Maddenation
Trickiness
I have always been fascinated with “tricky” problems; the ones that are not necessarily hard, but misleading because they seem deceptively simple and cause your brain to avoid careful thinking and short circuit to the wrong answer. In a word: tricky.
Sometimes tricky problems can be hard too, or controversial. Often they depend on semantics and can cause people to argue about them indefinitely. For example, consider the let’s make a deal problem that asks if you should change doors after Monty has shown you the goat behind one of the doors you didn’t select. Some people refuse to believe that switching helps, and usually demand to stop talking about the problem before they can be convinced.
Probability problems are almost always tricky, and often counter intuitive. A famous one is the “birthday” problem where it only takes 23 random people to have a better than 50:50 chance of coinciding birthdays. With 41 people, the odds are better than 90%. The tricky part for me was how you solve the puzzle. You have to go at it “backwards” by figuring out the probability that there are no two birthdays the same. You start with the 1/365 chance of a “hit” with the second person, which means the chance of a non-hit is 364/365. When the third person comes in, the chance that her birthday is different from both of the others’ is 363/365, so overall, the probability of 3 non-hits is (364)(363)/3652. Keep multiplying by (366 – n)/365 for each new person you add and by the time you reach n = 23, the probability (of not matching) drops below 50%.
Working the problem “forwards” is way, way too tedious. It starts to get complicated on the third person where the number of total possibilities is 3653, or 48,627,125. How many of those result in matching birthdays? That’s the tricky part. By starting with a smaller number of possible “birthdays” (like 4) you can examine all 64 possibilities and then extrapolate logically to 365 and figure out that the number of matches = 365[365 + 2(365)]. The probability of a match is the ratio of those numbers, which works out to be 1 – (364)(363)/3652. But enough of that.
Here’s a simpler tricky problem that I thought of during my ride from Fort Washington, PA to Whippany. The trip divides roughly into 3 segments, a slow part, and medium part, and a fast part. Let’s say for the sake of argument that I can average 25 mph for the first third of the trip, 50 mph for the second third, and 75 mph for the last third. What is my average speed? 50 mph right? Wrong! Go figure.
There are many other tricky problems out there, and I’d like to hear about them. Did you hear about that crazy locker problem they talked about on Regis and Kelly a few weeks ago? Now that’s tricky!
Dad • Puzzles • 12/19/08 • 0 comments
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