E27. A puzzling sequence

Someone gave me the following puzzle at a party. I don't know the source. It involves nothing beyond elementary school mathematics.

Find a simple pattern in the following sequence, and give the next row:

1
1 1
2 1
1 2 1 1
1 1 1 2 2 1
3 1 2 2 1 1

E26. Final digit of a power

The following is adapted from Posamentier and Salkind's Challenging Problems in Algebra, published by Dover Books.

(a) Prove that for any positive integer n, the final digit of n⁵ is the same as the final digit of n.
(b) Without using a computing device, find all positive integers whose 5-th power is a 7-digit number that ends in the digit 7.

Don't Be a Math Teacher

I have enjoyed teaching math for several decades and currently teach K–12 math teachers about education and mathematics. So you'd think that I would recommend that a mathematically-inclined person should give strong consideration to becoming a math teacher. You'd be wrong.

If when I began my career the state of the mathematics teaching profession, both at the school and university levels, was what it is today, I don't think I would have gone into it. Math teaching now has the following difficulties.

First, digital technologies have resulted in students with shorter attention spans and more dependence on instant gratification. You would be surprised how many high school students are unable to add or multiply two two-digit numbers without a calculator, and in many cases unable to add or multiply even one-digit numbers. This matters, in much the same way that it would matter if students couldn't read words of more than one or two syllables, and depended on print-to-voice readers to “read” literature. Also, most students faced with a problem that requires more than five minutes of work, most will say that they can't do it. A Korean student that I had in a Calculus course in an elite U. S. woman's college earned an A+ told me privately that she felt she was not good at mathematics but was best in the class because the rest of the girls were lazy. I think we will only see more and more students who are unwilling to do the hard work required to become good at math.

Second, in the United States, virtually all K–12 students must pass tests in order to advance in grade or to graduate. Teachers must teach to the test. While teaching to the test is, in my opinion, a good thing for formative assessments, teaching to the test for these high-stakes one-size-fits-all tests at best limits the ability of the teacher to approach the curriculum in innovative ways and at worst results in much time wasted with students taking practice tests and learning test-taking tricks that have nothing to do with mathematics. And of course, if the students do not do well on these tests, the teacher's job is on the line.

Third, computers are in the process of making math teaching obsolete. If a student can learn a subject online, where they can have their instruction completely personalized and can have their homework graded instantly, why should a school system invest in expensive and inefficient human beings to do the teaching? Most teachers will be replaced by low-paid “facilitators” of online courses who do not need to even be trained in mathematics. Instead of there being thousands of college instructors nationwide teaching Calculus I, a dozen or so master teachers will make videos and consult with courseware designers to put together online courses. Math teaching, as we know it, will exist only at the graduate school level where economies of scale do not apply, or at pricey private schools.

So my career advice is only to become a math teacher if it is really a calling for you. Good teaching will be harder than it has ever been, with fewer rewards and little in the way of job security.

Fun with divergent series

A friend sent me a link to a Numberphile presentation in which Tony Padilla and Ed Copland show that the infinite series S = 1 + 2 + 3 + 4 + ... has (in some sense) the value - 1/12. It seems that this result is used in superstring theory in physics and goes all the way back to Euler in 1740. It has generated a lot of heat (and some light) on one of the math forums I follow, so I looked into it and have written it up a bit.

How is 1 + 2 + 3 + 4 + ... = -1/12 ?

E 25. Dividing cards

Here is a problem from Math Charmers, a delightful book by Alfred S. Posamentier.

You are in a completely dark room, seated at a table. There are twelve playing cards laid out in front of you. You know that five of them are face up and the remaining seven are face down, but you have no way of determining whether a card is face up or not. Your task, should you decide to accept it, is to divide the cards into two piles so that each pile has the same number of cards face up as the other. You may, of course, turn cards over.

While at first this sounds impossible, there is a simple solution.

Ha-ha is like Aha!

Yesterday there was an interview on NPR Science Friday with a writer from The Simpsons and Futurama who sneaks in lots of references to advanced mathematics and physics into the cartoons. The references would not be recognized by 99% of the audience. Of course, those who get the references feel really good about being in on a secret. The references are often connected to the plots of the episodes.

For example, an episode in which the theme is "everything becomes easy" has a board filled with mathematical equations including "NP = P". In another episode there is an equation written down which, if true, would be a counterexample to Fermat's Last Theorem. The left and right hand sides of the equation evaluate as equal on a 10-digit calculator, but of course they are not exactly equal.

The interviewer asked the screenwriter if he could guess why there seemed to be some connection between comedy and mathematics and he made a suggestion that I don't remember exactly. I gave my own answer to this question in this blog a couple of years ago.

Back then I mentioned that the "ha-ha" moment in a joke is somewhat like the "aha!" moment in discovering, or appreciating, a mathematical proof, in that both often depend on recognizing the likeness between seemingly dissimilar things. In humor, if one doesn't understand a reference and therefore doesn't understand why something is funny, detailed explanations will almost never make the joke funny. Similarly in mathematics, a proof is very satisfying when it ties together ideas that are well known but seemingly unrelated. Imagine presenting the standard proof that √2 is irrational to a group of intelligent but mathematically ignorant college freshmen. A candid student might respond, "I follow what you did, but what is √2 and what is an irrational number, and why does it matter?" (Unfortunately, this kind of classroom experience is all too common.) By the time that the professor answers all these questions, the aha! moment that the professor was trying to elicit has disappeared forever.

A8. Acute triangles with given side lengths

This is a neat problem from last year's Putnam Examination. It was published (with answer given) in the most recent MAA Monthly.

Given 12 real numbers d₁, ... , d₁₂ on the interval (1, 12), show that there exist distinct indices i, j, k such that there is an acute triangle with side lengths d_i, d_j, d_k.

I will post a hint as a comment in a few days.