What’s It Gonna Take For Somebody To Win A Chess Game?

Game 5 of the World Chess Championship began under a cloud. Not a literal cloud, though there were those in London, too. Rather it was the lingering hubbub of a published and deleted video. Since that video was released, a prominent chess writer resigned and, oddly, the event’s organizing body announced that it had hired a security firm that was ready to sweep for illicit electronic devices and deploy polygraphs on the players if necessary. Was the latter related to the video? To some other bit of intrigue yet to fully emerge? Or just because chess’s governing body is, how do you say, filled with plenty of intrigue of its own?

I have no answers for you. But I do have some chess to relay. To catch you up if you’re just joining us: Magnus Carlsen of Norway is seeking his fourth world title. His challenger Fabiano Caruana of the U.S. is trying to become the first American world champion since Bobby Fischer in 1972. The pair began the day’s game tied 2-2 in the best-of-12 title match.1 It didn’t end much differently.

The two grandmasters started Thursday’s game with the Rossolimo variation of the Sicilian Defence — the third time they’ve opened with that sequence of moves in the match’s five encounters. But then came a lightning bolt that briefly illuminated the match. It was known as “6. b4!?”

Caruana’s sixth move — his white pawn to b4 — electrified the encounter. This is what the board looked like after it struck.

r1bqk1nr/pp1p1pbp/2n3p1/1Bp1p3/1P2P3/5N2/P1PP1PPP/RNBQR1K1 b kq – 0 0
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This is a rare move in this position at the game’s highest levels, and it’s an aggressive one — one often reserved for speed-chess games, rather than the lengthy, classically timed games of a world championship. Carlsen had faced this move with the black pieces only once before, according to ChessBase — in a 2005 game against the Dutch grandmaster Daniël Stellwagen, when Carlsen was just 14. (That game ended in a draw.) Given Carlsen’s prodigious memory for positions, it would be no surprise if he remembered that game well. And he claimed not to be troubled.

“To be honest, I was pretty happy about the opening,” Carlsen said after the game.

Lichess’s analysis tool calls that sixth move the “Sicilian Defense: Nyezhmetdinov-Rossolimo Attack, Gurgenidze Variation.” Gurgenidze was the Georgian grandmaster Bukhuti Gurgenidze, and “one of the most original and striking players of the Soviet era,” wrote ChessBase upon his death in 2008. The early part of Thursday’s game was striking, too. Grandmasters called it the sharpest opening they’d seen in world championship history.

Generically, this sort of move, a pawn to b4, is called a wing gambit, and it can be ventured in a few different openings. White sacrifices a pawn to potentially gain an advantage in the center of the board and in the mobilization of his pieces — the claiming of territory and the arming of his troops. Indeed, it was perhaps the first time in the match that the player with the white pieces had been able to sustain anything one might be able to call an attacking advantage.

Yet Carlsen was able to parry the threats. He appeared calm throughout the game, occasionally throwing one arm over the back of his chair, ever so suave in his gray suit.

By Caruana’s 19th move, he was perhaps regretting that his brief advantage had fizzled. And indeed it had. He spent nearly 32 minutes on that move, head often in both of his hands, pondering the board. Carlsen and Caruana agreed to a draw after 34 moves and just over 3 hours, in the position below. The match now sits level, 2.5-2.5.

8/5R2/5bp1/3rpk1p/6P1/4B2P/5P2/5K2 b – – 0 0
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Given the way the championship’s scheduling works, Carlsen will play with the white pieces — and its first-move advantage — for the next two games. It will prove a critical gauntlet for Caruana’s title hopes. Here’s a visualization of how things have gone, and we’ll keep the chart below updated throughout the match.

Game 6 begins Friday at 3 p.m. Greenwich Mean Time — that’s 10 a.m. Eastern. I’ll be covering it here and on Twitter.

Villanova Won A Title. Now It Must Start Over From Scratch.

The Villanova Wildcats produced one of the most dominant seasons in NCAA history last year, going 36-4, including a complete dissection of a strong Michigan team to win the championship game. The Wildcats scorched teams on offense, ranking No. 1 in the country in offensive efficiency and effective field goal percentage, according to college basketball stats guru Ken Pomeroy. This helped them beat the Wolverines by 17 points.

But the team that is defending that title — currently ranked eighth heading into Wednesday’s rematch with Michigan — is hardly recognizable eight months later, as four of coach Jay Wright’s stalwarts from a season ago are now in the NBA.

The departure of Jalen Brunson and Mikal Bridges — a pair of juniors left over from the 2015-16 national title-winning team — has left a crater in Wright’s lineup. Along with the exits of Donte DiVincenzo and Omari Spellman, the outgoing quartet combined for a whopping 26.1 win shares last season1.

It’s typical for reigning national champions to lose a large chunk of their talent the following season, especially in the one-and-done era. And while the Wildcats may not have lost the most win shares of past champions, the immediate exodus of talent will have huge consequences for their prospects to repeat as champions this season. This is perhaps a long way of saying winning back-to-back titles, or even coming close, has become very difficult in college basketball — and for good reason.

Villanova’s departures have left a sizable hole

Total win share of players who left NCAA championship teams the season after their championship, since the beginning of college basketball’s one-and-done era

Departing players
Season Champion Number Win Share
2017-18 Villanova 7 26.1
2016-17 North Carolina 7 21.8
2015-16 Villanova 5 11.4
2014-15 Duke 4 25.5
2013-14 Connecticut 7 19.8
2012-13 Louisville 4 12.3
2011-12 Kentucky 7 36.3
2010-11 Connecticut 5 13.4
2009-10 Duke 6 20.7
2008-09 North Carolina 9 26.6
2007-08 Kansas 9 35.3
2006-07 Florida 10 33.7
2005-06 Florida 2 1.5

Source: Sports-reference.com

Billy Donovan’s 2007 Florida Gators are the only team in the past 45 years to repeat as NCAA men’s basketball champions, after John Wooden’s UCLA Bruins capped off seven consecutive titles. Back then, Wooden had the luxury of coaching future NBA Hall of Famers Kareem Abdul-Jabbar2 and Bill Walton for three seasons, something that is largely unheard of in today’s game.3

And when it comes to the one-and-done era, the Gators are an anomaly themselves, as Donovan managed to persuade the likes of Joakim Noah, Al Horford and Corey Brewer to remain in Gainesville for their junior years before winning another title and then moving to lengthy careers in the NBA.

For most champions, winning a national title usually means saying goodbye to their best talent — the nation’s top freshman are forced to use college as a stopgap for a year before jumping to the NBA, and upperclassmen often ride their team’s success to test the NBA’s waters. For his part, Wright did well to keep Brunson and Bridges in Philadelphia for another two years after winning their first title, which built a bridge to that second championship.

But the team that cut down the nets last year has been gutted, particularly on the offensive side. Among the top four players of each champion since 2006, when the one-and-done began, Villanova’s departed quartet leave the greatest offensive hole for a reigning champion, a hole that might be too great to overcome.

Villanova fans might choose to view things in a more optimistic way, instead thinking themselves as fortunate that they only lost four players, especially seeing the Wildcats of Kentucky lose an unimaginable six players after their championship in 2012 and then stumbling into the NIT a year later. Nova’s relatively tiny rotation last year — Villanova ranked 302nd in total bench usage, according to KenPom — could be a blessing in disguise as the likes of Eric Paschall and Phil Booth are still available to make the leap to the top of the college ranks and potentially beyond.

Still, Villanova fans thinking of a repeat might want to curb the enthusiasm.

Any team not named Duke, Kentucky or Kansas — whose recruiting prowess means a revolving door of NBA-bound super freshmen — has struggled to be relevant again immediately. If you look past these three blue bloods, Louisville is the only reigning champion to reach the Sweet 16 the year following championship in the last dozen years. It’s why the 2010 North Carolina Tar Heels couldn’t even make the NCAA Tournament after winning the whole thing a year earlier. It’s why last year’s Tar Heels were swept aside in the second round of the tournament by Texas A&M.

Back-to-back has become a pipe dream

How men’s NCAA champions have fared the following season in college basketball’s one-and-done era

Season after championship …
Season Champion Wins Losses Postseason
2017-18 Villanova 2
0 ?
2016-17 North Carolina 26
11
2nd Round
2015-16 Villanova 32
4
2nd Round
2014-15 Duke 25
11
Sweet 16
2013-14 Connecticut 20
15
NIT
2012-13 Louisville 31
6
Sweet 16
2011-12 Kentucky 21
12
NIT
2010-11 Connecticut 20
14
2nd Round
2009-10 Duke 32
5
Sweet 16
2008-09 North Carolina 20
17
NIT
2007-08 Kansas 27
8
Sweet 16
2006-07 Florida 24
12
NIT
2005-06 Florida 35
5
Champion

The NCAA Tournament’s First Four was known as the First Round until the 2015 tournament.

Source: Sports-Reference.com

Joining senior Paschall, who’s being touted as a first-round pick in the 2019 NBA Draft, and redshirt senior Booth, who netted 23 in his season debut last week, is the 12th best recruiting class, according to ESPN. Five-star recruit Jahvon Quinerly is considered one of the best freshman point guards in the nation, and four-star forwards Cole Swider and Brendan Slater both also have a place on the ESPN 100. Whenever this is enough for Wright’s team to make waves again in March is a question for the season ahead. However, with currently the fifth-best ranked recruiting class for next year, Wildcats fans may have another title-winning team in the not-too-distant future, maybe just not in the immediate one.

The Biggest Blunder Of The World Chess Championship Is A Deleted YouTube Video

Game 4 of the World Chess Championship in London began with a fitting surprise: the “English Opening.”

Magnus Carlsen of Norway, the three-time defending world champion and world No. 1, began the game by pushing the white pawn in front of his left bishop to the c4 square — a relatively rare move at the game’s highest levels. Fabiano Caruana, the U.S. challenger and world No. 2 trying to become the first American world champion since Bobby Fischer in 1972, pushed a black pawn to e5. And with that, the English had come. Few bells would be rung for the rest of the game.

Game 4 ended in a draw, just as the three previous games had. It was an uninspired 34-move, 2.5-hour episode. The match for the game’s highest prize remains level, at 2 points apiece in a race to 6.5.1 The boring result failed to overshadow the real drama of the day: the Zapruder film of this world championship.

But first, the chess.

“Carlsen is trying to avoid that really annoying Petroff,” Robert Hess, a grandmaster, said during a broadcast on Twitch. The Petroff Defence is one of Caruana’s favorite chess tools when he has the black pieces, but he can deploy it only when white cooperates by opening with a pawn to e4. Carlsen’s opening move, therefore, was preventive — or “prophylactic,” as chess players like to say. (Bards of the game, one and all.)

The pattern of pieces that developed on the board is called, rather delightfully, a “Reverse Dragon.” The Sicilian Defence has a variation called the Dragon — named after the resemblance of the pawns to the constellation Draco — except in this case its colors were reversed. But the position breathed no fire on Tuesday.

After 10 moves, the game was an exact match of a game that Caruana played against Wesley So, another top American grandmaster, earlier this year — the only such game that had ever featured this position, according to ChessBase. Given how recent and high-profile it was, this was a game that Caruana and Carlsen almost certainly both remembered well.

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After that, the two entered previously unseen territory. But the uncharted wilderness did not provide much in the way of excitement. Carlsen and Caruana agreed to a draw after 34 moves, in the position below. Despite the many pieces on the board, the grandmasters’ expertise told them that there was only one way this was likely to go.

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Here’s how things have gone on the chessboard thus far, and we’ll keep the chart below updated throughout the match.

The day’s humdrum play was overshadowed by some excitement off the board, though. A chess-world controversy — or at least what qualifies as one — erupted. Before the game, the posh Saint Louis Chess Club posted, and quickly deleted, a YouTube video appearing to show aspects of Caruana’s pre-championship preparation sessions. The club is Caruana’s de facto office and was founded by the billionaire retired financier Rex Sinquefield, who also helps fund Caruana’s chess career — both the club and Caruana are totems of Sinquefield’s deep pockets and deep love of chess. A chess columnist named John Hartmann tweeted this screenshot from the video.

It shows a laptop screen, complete with chess ideas in progress — a “Fianchetto Grunfeld,” various Queen’s Gambits Declined (an opening that was played in Game 2), and a number of ideas related to the favorite Petroff. It also shows a number of games from the 2016 world championship, to which Caruana was surely paying close attention.

A clip of the video shows Caruana leafing through a book of Carlsen’s past championship games and then handing it across a chessboard to his grandmaster coach and “second,” Rustam Kasimdzhanov, while grandmaster Alejandro Ramírez sits nearby.

Scandalous, I know.

Some chess commentators suggested that it was a huge blunder; others suggested that it was a deft piece of a disinformation campaign. The Saint Louis Chess Club did not respond to my emailed requests for comment, and Caruana’s manager declined to comment. Caruana himself declined to comment at a post-game news conference, and Carlsen claimed that he hadn’t seen it but was aware of its existence.

World Chess Championship preparation is always closely guarded. Kasimdzhanov warned me last spring that anything I might write about Caruana’s prep would be pored over by his Norwegian opponent and his team of hired chess guns for any shred of usable information. A player readying for a championship match typically enters seclusion with a small, handpicked crew of grandmaster aides and other associates. Even revealing their identities could be risky, I was told, because different grandmasters have different chess tendencies, and revealing the grandmasters might signal a game plan to the opponent. When Fischer was readying for his championship match, at a resort in upstate New York where prizefighters trained, he told an interloping New York Times reporter to “shove off” and stopped answering the phone.

In April, FiveThirtyEight was promised access to Caruana’s training camp by Caruana’s managers — a promise that was rescinded in July. That change of heart was not unique to this website. At least one other major online outlet was promised exclusive access to his training camps — another promise that was rescinded in July.

But the players will have to play their moves in public eventually. Perhaps one of them will even win a game of chess.

The match rests on Wednesday, but Game 5 begins Thursday at 3 p.m. Greenwich Mean Time — that’s 10 a.m. Eastern. I’ll be covering it here and on Twitter.

Can You Stave Off A Cold With Willpower?

cwick (Chadwick Matlin, deputy editor): :sniffle: Maggie, Anna and Christie, I’ve gathered you all to discuss the kind of question that could change EVERYTHING (and one well-timed for flu season): When I know it’s really important for me not to be sick for a day or two (say, if there’s an election coming), can I stop myself from feeling sick? Does my mind really have power over my matter? Is my brain that dope?

You’re not doctors, but in my book, you’re close enough: journalists who talk to doctors. So tell me: Do I have the power???

slackbot: I’m sorry you aren’t feeling well. There is Advil and Tylenol in the cabinet in front of Nate’s office/Vanessa’s desk.

cwick: Readers, that’s an auto-response triggered every time one of us says “sick” anywhere in our chat app. But if you ever need an Advil in the FiveThirtyEight office, now you know where to find it.

christie (Christie Aschwanden, lead writer for science): Hi, Chad! I love it when an editor sends writers off to get an answer to his personal medical problem.

maggiekb (Maggie Koerth-Baker, senior science writer): I get where he’s coming from, though. This is a pretty common belief. I mean, my super skeptic engineer husband is convinced that he saved all his illnesses for the end of the semester in college. Last test done. And then the deluge.

cwick: I suffered through a 102-degree fever the day after my wedding. I don’t think anything you say will convince me that I WASN’T holding it at bay the whole weekend.

christie: Because I like you, Chad, I asked a few experts. I have a few pet theories myself. But first, the expert opinion: “Sorry. No evidence of the willpower effect!” That’s according to Michael Osterholm, director of the Center for Infectious Disease Research and Policy at the University of Minnesota.

anna (Anna Maria Barry-Jester, senior reporter): Was that a “no evidence for or against” the willpower effect, Christie?

christie: I think he meant there was no evidence that there is a willpower effect.

And Marc Lipsitch, an epidemiologist at Harvard, said the same thing that several others did: “I don’t know of any research one way or another about these things.”

anna: (Thank you for that prescient answer to my at-that-point unasked question, Mr. Lipsitch.)

christie: One of my theories: It’s all about attention and focus. When you’re stressed about something else or hyperfocused on getting through finals week or meeting a deadline or getting out a grant proposal, all of your attention is going to that other task. When you get through it, you have more attention to pay to your symptoms, and you can kind of bask in them. So it’s not that you’re not sick (while you’re supposedly holding it off), it’s just that you’re in denial or you’re ignoring the symptoms.

anna: I can buy that (though I assume you’re talking about mild colds here).

christie: Yeah, I’m talking about the usual winter crud. Not the kind of illness that you can’t possibly ignore, like a heart attack.

anna: The one interesting thing I found was some research about colds and placebo. But before we get to that, there seems to be very little research on warding off illness altogether or delaying it and a lot more on reducing symptoms or shortening how long you are sick. Part of the problem there is that mild colds come and go, making them difficult to study.

christie: Good point, Anna. And that’s also what makes placebos so good at treating them.

anna: But a study from 2011 really intrigued me. It’s one study and it’s small … It looked at echinacea, an herbaceous flower commonly used to treat colds (the National Institutes of Health says some preparations may potentially treat colds, but the evidence is weak). The study essentially found that people who believed in the power of echinacea and were given a placebo pill had shorter colds (by a lot, 2.5 days!) than those who didn’t believe. Placebo: It’s a hell of a drug.

christie: If you take the placebo when you are feeling your worst, which is usually the low point no matter what, then what is really just the illness’s natural course appears like an effective treatment. (When it’s at its worst, it can only get better.)

anna: What about my favorite cold placebo, vitamins in the form of a fizzy additive to water? I take that before I get on a plane. Or ride the NYC subway.

christie: Oh, the fizzy vitamin water? People love those! And they’re a great placebo. But … a Cochrane review found that vitamin C did not reduce the incidence of colds and that “trials of high doses of vitamin C administered therapeutically, starting after the onset of symptoms, showed no consistent effect on the duration or severity of common cold symptoms.”

anna: Oh, I know that there isn’t particularly convincing evidence about the effect of vitamin C on colds. It is literally my favorite placebo, not pharmaceutical. The placebo effect is real.

christie: I wonder how they measured the duration of the cold in the echinacea study. Because if it’s derived from a self-reported “how do you feel?” question, there’s a lot of wiggle room. If you expect to feel better, you might in fact rate yourself as feeling better. Because how you feel is a rating of your experience. It’s open to suggestion.

anna: “Duration was defined as total time elapsed from enrollment until the last time answering yes to the question, ‘Do you think you still have a cold?’” I think that’s part of why that study intrigued me, Christie. Because it suggested that how people felt depended on whether they thought the treatment could work. I don’t see why that couldn’t in theory translate to keeping a cold at bay, as well.

christie: Agree, Anna. How we actually feel is a conglomeration of a lot of things, which include how we expect to feel.

cwick: So are we really talking about two different questions: Can I stop myself from feeling sick for a bit? (No, you dummy.) Can I make myself feel better once I am sick? (Maybe if you truly believe.)

maggiekb: I’m going to add one more question here, Chad. Instead of asking whether you can hold off your illness until your stressful life events are done, what about the question of whether stressful life events make you sick. And the answer to that, kind of surprisingly, is … well … maybe … yeah.

There’s a line of research — enough individual research papers to make a meta-analysis — that suggests stressful life events can make you more susceptible to things like the common cold.

christie: That’s a good point, Maggie. And that leads me to another theory: When you’re stressed about that thing in your life, you may be more susceptible to illnesses. You may be skimping on sleep, not eating well, drinking too much, skipping exercise, etc. The illness just catches up to you.

maggiekb: The studies on this are super interesting because they actually involve getting these sample groups of people, doing inventories about their stressful life events, and then exposing them all to cold virus intentionally. So for instance, in one 2012 study, the people who reported stress and stressful life events were twice as likely to get sick.

cwick: Wait, what does purposeful exposure to cold virus look like? Do I get misted? Do I get slobbered on? Does someone wipe their snotty hand on my face?

anna: In the nose, Chad.

cwick: 👃!!!!

christie: In one study, Chad, they gave nasal drops containing rhinovirus. Turns out, people who were sleep deprived were most likely to get sick.

maggiekb: Misting in some of these studies. Swabbing in others.

cwick: My nose is twitching just thinking about it.

anna: Which is to say, don’t pick your nose after riding the subway.

christie: And don’t touch your nose, eyes or mouth ever during flu season if you want to stay well. Seriously, good hygiene practices are your very best defense. Wash your hands with soap and water. Don’t cough on people (or get coughed on).

maggiekb: It’s not just cold/flu, either. There are some papers that show connections between stress and worse outcomes for HIV/AIDS. Including higher levels of virus in the bloodstream, an increased risk of picking up other infections, and increased risk of death.

I’m kind of fascinated now by this researcher at Carnegie Mellon who has basically made it his life’s work to figure out WHY stress is able to increase your risk of contracting a communicable disease. And you guys are going to love this … he’s the American Psychosomatic Society’s 2018 distinguished scientist.

christie: Wow, there’s an American Psychosomatic Society?

anna: Pardon, what?

cwick: Cheap joke: They dreamed it into reality

maggiekb: Anyway, his theory is that the receptors that bind to stress hormones can become resistant to those hormones. The more resistance, the worse your body is at suppressing inflammation. The more chronic inflammation, the worse things work … in terms of preventing illness.

anna: Chad, you really just want to know if you can will yourself into not getting sick, right? Do you care at all about what you can do to make that sickness … shorter, less … sick?

cwick: IDK, once I am sick, I sort of revel in complaining about it. If you shorten my colds, what would I have to talk about?

christie: Go ahead and joke, Chad, but I think we should be careful about stigmatizing the connection between mind and body health.

maggiekb: It’s not hokum. At least not completely.

christie: The mind is a crucial part of the body, so it’s not surprising that one’s psychological state of being can affect physical symptoms. Instead of joking that it’s “all in their heads,” we might be better off trying to harness this connection.

anna: Related: A recent New York Times magazine story delves into the new science of placebo and after reading it, it’s hard not to feel like a whole new era of treatment could be upon us. But it means completely revamping how we think about things like placebo.

christie: You know where this is leading, right, Chad? To my perpetual mantra: It’s complicated, and science is hard.

cwick: When will science get easier?

christie: Never! Sorry.

cwick: Lame.

The World Chess Championship Is Deadlocked After Game 2

Heavy rain showers, a gentle breeze and 57 degrees in London, the BBC reported this morning. The top American grandmaster Fabiano Caruana was unprepared for such weather, arriving for Game 2 of the World Chess Championship sporting a wet blazer.

Never mind the sartorial dampness, however. He arrived excellently prepared for the chess.

His opponent is Magnus Carlsen of Norway, the three-time defending world champion and world No. 1. Caruana, the world No. 2, is vying to become the first American world champ since Bobby Fischer won the title in 1972. The pair had already played an epic, 7-hour, 115-move draw in Game 1 on Friday, and the best-of-12-game match sat level, 0.5 points apiece.1

The result in Game 2 on Saturday was the same — but the path to it was much shorter. After a 49-move, three-hour draw, the grandmasters are level at 1 point apiece, and the championship remains deadlocked.

Carlsen and Caruana began Game 2 — Carlsen with the white pieces, Caruana with the black — in an opening called the Queen’s Gambit Declined. Everything went according to well-established chess theory until Caruana’s 10th move, which can be rendered in chess notation as “10…Rd8!?”

The “!?” denotes an “interesting move.” (The “10” means it’s the 10th move, the “…” means we’re talking about the black pieces, the “R” means the rook, and the “d8” indicates the square the piece is bound for.)

“Surprise is very important,” said Judit Polgar, a grandmaster commentating the match.

The surprise appeared to pay off, as Carlsen, playing the white pieces, thought for 17 minutes, head in his hand, looking rather perturbed. This is what he saw:

r1br2k1/pp3ppp/2n1pn2/q1bp4/2P2B2/P1N1PN2/1PQ2PPP/3RKB1R w K – 0 0
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Tick tock.

Tick tock.

“Magnus almost touched a piece!” yelled Anna Rudolf, another match commentator, capturing quite faithfully the causes of — and level of — excitement that accompanied Game 2 of the World Chess Championship.

Carlsen eventually did actually touch a piece, breaking the unbearable tension, moving his bishop to e2. And by this point, this particular chess position had only ever been seen once before in a competitive game according to ChessBase — in an obscure contest between two English players in 2014 in Aberystwyth, Wales.

Nevertheless, Caruana appeared ready for everything, moving quickly through the novelty. He was prepared for the chess, if not the rain.

This trend continued. Carlsen thought for another 12 minutes before making his 12th move, 9 minutes for his 15th move, 15.5 minutes for his 16th move, 9 minutes for his 19th move and, well, you get the idea. In many ways, the roles were reversed from the first game. Caruana, whose own time had ticked down to mere seconds in Game 1, opened up an hour advantage on the clock in Game 2, and looked rather comfortable.

Carlsen faced another ?! — er, I mean interesting — position while contemplating his 17th move, shown below.

r1br2k1/pp3pp1/3bp2p/q2nN3/P4B2/2P1P3/2Q1BPPP/3R1RK1 w – – 0 0
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The chess computer engine that keeps me warm during the games suggested a rather dramatic sacrifice: taking black’s pawn on f7 with the white knight. Black’s king would take the white knight, but white would gain exciting but complex attacking chances. Carlsen opted against the gambit — instead developing a bishop to f3 — perhaps because his clock was loudly ticking in his ear.

Caruana started scratching his own head at this point — but the tension soon slackened. By move 26, the queens and remaining bishops had been traded off the board, leaving the two in another endgame involving only rooks and pawns, just like they had on Friday. This version, however, ended much more quickly than the marathon that came before.

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Carlsen and Caruana played around here for a bit, but shook hands and agreed to draw after 49 moves.

The match now sits at 1-1 in this race to 6.5, and will likely stretch to the end of the month. We’ll be keeping the chart below updated throughout.

Sunday is a rest day, and Game 3 begins Monday at 3 p.m. Greenwich Mean Time — that’s 10 a.m. Eastern. I’ll be covering it here and on Twitter.

The World Chess Championship Opened With A Wild Draw

It was 55 degrees and lightly raining in London, just as I imagine it always is, as the grandmasters Magnus Carlsen of Norway and Fabiano Caruana of the United States sat down to begin playing for the 2018 World Chess Championship. The gray meteorology belied the volcanic and lengthy chess on Day 1 of the World Chess Championship. Over seven hours and 115 moves, the players fought a fiery and oscillating battle to open the match, which will likely stretch to the end of the month. The American was lucky to emerge, largely unscathed, with a draw.

The players’ venue is in central London, in a place called The College, about a 15-minute walk north of the Thames. Carlsen, 27, and Caruana, 26, are ranked No. 1 and No. 2 in the world, respectively. Carlsen is the three-time defending world champion. Caruana is vying for the first American title since Bobby Fischer beat Boris Spassky in 1972.

Following a drawing of lots, Carlsen chose to begin the best-of-12-game match on defense with the black pieces. Wins here are worth 1 point, draws a half-point for each, and losses zero points. If the 12th game ends with each player having 6 points, a series of tiebreaker games will ensue. And that’s exactly what happened at the last world championship, in 2016, when Carlsen edged out Sergey Karjakin of Russia. There may still be a lot of chess left.

But there was a lot of chess Friday, too. Carlsen began with the combative Sicilian Defence, and the two entered into something called its Rossolimo Variation. Carlsen was likely pleased — he had beaten Caruana in this very variation in an attacking game in 2015.

After move 9, Caruana went into what the official match broadcast called the first “deep think” of the match, and the game ground to a near-halt as the position ventured into uncharted territory. By Caruana’s 11th move, a board like this had never been seen before at the game’s high levels, according to ChessBase’s database.

Carlsen donned a puffy jacket. Caruana removed his blazer.

Time soon became an issue. The players each get 100 minutes for their first 40 moves, with 30 bonus seconds after each move. After that, they get 50 minutes for the next 20 moves and 15 minutes for any moves after that. Nevertheless, while contemplating his 22nd move, Caruana’s clock ticked down to less than 10 minutes. Carlsen, across the board, moved quickly. With his 25th move, Caruana’s clock had dropped to six minutes. On his 32nd move, less than two minutes. On his 34th move, 6 seconds. If his clock had hit zero, he would have forfeited the game instantly.

Forestalling the end, Caruana fought for his life in the lower right corner of the board. The time pressure — and the pressure from the powerful pieces controlled by the best player in the world — were obviously too much to handle. It was over, and the players would surely head to an early dinner. The computer engines and the chess cognoscenti assessed Carlsen’s position as “surely winning” and Caruana’s as “sad.”

Carlsen’s troops were standing over Caruana’s king, ready to kill and take a devastating early lead in the championship.

But black slipped. The Norwegian champ captured a juicy-looking pawn he oughtn’t have, giving the American a chance to scurry to safety with seconds to spare. You can see what happened below. Carlsen, playing black, captured the pawn on c3 with his bishop. The better move, according to a chess engine whirring on my laptop, would have been to venture deep into the American’s territory, moving the black queen to g1. Carlsen’s advantage, according to the computer, dropped from roughly three pawns to roughly nothing.

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By the 43rd move, the two grandmasters had traded queens, clearing the board significantly, and they entered an intricate endgame that stretched on — for hours.

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The rooks and kings traveled around for what seemed like an eternity. But the lava that had flowed over the game earlier in the day had finally cooled — and finally (finally!) hardened into a draw. Here’s the whole dang thing, compressed into less than a minute:

Despite the draw, Robert Hess, an American grandmaster, called it a “dominant start for Magnus.”

The championship is level at a half-point each in this race to 6.5. We’ll keep the chart below updated throughout the match.

Game 2 begins tomorrow at 3 p.m. Greenwich Mean Time — that’s 10 a.m. Eastern. I’ll be covering it here and on Twitter.

Read more: The American Grandmaster Who Could Become World Champion

Something Looks Weird In Broward County. Here’s What We Know About A Possible Florida Recount.

The Florida U.S. Senate race is still too close to call. According to unofficial results on the Florida Department of State website at 11:45 a.m. Eastern on Friday, Nov. 9, Republican Gov. Rick Scott led Democratic Sen. Bill Nelson by 15,046 votes — or 0.18 percentage points. We’re watching that margin closely because if it stays about that small, it will trigger a recount. It’s already narrowed since election night, when Scott initially declared victory with a 56,000-vote lead.

The changing margin is due to continued vote-counting in Broward and Palm Beach counties, two of Florida’s largest and more Democratic-leaning counties. On Thursday evening, the supervisors of elections in the two counties told the South Florida Sun Sentinel that vote counting there was mostly complete. Under Florida law, counties have to report unofficial election results to the secretary of state by Saturday at noon, but Nelson’s campaign is suing to extend that deadline. Scott’s campaign and the National Republican Senatorial Committee are also suing both counties for not disclosing more information about the ongoing count, and Scott called on the Florida Department of Law Enforcement to investigate Broward’s handling of ballots.

Unusually, the votes tabulated in Broward County so far exhibit a high rate of something called “undervoting,” or not voting in all the races on the ballot. Countywide, 26,060 fewer votes were cast in the U.S. Senate race than in the governor race.1 Put another way, turnout in the Senate race was 3.7 percent lower than in the gubernatorial race.

Broward County’s undervote rate is way out of line with every other county in Florida, which exhibited, at most, a 0.8-percent difference. (There is one outlier — the sparsely populated Liberty County — where votes cast in the Senate race were 1 percent higher than in the governor race, but there we’re talking about a difference of 26 votes, not more than 26,000, as is the case in Broward.)

To put in perspective what an eye-popping number of undervotes that is, more Broward County residents voted for the down-ballot constitutional offices of chief financial officer and state agriculture commissioner than U.S. Senate — an extremely high-profile election in which $181 million was spent. Generally, the higher the elected office, the less likely voters are to skip it on their ballots. Something sure does seem off in Broward County; we just don’t know what yet.

One possible reason for the discrepancy is poor ballot design. Broward County ballots listed the U.S. Senate race first, right after the ballot instructions. But that pushed the U.S. Senate race to the far bottom left of the ballot, where voters may have skimmed over it, while the governor’s race appears at the top of the ballot’s center column, immediately to the right of the instructions.

Sun Sentinel reporters talked with a ballot expert, who said that some voters may not have noticed the Senate race (perhaps thinking it was just part of the ballot instructions) and started filling out their ballot with the governor race instead. That theory is supported by a data consultant who’s worked for several political campaigns in Florida, who found that the parts of Broward County that fall in the 24th Congressional District did see higher levels of undervoting than other parts of the county. That might be because the 24th District was uncontested, which according to Florida law means that the congressional race did not appear on the ballot at all. As you can see in the sample ballot above, the congressional race would also appear in the lower-left corner on many ballots, along with the Senate race. In districts where there was no congressional race on the ballot, however, that corner would have looked even emptier, perhaps making it easier for voters to inadvertently skip over the Senate race.

An alternative explanation is that an error with the vote-tabulating machines in Broward County caused them to sometimes not read people’s votes for U.S. Senate. If that’s true, we would probably only find out if there is a manual recount. According to Florida law, any election that’s within half a percentage point (as this one currently is) triggers a machine recount; then, after the machine recount, if the race is within a quarter of a percentage point, it goes to a much more complex manual recount — a.k.a. each ballot is recounted by hand. As long as the machine recount doesn’t change the Senate results too much (barring a surprise in the remaining ballots in Broward and Palm Beach), it looks like that’s where we’re headed. In addition, Republican former Rep. Ron DeSantis and Democratic Tallahassee Mayor Andrew Gillum are separated by just 0.44 points in the governor’s race, so that could go to a machine recount, too.

But recounts rarely change the outcomes of elections. A FairVote analysis found that the average recount from 2000 to 2015 shifted the election margin by an average of just 0.02 percentage points. The largest margin swing was 1,247 votes — coincidentally also coming in Florida, in the 2000 presidential race. If Nelson is going to stage a comeback in the Sunshine State, he’ll almost certainly have to close the gap between him and Scott even more in the next couple of days.

Todd Gurley Is In The Right System At The Right Time

Todd Gurley is off to one of the hottest starts in NFL history. After rushing for a league-leading 623 yards and nine touchdowns — plus 247 receiving yards and two more TDs through the air — Gurley has accumulated the fifth-most adjusted yards14 from scrimmage through six games since the 1970 AFL-NFL merger, joining former Rams greats Marshall Faulk and Eric Dickerson near the top of the list. The Rams are 6-0 on the young season, and Gurley’s breakneck performance is often cited as a catalyst for the team’s success. He has even been in the early discussion for league MVP.

But is that really warranted? Does the Rams offense truly run through Gurley, or should we be giving head coach Sean McVay more of the credit?

One approach to answering that question is to look at how McVay’s scheme affects Gurley’s performance. So far this year, the Rams have run nearly every offensive play from what is called the “11” personnel: one running back, one tight end and three wide receivers. According to charting from Sports Info Solutions, the Rams have run 95 percent of their offensive plays from this package — 32 percentage points more than the league average of 63 percent. And while heavy utilization of three wide receiver looks isn’t new to McVay — the Rams ran 81 percent of their plays out of “11” in 2017 — 2018 is a massive outlier. McVay appears to have concluded that the deception afforded the offense by lining up with the same personnel package each play is greater than the constraints it places on his play calling.

The Rams rarely stray from their favorite look

NFL teams by the share of their plays run in each of the three most popular personnel packages, 2018

Personnel package
Team 11: ONE RB, ONE TE, three WRs 12: ONE RB, TWO TEs, TWO WRs 21: Two RBs, one TE, two WRs
L.A. Rams 95% 2% 0%
Green Bay 77 14 1
Miami 77 8 1
Seattle 77 9 5
Indianapolis 72 18 3
Cleveland 70 16 1
Jacksonville 70 10 6
Cincinnati 69 20 2
Washington 69 17 0
Oakland 68 13 7
N.Y. Giants 67 23 4
Tampa Bay 67 14 7
Arizona 66 19 4
Denver 66 13 11
Buffalo 64 20 10
Chicago 64 20 10
Houston 62 34 0
Minnesota 62 23 9
Detroit 61 10 5
Pittsburgh 61 15 7
New Orleans 60 13 12
Carolina 59 14 8
Kansas City 59 22 9
Dallas 57 18 6
Atlanta 56 14 13
L.A. Chargers 56 17 10
Philadelphia 54 36 0
Tennessee 53 35 2
N.Y. Jets 52 24 0
New England 49 9 28
Baltimore 48 26 1
San Francisco 40 8 41
Average 63 17 7

Source: Sports Info Solutions

There are other benefits from repeatedly giving the opponent the same look, however, and they affect Gurley’s performance in important ways. When a team can spread a defense out laterally across the field, it opens up the middle and makes running the ball easier. Running backs with at least 20 carries averaged 4.75 yards per carry against six men in the box from 2016 to 2018.15 That’s well over half a yard higher than the average of 4.09 yards per carry when that same group of runners faced seven defenders near the line of scrimmage. Against eight-man fronts, the average gain falls to 3.59. Facing a loaded box makes running much more difficult.

McVay is no rube. He likely realizes that if you are going to run in the NFL, you should do so against a light box. Even better, this is something he can control. An offense exerts quite a bit of influence over how many box defenders it faces by how many wide receivers it chooses to deploy. When offenses play three wideouts, NFL defensive coordinators will typically match body type with body type and send a nickel defensive back in to cover the third receiver, leaving six defenders in the box.

As a consequence, Gurley has faced more six-man fronts on his carries than any other running back in football since McVay took over as head coach of the Rams. It has paid serious dividends. So far this season, Gurley is crushing it against those fronts, averaging 5.5 yards per carry. But against a neutral seven-man front, he’s been below league average at just 3.7 yards per attempt.

Gurley thrives when there are fewer defenders

Number of carries and yards per carry against a standard defense of six men in the box, 2017-18

Player No. of carries yards per carry
Todd Gurley 202 5.12
Kareem Hunt 113 4.91
Lamar Miller 112 4.42
LeVeon Bell 103 4.45
Melvin Gordon 101 4.73

Source: Sports Info Solutions

Gurley is basically the same back he has always been since he came into the league. If you use broken and missed tackles as a proxy for talent,16 you can see that Gurley makes defenders miss when running against six-man fronts far less than expected. He thrives, like most running backs, when he’s allowed to hit open holes and get to the second level relatively unscathed.

So Gurley is the beneficiary, not the proximate cause, of the Rams’ offensive resurgence under McVay. Gurley has been put in a position to succeed and has taken full advantage. Crucially, while the Rams have benefited from being smart in their offensive schemes and decision-making, it’s likely that many teams could emulate them and achieve similar success on the ground. Spreading a defense out and running against a light front is not a particularly novel idea. The commitment shown by running 95 percent of your plays out of a formation that encourages that result, however, is quite innovative. McVay pushes winning edges better than any coach in the NFL — and he, not his running back, is the principal reason that the Rams are currently the toast of the league.

Check out our latest NFL predictions.

How Worried Should Real Madrid, Bayern Munich And Barcelona Be?

The big leagues in continental Europe have been dominated by their superpowers for years. Bayern Munich won the German Bundesliga title each of the past six seasons. In 13 of the past 14 seasons in Spain’s La Liga, either Barcelona or Real Madrid has taken the crown. The trio has also combined to win the past six Champions League titles. But right now, you can say something about these teams that’s been largely unthinkable for nearly a decade: They look vulnerable.

Bayern is sixth in the Bundesliga, 4 points behind leaders Borussia Dortmund and trailing smaller clubs like Werder Bremen and Hertha BSC as well. Sevilla currently tops La Liga, with Barcelona and Madrid trailing close behind, but the two Spanish giants have each won just four of eight matches this season. Over the past decade, both teams have typically won at least 28 of their 38 matches per season, and the lowest win total either has posted was 22. These numbers are well off their pace.

How worried should the superpowers of soccer be? The Soccer Power Index suggests reason for both confidence and concern. At the start of the season, Bayern was projected as 82 percent favorites to win the title. That has fallen, but only to 70 percent. Real Madrid has seen its La Liga title chances drop from 41 percent to 37 percent, but Barcelona’s have actually increased to 47 percent from opening at 43. For now, it seems likely that these teams have enough of a head start in talent that they can still win their domestic leagues.

The Champions League may be another story. At the beginning of the year, the continental big three plus Manchester City were dead even with one another at the top of the projections. Now City leads, Juventus has caught up, and the gap to Liverpool and Paris Saint-Germain is narrowing. And this is particularly striking because all three clubs are still massive favorites to progress out of their groups. What’s changed is that the Soccer Power Index is starting to downgrade its projections.

The early season struggles of Barcelona, Bayern Munich and Real Madrid are not merely a matter of a few bad bounces. Expected goals, a measure of the quality of scoring chances created and conceded, shows that this is no fluke of hot or cold shooting — these sides’ underlying production numbers are off, too. The following chart shows the goal difference and expected goal difference for Barcelona, Bayern and Real Madrid in their first 10 matches of the season between domestic and Champions League play since 2010-2011, according to data analytics firm Opta Sports. For all three clubs, these are among their slowest starts to the season ever.12

Among all these starts to the season, the only one that was significantly worse than the three this year was Bayern Munich’s in 2010-11, when the club ended up in third place in the Bundesliga on 65 points.

These numbers suggest three things. First, Bayern has been better than its table position suggests. Its goal difference is the second-worst of any of these clubs since 2010, but its expected goal differential is merely eighth-worst. Barcelona’s good goal difference, by contrast, is covering up problems in the underlying numbers. And Real Madrid is simply in trouble.


In its last three league matches, Bayern has taken only 1 point — a draw against Augsburg — and scored just one goal while conceding six. However, its expected goals difference for those matches is roughly 4.8 to 2.5. These are performances typically good enough to win in the Bundesliga, and the points should come.

But even the expected goals numbers do not reflect outright dominance. Bayern has struggled to produce spectacular attacking numbers. In particular, 30-year-old striker Robert Lewandowski is having a surprisingly down season, which comes on the heels of a surprisingly down World Cup. After scoring 27, 29 and 37 nonpenalty goals in the past three seasons between domestic and Champions League competition, with underlying numbers to match, the Polish forward has scored just two nonpenalty goals this season. His expected goals per 90 minutes has been more than 0.8 each of the last three seasons, and it’s down to 0.28 now. Arjen Robben and James Rodriguez have carried the shooting load for Lewandowski so far, but that has meant a decline in their creative passing numbers, which has weakened the whole team. It is possible that this is just an early season slump or World Cup-related fatigue, and Lewandowski will snap out of it. If he doesn’t, Bayern could be in for a disappointing year.

Striker problems also have beset Real Madrid, but for them it’s even worse. Real sold Cristiano Ronaldo over the summer and shocked observers by simply not replacing him. The club eventually purchased Mariano from Lyon, but no one expected that to be a like-for-like replacement. In nearly the same number of minutes last season, against weaker competition, Mariano attempted 130 shots, exactly half of Ronaldo’s 260. Mariano has yet to start a match this season for Real; Gareth Bale and Karim Benzema have now been promoted to the point men in the attack after serving as Ronaldo’s support crew for years. The results have been as expected.

Losing Cristiano Ronaldo has zapped Real’s offense

Real Madrid’s expected goals through the first 10 matches of its season, 2010-19

Season Expected goals through first 10 matches
2018-19 19.4
2017-18 28.6
2016-17 24.9
2015-16 27.7
2014-15 26.5
2013-14 25.3
2012-13 30.5
2011-12 32.4
2010-11 25.0

Bars in orange indicate Real Madrid seasons with Ronaldo.

Source: Opta Sports

Real has consistently produced about 2.5 or more expected goals per match over its first 10 games of the season, and that number is under 2.0 per match this year. The attack is no longer elite, and it’s hard to see how Real can improve without an injection of talent. Real Madrid looks headed for year in the wilderness as merely one of Europe’s 10 to 15 best teams rather than a top Champions League contender.

For Barcelona, the problems are more complicated but perhaps no less severe. And unlike with Bayern and Real, they do not start at the top. Lionel Messi is still Lionel Messi, with 11 goals and four assists. Rather, Barcelona is struggling in the midfield, and that’s leading to defensive problems. Last season, Barcelona conceded just 29 goals, second-fewest in La Liga. It is hardly unusual for the Catalan side to put up dominant defensive numbers, but last year’s effort involved a change in tactics from manager Ernesto Valverde.

In 2017-18, Barcelona relaxed the high press that had been a feature of its play at least since Pep Guardiola’s reign ended in 2012. With midfielders content to allow opposition teams to hold possession in less dangerous areas, Barcelona broke up only about 48 percent of new open-play possessions for the other team before they completed three passes. This year, Valverde has brought the old press back, and Barca is breaking up 55 percent of new opposition possessions.

This has not worked to their advantage. The 2017-18 team conceded shots at a reasonably high rate — 444 shots, seventh fewest in La Liga. But it prevented quality chances by keeping numbers back and not allowing passes in behind the defense. Barcelona’s 0.087 expected goals per shot was second-best in La Liga after only Atletico Madrid. Valverde drilled his team to defend deeper rather than dominate midfield, and it worked. This year, the new style is having the opposite effect. Barca’s expected goals per shot conceded has exploded to 0.148, the worst in La Liga.

Barcelona’s midfield depends on two 30-year-olds, Sergio Busquets and Ivan Rakitic, and last year Valverde’s tactics already suggested he knew he needed to cover for their deficiencies in the press. The results of the new, more aggressive midfield tactics confirm he was right to pull back.

So Barcelona’s problems seem fixable, at least compared to those of Real Madrid and Bayern. If Valverde can accept once more the limitations of his midfield and play the more basic, defensive style he rolled out last season, there should be more than enough talent in the forward line to carry Barcelona deep in the Champions League. But if the team persists with this press, the Catalan side may end up in just as much trouble as Bayern and Real.

Check out our latest soccer predictions.

So Your Archipelago Is Exploding. How Doomed Is Your Island?

🚨🚨🚨 “The Riddler” book is out now! It’s chock-full of the best puzzles from this column (and, fret not, their answers) and some riddles that have never been seen before. I hope you enjoy it, and thank you for riddling with us these past three years. 🚨🚨🚨

Welcome to The Riddler. Every week, I offer up problems related to the things we hold dear around here: math, logic and probability. There are two types: Riddler Express for those of you who want something bite-size and Riddler Classic for those of you in the slow-puzzle movement. Submit a correct answer for either,30 and you may get a shoutout in next week’s column. If you need a hint or have a favorite puzzle collecting dust in your attic, find me on Twitter.

Riddler Express

From Philip Ruo, a puzzle of pondering playing perfection:

The NFL season is in full swing, and only one undefeated team remains — the 6-0 Los Angeles Rams. In theory, though, given the current NFL scheduling scheme — or at least what Wikipedia says it is — what is the largest number of teams that could finish a regular season 16-0?

Submit your answer

Riddler Classic

From Ricky Jacobson and Ben Holtz, geological disaster looms beneath:

You live on the volcanic archipelago of Riddleria. The Riddlerian Islands are a 30-minute boat ride off the shores of the nearby mainland. Your archipelago is connected via a network of bridges, forming one unified community. In an effort to conserve resources, the ancient Riddlerians who built this network opted not to build bridges between any two islands that were already connected to the community otherwise. Hence, there is exactly one path from any one island to any other island.

One day, you feel the ground start to rumble — the islands’ volcanoes are stirring. You’re not sure whether any volcano is going to blow, but you and the rest of the Riddlerians flee the archipelago in rowboats bound for the mainland just to be safe. But as you leave, you look back and wonder what will become of your home.

Each island contains exactly one volcano. You know that if a volcano erupts, the subterranean pressure change will be so great that the volcano will collapse in on itself, causing its island — and any connected bridges — to crumble into the ocean. Remarkably, other islands will be spared unless their own volcanoes erupt. But if enough bridges go down, your once-unified archipelagic community could split into several smaller, disjointed communities.

If there were N islands in the archipelago originally and each volcano erupts independently with probability p, how many disjointed communities can you expect to find when you return? What value of p maximizes this number?

Submit your answer

Solution to last week’s Riddler Express

Congratulations to 👏 Sarry Al-Turk 👏 of Toronto, winner of last week’s Riddler Express!

Last week, you learned about a girl who loves to sing “The Unbirthday Song” to people — as one does. But she can only do that, of course, if it’s not the person’s actual birthday. If she kept singing it to random people until it happened to be someone’s birthday, how long would her singing streak go before it became more likely than not that she would encounter someone whose birthday it is?

Care for those vocal cords, child: It’s 252 people.

The probability that it’s not an individual person’s birthday is 364/365. The probability that you sing to N people in a row without it having been anyone’s birthday is \((364/365)^N\) — since the birthdays are independent events, we can multiply that fraction over and over to match the number of people. We want to find the number N such that that probability is larger than 0.5.

We could just stick that straight into a computer solver, but it’s Friday, so let’s have some fun and do a little algebra.31 First, we can take the logarithm of both sides, to get N out of the exponent, and then we can rearrange things a little bit:

\begin{equation*}(364/365)^N > 0.5\end{equation*}

\begin{equation*}N\log(364/365) > \log(0.5)\end{equation*}

\begin{equation*}N > \log(0.5)/\log(364/365)\end{equation*}

That numerator, log(0.5), equals about -0.3, and that denominator, log(364/365), equals about -0.001. That fraction equals about 253. So the singing streak is expected to go 252 people before it became more likely than not that a birthday boy or girl would be encountered.

And a very merry unbirthday to you, dear reader — unless, of course, it is the big day.

Solution to last week’s Riddler Classic

Congratulations to 👏 Clemens Fiedler 👏 of Krems, Austria, winner of last week’s Riddler Classic!

Last week, a farmer wanted to tether a goat — as one does. Specifically, the farmer wanted to tether the goat to the fence that surrounded his circular field such that the goat could graze on exactly half the field, by area. The field had a radius R. How long should the goat’s tether be?

The tether should be a bit longer than the radius — specifically, it should have a length of about 1.159R.

Solver Russell No-last-name-given showed us what this looks like, pictured below. The field is green with radius R; the goat’s would-be grazing area is gray, with radius r.

The picture is all fine and good, but what about that math? It looks a little nasty. And it is a little nasty, I’m afraid. But, hey, it’s nearly Halloween, so let’s surrender to the nastiness and fear.

The area on the field available to the goat is the intersection between two circles. And there just happens to be a whole body of knowledge about such circle-circle intersections. One way to think about it is to consider the two different shapes that the goat’s tether allows it to graze in. The first is the big “pizza slice”, defined in the image above by the two diagonal green radii and the bottom arc of the gray circle, and the second are the narrower areas between the pizza slice and the fence.

Hector Pefo broke this math down a little bit more for us in his diagram (this time, the goat is on the southern end of the field):

And finally, Laurent Lessard showed us a way to get to this same solution with calculus.

Either way, I hope you enjoy it, goat. Nom nom nom.

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