2: I won’t tell you ahead of time, so Quiz Day will be a surprise
Part 1 is useful information -- the day of the quiz is constrained to be one of five days.
Part 2 is useless information -- you can’t use this information to add any extra constraints. If you don’t think about it, you will be surprised. If you do think about it, you will still be surprised.
The most important lesson: Always be ready for a math quiz!
Why is this interesting?
This problem is sometimes called the “Unexpected Hanging Paradox” ... A judge sentences a criminal to be executed by hanging within a week. on a day that will be a surprise. Math quizzes sometimes feel like an unexpected hanging, don’t they?
Here is the real lesson: There are many people who talk without really giving you any information. Not all statements are “informational.” You have to listen carefully and think... did what I hear just “add to my knowledge” or was it just an empty statement that tried to confuse me?
This is an important idea in science, business, and politics.
Scientists sometimes create a lot of data (from things like medical trials), but the data “might not mean anything” -- there might be no good conclusions you can use. (Should I stop eating Halloween candy? should I avoid glutens? should I take over-the-counter vitamins? We’re not sure.)
Businesses make decisions about hiring people and purchasing equipment. (Do I really need someone who is a black-belt Excel spreadsheet developer? Should we buy a bigger computer server, or should we rent time on the cloud?)
Poltics is an ideal domain for “empty statements”: “We plan to cut the “average” tax rate by 10%.” (Will my tax rate go down? Not likely... your tax rate might actually go up. If we lower the tax rate for a small set of high earners, it looks like the average will decrease, even if most people are paying more.
References
This paradox was discussed by many mathematical logic experts in the 1940s.Martin Gardner wrote about the Unexpected Hanging Paradox in one of his regular Scientific American columns:
- “Mathematical Games,” Scientific American, March 1963.
Parts of this article have been reprinted and extended in other books by Martin Gardner:
- The Unexpected Hanging and Other Mathematical Diversions (University of Chicago Press, 1991)
- Chapter 1 (“The Paradox of the Unexpected Hanging”) of the book Knots and Borromean Rings, Rep-Times, and Eight Queens by Martin Gardner (Cambridge University Press, 2014)
- Wikipedia article
- Resolving the unexpected hanging paradox by Alex Mennen (2011 article)
Don’t expect a “solution” to the paradox. It is just one of those great problems that get people talking about what we mean when we say we “expect” something.
Triangular paintball duel
Another interesting paradox is the 3-way duel.
Andy, Mark, and Steve decide to have a paintball duel tournament. The shooting order is determined by drawing cards. Each player takes turns -- tries to shoot at one of the other two players. If a player is hit, he leaves the game. The game continues until only one player is left.
- Andy is an ace shooter -- 100% accuracy.
- Mark is also very good -- 80% accuracy.
- Steve only hits his target 50% of the time.
Assuming each player chooses his best strategy, who has the best chance of winning?
Unusual strategy for Steve
It’s hard to believe at first, but “50%” Steve has the best chance to win!
- If Steve shoots first, he should aim to miss!
- Andy or Mark won’t aim at Steve in the first round (Why?)
- Andy or Mark will eventually be knocked out
- If it is Andy, Steve has a 50% of winning
- If it is Mark, Steve has a better than 50% chance of winning (Why?)
Wow!! This only works because Steve is a lesser threat (and therefore a lesser target).
Here are the chances of winning:
- Steve = 52.22%
- Andy = 30.00%
- Mark = 17.78%
Why won’t Andy or Mark aim at Steve in the first round?
- If Andy shoots at Mark (and hits), there is a 50% chance that Steve will hit him
- But if Andy shoots at Steve (and hits), there is an 80% chance that Mark will hit him
- So for Andy, aiming at Mark reduces Andy’s chance of winning to 20%
- Similar argument for why Mark won’t aim at Steve (if Mark hits Steve, Mark’s chance of winning drops to 0%)
This problem is based on one of the problems in Chapter 5 of the book Origami, Eleusis, and the Soma Cube by Martin Gardner (Cambridge University Press, 2008)
Lessons from this problem
In a world where each person is “out for himself,” the strongest person is the biggest target
Andy and Mark could do better -- by forming an alliance (as often happens in reality-show competitions, foreign affairs, and politics)
- Andy and Mark promise each other to go after Steve first
- But they also promise to flip a coin to determine who will shoot next after Steve is eliminated
If Andy and Mark follow their agreement, the winning odds change:
- Steve = 2.03%
- Andy = 49.00%
- Mark = 48.96%
There is a lesson here for politics and international relations: if you follow a go-it-alone strategy because you are strong, you actually increase your chances of losing to another person (or country) that is a lesser threat.
It is a much better strategy to form alliances -- especially with the other strong players. If you can negotiate reasonable rules for the alliance, all of the members of the alliance can improve their chances of winning!
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Last modified: Oct. 18, 2018
Dennis Mancl - http://manclswx.com