Three years and three weeks ago I started my first blog. I wasn’t quite sure what to call it, so I went to John for advice. I had several names in mind, but John quickly zeroed in on one: Quantum Frontiers. The url was available, the name was simple and to the point, it had the word quantum in it, and it was appropriate for a blog that was to provide a vantage point from which the public could view the frontiers of quantum science. But there was a problem; we had no followers and when I first asked John if he would write something for the blog, he had said: I don’t know… I will see…maybe some day… let me think about it. The next day John uploaded More to come, the first real post on Quantum Frontiers after the introductory Hello quantum world! We had agreed on a system in order to keep the quality of the posts above some basic level: we would send each other our posts for editing before we made them public. That way, we could catch any silly typos and have a second pair of eyes do some fact-checking. So, when John sent me his first post, I went to task editing away typos. But the power that comes with being editor-in-chief corrupts. So, when I saw the following sentence in More to come…
I was in awe of Wheeler. Some students thought he sucked.
I immediately changed it to…
I was in awe of Wheeler. Some students thought less of him.
And next, when I saw John write about himself,
Though I’m 59, few students seemed awed. Some thought I sucked. Maybe I did sometimes.
I massaged it into…
Though I’m 59, few students seemed awed. Some thought I was not as good. Maybe I wasn’t sometimes.
When John published the post, I read it again for any typos I might have missed. There were no typos. I felt useful! But when I saw that all mentions of sucked had been restored to their rightful place, I felt like an idiot. John did not fire a strongly-worded email back my way asking for an explanation as to my taking liberties with his own writing. He simply trusted that I would get the message in the comfort of my own awkwardness. It worked beautifully. John had set the tone for Quantum Frontier’s authentic voice with his very first post. It was to be personal, even if the subject matter was as scientifically hardcore as it got.
So when the time came for me to write my first post, I made it personal. I wrote about my time in Los Alamos as a postdoc, working on a problem in mathematical physics that almost broke me. It was Matt Hastings, an intellectual tornado, that helped me through these hard times. As my mentor, he didn’t say things like Well done! Great progress! Good job, Spiro! He said, You can do this. And when I finally did it, when I finally solved that damn problem, Matt came back to me and said: Beyond some typos, I cannot find any mistakes. Good job, Spiro. And it meant the world to me. The sleepless nights, the lonely days up in the Pajarito mountains of New Mexico, the times I had resolved to go work for my younger brother as a waiter in his first restaurant… those were the times that I had come upon a fork on the road and my mentor had helped me choose the path less traveled.
When the time came for me to write my next post, I ended by offering two problems for the readers to solve, with the following text as motivation:
This post is supposed to be an introduction to the insanely beautiful world of problem solving. It is not a world ruled by Kings and Queens. It is a world where commoners like you and me can become masters of their domain and even build an empire.
It has been way too long since my last “problem solving” post, so I leave you with a problem from this year’s International Math Olympiad, which took place in gorgeous Chiang Mai, Thailand. FiverThirtyEight‘s recent article about the dominance of the US math olympic team in this year’s competition, gives some context about the degree of difficulty of this problem:
Determine all triples (a, b, c) of positive integers such that each of the numbers: ab-c, bc-a, ca-b is a power of two.
Like Fermat’s Last Theorem, this problem is easy to describe and hard to solve. Only 5 percent of the competitors got full marks on this question, and nearly half (44 percent) got no points at all.
But, on the triumphant U.S. squad, four of the six team members nailed it.
In other words, only 1 in 20 kids in the competition solved this problem correctly and about half of the kids didn’t even know where to begin. For more perspective, each national team is comprised of the top 6 math prodigies in that country. In China, that means 6 out of something like 100 million kids. And only 3-4 of these kids solved the problem.
The coach of the US national team, Po-Shen Loh, a Caltech alum and an associate professor of mathematics at Carnegie Mellon University (give him tenure already) deserves some serious props. If you think this problem is too hard, I have this to say to you: Yes, it is. But, who cares? You can do this.
Note: I will work out the solution in detail in an upcoming post, unless one of you solves it in the comments section before then!
Update: Solution posted in comments below (in response to Anthony’s comment). Thank you all who posted some of the answers below. The solution is far from trivial, but I still wonder if an elegant solution exists that gives all four triples. Maybe the best solution is geometric? I hope one of you geniuses can figure that out!