Welcoming a New Book Into the World

by admin on April 27, 2022

What I did during the pandemic.

One of the best things about being a writer is the moment when you first hold a book in your hands that has your name on the cover. I think that it may be similar to becoming a parent. You spend months and months anticipating something — and when that “something” finally arrives, it’s so tiny compared to the amount of work you put into it! Also, specifically, it’s a little bit like becoming a father, because the actual nitty gritty (putting the ink onto the paper, binding and trimming it) is done by other people and the process is kind of hidden from me. All I get to see is the final result.

Yesterday I got the chance to hold a new “baby” for the eleventh time. If you’ve lost track, here are their names: What’s Happening in the Mathematical Sciences, volumes 6-12; The Book of Why; The Universe in Zero Words; Visualizing Geology; and The Big Splat, or How Our Moon Came to Be. The new arrival is Volume 12 of the What’s Happening series, which is published by the American Mathematical Society. You can order it from the AMS or through Barnes and Noble. Strangely, it does not show up yet on the Amazon website. I guess Jeff Bezos is asleep at the wheel.

I’ll probably always think of this one as my “pandemic book.” I started working on it officially on October 1, 2020, and for the next twelve months it was almost the only thing that I worked on. A large fraction of the book, three out of eight chapters, is devoted to the way that mathematicians responded to the pandemic. The chapters progress roughly from large scale to small scale. The first is about epidemiological models in the general population (don’t call them “forecasts”). The second talks about what happens when you have subpopulations with different characteristics — say, college students or prisoners or vaccinated people. When is it safe to open up a college? Why is it impossible to contain an epidemic if you have a massive incarcerated population? Finally, the third chapter talks about COVID-19 within the human body. How does the immune system work and why does it not work in people with severe cases? What are some targets in the virus itself that we can use to develop anti-COVID drugs? Many of these questions seem biological, but every one of them involves mathematical models.

When the COVID epidemic began, I initially had great zeal to write articles about it, but that zeal waned fast for two reasons. One was that the epidemic itself was such a depressing experience, as the lockdowns took away human contact and made every day similar to every other one. The second was that information about COVID was changing so fast, so something that seemed exciting and important today might be wrong, debunked, forgotten in a month or two. It was difficult for me to see the lasting value in the articles I was writing.

What’s Happening gave me a chance to step back and recalibrate. What can I say about this epidemic that might have permanent value? And What’s Happening also freed me from the pressure to produce something “newsy.” I was able to spend a month or two interviewing researchers in depth and understanding the intricacies of their models, which was not easy. The three COVID chapters were basically finished by the end of March last year. I was worried that they, like my initial forays into writing about the epidemic, might be outdated by the time the book finally came out (a year later!). But when I look at those chapters now, I think they held up remarkably well. Partly that is because of the honesty that mathematics imposes. If the model is mathematically correct and consistent, it will continue to mean something a month or a year from now. The only thing that might invalidate it is that we might find some of the assumptions that went into the model were wrong — but that’s not a mathematical error. In any case, it means we have learned something, and that is the way that science is supposed to progress.

So if you’re sick of reading politicized takes on the pandemic, take a look at these three chapters and learn about some pandemic science stories that you probably never heard before.

The best thing about the other five chapters of What’s Happening is that they are not about the pandemic! Life went on, even during the pandemic, and math did, too. Four of the chapters are about various topics in pure math, which are by turns fun and abstruse and sometimes both. The other chapter is about the mathematics of climate change. As in the chapters about the pandemic, the challenge was to say something new and interesting about a subject that has so much written about it in the popular media that has little value or consequence. Here again, mathematics is my guide. If the math is good, then there is surely something that we can learn from it.

So now, when somebody asks me what I did during the pandemic, I finally have something to show them!

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Analysis Without Words

by admin on April 7, 2022

Last year, after bingeing too many times on chess games played against the computer, I said, “No more.” I actually stuck to that resolution for about seven months, but then I updated the operating system on my Apple computer. The chess program I had on that computer (Shredder 12) no longer worked, so I pulled the trigger and updated to Shredder 13. And of course, having a newer and stronger version of the program, I had to try playing a few games against it.

After falling “off the wagon” for a couple days, I remember now why playing against the computer was such a bad idea, and so I will renew my vow not to do it again. However, one of my games did give me this entertaining position to look at. I’m playing White.

Position after 54. … Kxh5. White to move.

FEN: 5Q2/8/4K3/r6k/8/8/8/8 w – – 0 55

In the spirit of full honesty, let me say that this position did not occur in our game. I made a mistake on move 25 and lost, but then I was curious what would have happened if I had made the right move, so I played out the game from there and reached this position.

This is one of the Four Endgames of the Apocalypse that I’ve written about before: K+B+N vs. K, K+Q vs. K+R, K+Q+h vs. K+Q, and K+R+B vs. K+R. They’re the most difficult endgames that are likely to arise in human play, and I have played each of them from one side or the other in tournaments. However, I’ve never played the K+Q vs. K+R endgame against the computer as part of a game, rather than as an exercise. I was very curious about whether I would be able to win it.

Until computers came along, the conventional wisdom about this endgame was that Black has to keep his king and rook close to each other, in order to avoid combinations where the king and rook get forked. Unfortunately, this passive approach allows White to simply squeeze the K+R into a corner, reaching a standard zugzwang position where Black is forced to separate them against his will.

But when you play K+Q versus K+R against a computer, you will find that it does not hesitate to separate the king and rook, and it’s annoying as heck. The position above is a typical example — and a good one to remember if you ever find yourself defending this endgame with the rook. An optimal formation for Black is to put his king and rook on opposite colors and opposite sides of the board, and with the rook on the same rank (or file) as the king. For example, a5 and h5. On those squares they cannot be forked. If Black could just keep them there forever, he could draw the game.

So what is White to do? First, let’s take an inventory of the forking squares: e1, d2, c3, b4, c7, and d8. (Not b6, because the White king is in the way of any checks on the 6th rank.) In order to have access to the majority of these square, White’s queen should head toward the first four ranks. This immediately makes 55. Qf3+ the strongest candidate for his first moves. The other possible checks, on e8, f7, or h8, do not “head in the right direction.” Another way to think of it is that White’s king has a lot of influence on the fifth, sixth, and seventh ranks, preventing Black’s king from running in that direction. So it makes sense that the queen should mostly patrol the first four ranks, in order to give White influence over that part of the board. So our first move is 55. Qf3+. But we’re just beginning!

Another very interesting thing you’ll notice is that Black’s king has to be very careful about moving onto any dark squares. In this position he does not have to be careful yet. But for example, after 55. Qf3+ Kg5 56. Qe3+ Kh5 (Black should head for this “unforkable” square whenever he can) 57. Qe2+ now we already have a position where Black’s king cannot go to any of the dark squares: 57. … Kg5 or 57. … Kh6 allow a fork on d2, while 57. … Kh4 allows a fork on e1. It’s as if White had an invisible dark-squared bishop on the board, covering g5, h6, and h4! That is a huge help for the queen. Black’s only move is 57. … Kg6, and now after 58. Qd3+ Kh5 (back to the “unforkable” square), we have a position where White has to come up with something clever. A quiet move.

Position after 58. … Kh5. White to move.

FEN: 8/8/4K3/r6k/8/3Q4/8/8 w – – 0 59

Do you see the winning move? I do not believe that White can make progress in this position just by chasing the Black king with checks. Here, it’s time to bring his second piece into play, with 59. Kf6! Here are the points of this exquisite move:

  1. It threatens 60. Qh3 mate.
  2. It restricts Black’s king, so that the next time White plays a diagonal check, the king will not be able to play … Kg6 any more and will have to move to a dark square.
  3. The king is also safe from harassment by rook checks, primarily because the Q on d3 prevents … Ra6+. Note that d3 is the only square on the board (aside from f1) where the queen hits both h3 and a6.

Black has no good moves, and the game ends very quickly after 59. … Kg4 (or h4) 60. Qc4+ Kh5 (once more back to the “unforkable” square) 61. Qe2+, and as per point #2 above, Black is forced to a dark square and loses his rook to a fork.

I hope that this gives you a new appreciation for just how tricky it is to win this endgame. This whole variation hinges on White’s being able to maneuver his queen to the unique winning square, d3. When you’re playing this endgame against a computer, it will over and over again find the lines where you can’t win just by chasing the king around, and you have to find a quiet move or a zugzwang. Humans are not very good at that.

After the game, I was curious what it would look like if I drew a decision tree, a diagram that shows White’s best move against every Black move in every position, starting from the first diagram. How complicated would it be? Here is what I came up with.

Decision tree after 55. Qf3+.

Notice that I’ve put the first move, 55. Qf3+, at the center of the tree, and succeeding moves go outward from the center. On one hand, I thought this was interesting to look at. It’s a proof without words that White is winning. Every single variation is covered, and terminates either with checkmate or a fork of the king and rook. No words are needed! So, in one sense, it’s a much more efficient representation of White’s winning technique than this post, which has hundreds of words.

On the other hand, for a human it’s absolutely useless. You can’t apply this decision tree to any other position. By comparison, all those hundreds of words I’ve written may give you help in analogous positions. The importance for Black of the square h5; White’s idea of putting his queen on the side of the board where his king isn’t; the importance for White of checks on the white diagonals; the “invisible bishop” that prevents Black’s king from moving to dark squares — all of these concepts could be useful to you in other positions.

But there is one nice pedagogical point about the diagram. Notice how it falls into two halves, above and below the dashed line. Above the dashed line we have all the positions that White can get to just by playing queen checks and never touching his king. That top half is “level one.” But in order to win the position, when we get to the key position we need to go up to “level two,” changing the position by moving our king. Once we’re on level two, the decision tree simplifies dramatically and we can calculate our way to victory.


  1. When defending with K+R vs. K+Q, if you must separate the king and rook, try to aim for a position where they are as far apart as possible, on the same rank or file, and on opposite colors.
  2. When trying to win with K+Q vs. K+R, if your opponent has separated his king and rook as in part one, use checks to maneuver your queen to a side of the board that is away from your king, so that you have access to the largest number of forking squares and have “influence” over all parts of the board.
  3. If Black’s king and rook are separated, it becomes very dangerous for the king to move to the same color square as the rook. If you are playing White, you can visualize this as having an “invisible bishop” helping your queen.
  4. If Black defends perfectly, White cannot win with queen checks alone. At some point White will need to play a “quiet move” with his king in order to hem in Black’s king. If you are White, you should prepare for the quiet move by looking for “magic squares” for your queen that defend your king against any possible rook checks. (In this example, d3 was the “magic square.”)
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“There’s No Such Thing as an Even Trade”

March 12, 2022

First, let me say that the title of this post is not literally true. However, it’s something that I like to tell my students, and it’s not as far-fetched as it seems. It’s intended to correct a very harmful mindset that starts affecting players as soon as they learn the values of the pieces. According […]

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The World Changes Again

February 27, 2022

February, 2020. A previously little-known virus escapes from China, and within a few weeks it is everywhere. Even for those who were not infected by the coronavirus, it completely changed our view of what was possible. Most of us had never been through a worldwide pandemic that killed millions of people. Now we have, and […]

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The Student Draws the Master

February 13, 2022

The headline says it all! Yesterday, my student Atlee (~1500) scored his first draw against me in our weekly training game. I knew this was going to happen eventually, and I wondered what it would take for Atlee to draw or win. The short version: I blundered a piece but managed to fight back and […]

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Levels of Truth

February 1, 2022

As I mentioned before, I’ve been playing a training game each week with one of my students from the Aptos Library Chess Club, Atlee, who is about a 1500-strength player (I’m guessing). The games follow an interesting pattern. I win, I teach Atlee a lesson, then I come home and go over the game on […]

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Catastrophes in the Nimzo

January 16, 2022

When you play the move 1. d4 as White, you’re generally saying that you want the game to be a sumo match rather than a sword fight. You’re tired of all the sharp tactics of 1. e4, whether it’s the Sicilian Defense or the open games. You want to just get a solid, safe space […]

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New Year, New Opening!

January 12, 2022

Last weekend Gjon Feinstein and I met up for not only our first live chess games of the year, but our first since the pandemic began. Back in the long-gone days B. P. (before pandemic) we used to meet very frequently, at least once every two weeks. Times have sure changed. We saw each other […]

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Fairer Chess?

December 31, 2021

One of the first things that kids love to do, after they learn the rules of chess, is to tamper with the rules. I’m not sure why. For example, they say, “Let’s make every piece a queen!” Sounds like fun in principle, but what happens in practice is that every move is a capture and […]

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Best American Player of My Generation

December 26, 2021

Here is your trivia question for today: Who is the first chess player to simultaneously be #1 on both the USCF’s list of top players 50 and older and the list of top players over 65? Need a little hint? Okay, here it is. Although Larry Christiansen has had an incredible career, I think that […]

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