I’m hosting issue number 170 because I have a thing for the number’s largest prime factor, but it turns out there’s a reason for a Martin Gardner fan like me to appreciate the number itself: 170 is the number of steps1 needed to solve a classic mechanical puzzle called The Brain invented by computer scientist Marvin H. Allison, Jr., described by Martin Gardner in his Scientific American essay “The Binary Gray Code”, and still available from Amazon.
The Brain, aka The Brain Puzzle, aka The Brain Puzzler.
Here’s what Gardner says about The Brain:
It consists of a tower of eight transparent plasticdisks that rotate horizontally around their centers. The disks are slotted, with eight upright rods going through the slots. The rods can be moved to two positions, in or out, and the task is to rotate the disks to positions that permit all the rods to be moved out. The Gray code supplies a solution in 170 moves.Continue reading →
Back in the 1990s, when I was serving on a jury in a one-day trial, my mathematical temperament got me in hot water with my fellow jurors; fortunately, my outside-the-classroom mathematical training got me out of it. But that doesn’t come in until the end of the story.
The case featured a couple of surprising twists — which is in itself surprising, since even a single twist is unusual in a one-day trial. It had seemed at first like a very straightforward drunk-driving charge. The defendant went to a party, drank some alcohol, left the party feeling unwell, got into his car, drove off, and blacked out, though with enough advance warning of his impending unconsciousness that he was able to pull over to the side of the road and turn on his hazard lights before passing out. A police officer found him slumped over the wheel of his car. The officer smelled his breath and it smelled of alcohol. The District Attorney presented these facts confidently, as if this was going to be an open-and-shut case. But then, in the kind of surprise you see only on television, the defense attorney asserted (with medical records to support his assertion) that in fact the defendant was diabetic, that someone with diabetes can go into hypoglycemic shock if they ingest a little bit of alcohol on an empty stomach, that the breath of someone in hypoglycemic shock is often nearly indistinguishable from the breath of someone who is drunk, and that the amount of alcohol that the defendant had drunk at the party was, according to witnesses, well under the amount that would cause blood alcohol concentration to reach .08% (the legal definition of “too much”). Continue reading →
Many family restaurants offer paper placemats that entice children into solving puzzles as an alternative to kicking each other under the table, blowing bubbles in their beverages, and so on. I remember those placemats from my own childhood and I recall mazes in particular. The mazes weren’t large, but the designers, in their quest to keep us kids occupied as long as possible, would put long dead-ends near the start of the maze. I quickly hit on a strategy that the emphatically named Thomas T. Thomas also found, as he later recounted in his charming essay “Working Backward“:
One of the most valuable techniques in problem solving I learned in the third grade. But it certainly wasn’t a lesson my teacher intended.
I’ll explain that prescription shortly. But first, a confession:
The title of this essay, in which the word “flip” refers to the pedagogical innovation called the “flipped classroom”, is misleading because I’m not going to tell you how to run a flipped classroom. For those of you who haven’t heard, the flipped classroom approach to education is based on moving lecture-delivery out of the classroom and into the dorm room. No, not by having the professor show up in everyone’s dorm room to deliver content on an individual basis; that would be both impractical and improper. Instead, the professor prepares a video, and the students watch it on their laptops. That way, when the students show up in the classroom, they can focus on teacher-supervised activities that take their knowledge to the next level.
There’s an episode1 of a science-fiction television series in which space travelers land on a planet peopled by their own descendants. The descendants explain that the travelers will try to leave the planet and fail, accidentally stranding themselves several centuries in the past. Armed with this knowledge, the travelers can try to thwart their destiny; but are they willing to try if their successful escape would doom their descendants, leaving the travelers with the memory of descendants who, thanks to their escape, never were?
This is science fiction, but it’s also math. More specifically, it’s proof by contradiction. As Ben Blum-Smith recently wrote on Twitter: “Sufficiently long contradiction proofs *make me sad*! When you stick with the mathematical characters long enough, you start to get attached, and then they disappear, never to have existed in the first place.”
This will be an essay about things that seem to exist but which, when you study them deeply enough, turn out not to exist after all. Continue reading →
The kind of magic that grabs (and enchants) me is fantasy-literature magic, not stage-magic, but this month I’ll make an exception to talk about a new math trick inspired by Art Benjamin’s fun-packed book “The Magic of Math: Solving for x and Figuring Out Why”.
After you read this essay, you’ll be able to compute the answer to questions like “What’s the remainder when you divide 123456789 by 9?” or “What’s the sum of the digits when you multiply 123456789 by 9?” without setting pencil to paper, purely through the power of your mind. But before I explain that, I’d better address the question many of you are asking: why would anyone want to learn how to do this sort of thing? (Short answers: to stay sane, to balance their checkbook, and/or to become fame-ish.)
But first, a few words about numbers, rapidity, and mental math.
Here’s something you’ll never see in popular writing about musicians:
“Music. For most of us, the mere word conjures up memories of metronomes and endless scales, the student’s never-ending fear of playing a wrong note (or the right note at the wrong time), and the frowns of teachers from whom a curt ‘Good’ was the highest expression of praise. And yet there are some people who just can’t get enough of making music; they practice hour after hour, honing their skills and punishing their bodies, long after the stage of life when there are parents and teachers forcing them to do it. What strange quirk of character compels this behavior?”
As I said, nobody writes about musicians that way. And yet, when the subject is a mathematician, writers sometimes come up with passages like this:
“You and math – one of the greatest love/hate relationships of all time. What is it about the subject that excites us yet sends a chilling tingle down our spine at the same time? How can it be so precise, yet so fickle? We may never know the answers to these questions, but we do know that math is ubiquitous, though some of us may try to hide from it.”
This is from the introduction to a 2010 profile of mathematics editor Vickie Kearn, which I recently saw on the blog-site of Princeton University Press, where Ms. Kearn has worked as senior editor for many years. The intro continues in a similar vein. “While math may sometimes cause us to cry tears of despair, it has caused Vickie to cry tears of joy.” Continue reading →
The wizard’s-cap graphic that appears at the top of my blog as part of the logo is a piece of an infinite mathematical surface called the pseudosphere.
I don’t study the pseudosphere in my research, and I can’t say I have a lot of intuition about it; in fact I don’t especially like the thing. So why did I choose it to visually represent what this blog is about? Continue reading →
A uniquely French way to express contempt for someone is to call them an “espèce d’espèce” (see Endnote #1); literally, “a sort of a sort”. This month I’m going to tell you about a sort of a sort (or rather, a sort of sorting) that, from a practical standpoint, merits this degree of contempt: the procedure is ambiguous, is annoyingly slow, and doesn’t always sort things correctly. Yet there’s an unresolved mathematical mystery arising from the way that the procedure works better than it has any right to.