This is a followup to my last post. I promised some more problems in which there is an initial pattern that’s wrong. Here is one more. It is not nearly as good as the points-on-a-circle problem I discussed before, for reasons I’ll say below. But I’m brainstorming here, and hope you’ll join me, so anything’s better than nothing. (And actually I think it’s a cool problem in its own way.) Thanks Kate, jd2718, and Gilbert for contributing ideas so far.
The problem involves seeking a simple formula that produces prime numbers only. As you probably know, in spite of centuries of research no such formula is known to this day. There is some fun history around this. For example, Fermat believed that 2^k + 1 was prime whenever k was a power of 2. It is prime for k=1, 2, 4, 8 and 16. However for k=32, the number is 4,294,967,297 which was found by Euler to be equal to 641 * 6,700,417. Now, even in the age of computers, no other prime of the form 2^k + 1 has yet been found. Of course, I’d avoid putting kids in the position of having to calculate 4,294,967,297 or to show that it’s not prime.
Anyway, the idea is to get a class engaged in a search for such a formula. My idea for how the lesson goes would be to try out some examples with them to show them what is being sought. Like, maybe 4n + 3 which equals 3, 7, 11 for n = 0, 1, 2 but then fails for n = 3, or, starting with p_1 = 2 and then recursively doubling and adding 1 which gives you 2, 5, 11, 23, 47 and then the next one fails. (A closed form for this last one would be
3*2^n – 1 for n = 0, 1, 2, 3, 4.) This second formula is a good replacement for Fermat’s conjecture, because it gives you 5 primes before it fails, just like Fermat’s conjecture, but the primes are a reasonable size and the one that fails (95) is obviously not prime. Anyway, once they understand what’s being sought, the problem is to find such a formula. They will totally fail and they have no tools that will help them, so don’t let them stew too long. Then, show them a very pretty creation of Euler’s:
n^2 – n + 41. This quadratic polynomial is (amazingly) prime for n=0, 1, 2, …, 40. So there’ll be some initial excitement as this one seems to answer the question. But actually, it can’t possibly answer it. And the class may be able to see that the n=41 case will fail without actually doing the calculation. Even if the calculation is needed, it can still lead to a cool conversation.
Now this isn’t as rich as the points-on-a-circle problem because the inordinate primality of n^2-n+41 is sort of a mathematical accident; there isn’t a rich story behind it (at least not one I’ve ever heard), so once the pattern is noticed and then broken there’s nowhere to go. But it does at least give students some experience of the fact that if a rule holds for small cases, it doesn’t mean it always holds. And the breakdown at n=41 is accessible to reasoning alone, without calculating. So it’s a win for the power of mathematical reasoning over raw pattern-noticing.
Other ideas in this vein? (Problems where there is a “obvious” or “apparent” pattern or conclusion that is actually wrong?)
Research in Practice 
Here’s another problem that can cause a severe clash of intuition with reality. In all my years of posing the question, I have had only a handful of students who managed to get the correct answer, and most of those were lucky guesses that were revealed as such when the problem is pushed to involve three dice.
Here is the problem. Two dice are rolled. Given that at least one “1” appears, what is the probability that the dice roll is “snake eyes” (double 1)?
The most common answer, by far, is 1/6. Other popular answers are 1/36 and 1/3, with rare appearances by 1/18, 1/12, and the like. The disturbing thing is that all students who have studied Algebra II and Precalculus formulas for probability go straight to their “knowledge” (faulty though it is) instead of rolling actual dice and seeing that the occurence is rather rare–but certainly not as rare as 1 time in 36.
More disturbing than the 99.9% occurrence of wrong answers is the astonishingly high confidence that students place in their probability estimates. Worst of all, the “best” students, the ones in my honors calculus class, tend to be the most overconfident. What does this say about an economy confidently directed by the “elite, supersmart” Wall Streeters of the future? I’m guessing it will be just about as fouled up as the economy that crashed in 2008 on the advice of their predecessors.
By the way, nobody ever takes me up on my offer to bet actual money on the outcome of the two-dice problem. (It’s probably just as well–I might get parent complaints, or worse.) You see, if the student says the probability is 1/6, I can safely offer him odds of 7:1 if he thinks he can roll snake eyes 1/6 of the time when at least one “1” appears. And I’ll take lots of money off him in the long run.