The Bears Are Stumbling After a Playoff Run. Yes, Again.

Last season, the Chicago Bears were one of the NFL’s great success stories. Picked by our model to win just 6.6 games,1 with only a 19 percent chance of snapping the team’s seven-year playoff drought, Chicago instead flourished under second-year quarterback Mitchell Trubisky and first-year Head Coach Matt Nagy. The Bears went 12-4, won the NFC North and were mere inches away from beating the defending champions in the playoffs. Nagy won Coach of the Year honors, and going into 2019, Chicago seemed primed to build on its breakout — thanks to a staunch defense and further potential development from Trubisky.

But things haven’t really gone according to that plan … because, let’s face it, they almost never do in the Windy City. The Bears’ 2019 campaign has been beset by the usual maladies: kicking disasters, coaching miscues, poor quarterbacking, defensive regression, etc. After losing a potentially winnable slog against the Los Angeles Rams on Sunday night, Chicago’s playoff odds are down to 1-in-100.

We would ask how things got to this point, except that things always seem to get to this point for the Bears when seeking a return trip to the playoffs. Going back to the 1990s, Chicago has repeatedly tried — and failed — to build on its breakout seasons, most of the time leaving fans bitterly disappointed instead. And, barring a miracle, this season looks like it will be no different.

Let’s rewind a bit. The year was 1994, and the Bears were two seasons removed from a run of seven playoff appearances in eight years, including a legendary championship in 1985. But they had gone 12-20 in 1992 and ’93, during which time the team officially moved on from both its head coach — longtime boss Mike Ditka — and quarterback — Jim Harbaugh, who’d shown flashes of talent but played miserably in 1993. Led by second-year Head Coach Dave Wannstedt and a combination of Steve Walsh and Erik Kramer under center, Chicago turned things around with a 9-7 regular season in ’94 and added a road playoff win over the Minnesota Vikings before being dispatched by the eventual champion San Francisco 49ers.

Poised to make a postseason return the following year with Kramer — who actually had a tremendous 1995 season — starting full time, Chicago started 6-2. But the Bears ultimately missed the playoffs on a tiebreaker after going 3-5 in the second half of the season, undone by special teams and (of all things) defense. The franchise wouldn’t make it back for the rest of the decade.

That playoff drought was snapped in 2001, when a QB nobody thought much of — Jim Miller, who had thrown fewer passes for the 2000 Bears than the immortal Cade McNown or Shane Matthews — led a team nobody thought much of to a season nobody saw coming. That year’s Bears defense went from 20th in points allowed to first, a feat made even more incredible by the team’s middling rankings in more yardage-based metrics. The Cinderella story ended at home against the Eagles in the playoffs, but at least some observers saw another playoff trip in the cards for 2002.

Instead, the defense regressed back to 25th in points allowed, Miller didn’t last the season as starting QB, and Chicago crashed back to earth with a 4-12 record. The Bears would languish out of the playoffs for a couple more seasons after that, during which time the team burned through a succession of starting QBs that included Kordell Stewart, Craig Krenzel, Chris Chandler, Chad Hutchinson, Jonathan Quinn, Kyle Orton and Rex Grossman.

Those last two names are where the story takes a turn, though, as part of the exception to the overall rule.

The 2005 Bears were another one of those underdog Chicago teams with an untested QB (Orton, standing in for the injured Grossman) and a second-year head coach (Lovie Smith), about which little was expected aside from some hints of defensive promise. They started the year 1-3, as Orton played poorly — but then, out of nowhere, they rattled off an eight-game winning streak as part of a stretch with 10 wins in the season’s final 12 games. The team once again led the league in scoring defense — even coming by it more honestly than in 2001 — and it kept winning, even though Orton continued to struggle until Grossman’s return in Week 16. Despite going 10-5 as Chicago’s starter during the regular season, Orton was benched for the playoffs in favor of Grossman. But at home in the playoffs against the Carolina Panthers, Grossman didn’t fare much better than Orton had, and the Bears’ season ended in a 29-21 loss.

Looking ahead to 2006, it sure seemed like history repeating itself from a few years earlier.2 But the ’06 Bears actually did make it back to the playoffs — and even went as far as the Super Bowl before running into Peyton Manning and the Indianapolis Colts on a rainy Super Sunday in Miami. Though Grossman was a mediocre passer at best, the Brian Urlacher-led defense repeated its top-five performance in points allowed and helped the Bears make a rare postseason return before the wheels began to fall off in 2007.

That was the start of a new three-year playoff drought for the club, but in 2010 the Bears still had Smith as an experienced coach to go with QB Jay Cutler — who was acquired from Denver for Orton and picks, and who had something to prove after a terrible debut season in Chicago. Cutler’s own 2010 was up and down, but the Bears’ defense once again was terrific, allowing the fourth-fewest points in the NFL, and the team finished 11-5. In the playoffs, Chicago beat the Seattle Seahawks before losing at home to the Green Bay Packers in an NFC championship game that Cutler left early with a knee injury.

Would the Bears return to the playoffs in 2011? Of course they would not. Chicago started 7-3 before Cutler was injured again, and the team went 1-5 down the stretch with a combination of Caleb Hanie and Josh McCown replacing Cutler under center. The Bears rebounded to go 10-6 in 2012 but also became one of the rare NFL teams in history to win double-digit games and miss the postseason. After that, they’d never muster anything better than an 8-8 record for the rest of the Cutler era, eventually missing the playoffs in seven straight seasons from 2011 to 2017.

All of which brings us to 2019, when Chicago had its fifth chance since the mid-1990s to build on an unexpected playoff run with another postseason bid. And likely for the fourth time — 2006 being the sole exception — the team will fail to follow up on the breakthrough. The return rate for playoff teams since the NFL expanded to its current format in 1990 is 52 percent, and only 42 percent for teams coming from outside the postseason the year before. So naturally you would expect this phenomenon to happen multiple times to some team. But the Bears have done it so often, they’ve practically become the poster franchise for playoff teams dropping out of the field the next season.

The Bears can’t seem to shake playoff hangovers

In seasons when the Bears went from out of the playoffs to in, primary starting QB (with QB Elo rating) and scoring defense rank for that season and the following season, 1990-2019

Playoff Season Follow-Up Season
Years Starting QB ELO* Def. Rk Starting QB ELO* Def. Rk Playoffs?
’94-95 Steve Walsh -16 10th Erik Kramer +47 22nd
’01-02 Jim Miller -33 1st Jim Miller -28 25th
’05-06 Kyle Orton -99 1st Rex Grossman -46 3rd
’10-11 Jay Cutler -31 4th Jay Cutler -8 14th
’18-19 Mitch Trubisky +22 1st Mitch Trubisky -53 4th

*FiveThirtyEight rolling QB Elo rating, relative to an average starter, by season’s end. (Through Week 11 in 2019.)

Source: Pro-Football-Reference.com

The 2019 team happens to share some commonalities with Chicago’s previous stumbles. The defense, for instance, isn’t quite as dominant as it was last season — which seems to be a common flaw for these one-off Bears playoff bids. (Does the legacy of the ’85 Bears defense loom so large over the franchise that it must constantly try to rebuild in that team’s image?)3 There’s a lot of research showing that defensive quality, while extremely important for team success, is also extremely variable from season to season, even with talent like Khalil Mack on your side. So when your quantum leaps from obscurity into the playoffs are always reliant on dominating D, it’s no surprise to see a penchant for backsliding in subsequent seasons.

But an interesting difference this year, as compared with earlier failed playoff returns, is at quarterback. Most of the other seasons involved Chicago starting with a low-rated QB who won despite poor passing. Often, these QBs would improve (or be replaced by a better option) in the follow-up bid, but it wouldn’t be enough to overcome the defensive regression. Trubisky, however, entered the 2019 season with an above-average QB Elo rating, which could only be said about quarterbacks in 19 Bears seasons since 1950 — and only three QBs since 1998 (including Trubisky).

This appeared to be an argument in Chicago’s favor for returning to the postseason: that the 25-year-old’s ongoing development would provide insurance in case of a defensive slide. But instead, Trubisky’s underwhelming play (he ranks 27th in QB Elo out of the 34 current and/or primary starters in the league) has stalled the Bears’ rebuilding process and called into question his long-term future with the team.

We don’t know where the Bears go from here. But even if they make it back to the postseason, say, next season, they’ll need to prove it’s not a one-year wonder — unlike basically every other Bears playoff run in the past three decades.

Check out our latest NFL predictions.

The NCAA’s Graduate Transfer Trend Isn’t Slowing Down

When Cameron Johnson left Pitt after three years for North Carolina, he set off a minor firestorm of controversy. But Johnson parlayed his two years as a graduate transfer for the Tar Heels into a lottery pick, and the 11th overall selection in this year’s NBA draft is now averaging 8.2 points in nearly 17 minutes a game for the Phoenix Suns.

Kiara Leslie left Maryland for North Carolina State after graduating in three years. She had missed a season for the Terrapins with an injury and had two seasons of eligibility remaining. Leslie was drafted 10th overall by the Washington Mystics this summer, and though she missed the Mystics’ season with a knee injury, she’s still a WNBA champion.

Johnson and Leslie are paradigms of the growing trend of athletes transferring as graduate students to keep their pro dreams alive. In 2013, 38 men and 13 women transferred as graduates. In 2018, those transfer numbers had skyrocketed to 121 men and 48 women — increases of 218 percent and 269 percent, respectively.

When athletes switch schools, they typically have to sit out a year before playing for their new team. But earlier this decade, following quarterback Russell Wilson’s high-profile transfer from North Carolina State to Wisconsin, the NCAA appeared to relax its rules for graduate transfers — those going straight from undergraduate studies into a master’s program — allowing athletes to play immediately, so long as they had completed their undergraduate degrees and had at least one year of NCAA eligibility remaining.

The rapid increase in the number of graduate transfers has led to intense scrutiny from schools whose programs have not benefited from the trend and from coaches who see it, in the words of Purdue men’s basketball coach Matt Painter, as “free agency.” The NCAA looked into curtailing the movement by proposing a rule change1 this past offseason that would have taken away a scholarship from the school gaining the graduate transfer if the student didn’t obtain his or her secondary degree within a year. This would have been a clear disincentive for schools to accept graduate transfers, since many graduate degrees take two years to complete. But the proposal was struck down earlier this year by the NCAA Division I Council.

For athletes, the transfer rule provides the flexibility of another year of eligibility in deciding when to go pro, as well as another year to sharpen their skills in front of pro scouts. Leslie recorded career highs in points, assists, rebounds and steals in her final year with the Wolfpack, while Johnson finished tied for ninth in box plus-minus last season in Division I.2

But coaches are decidedly split. Baylor women’s head coach Kim Mulkey is one coach who seems to have been won over. The rule “can devastate a program when you lose a graduate transfer that you spend all this time with,” she said during the NCAA Tournament last season. “But yet, there’s also those programs that reap those rewards on the other end. And we understand it could happen to us.” Led by Chloe Jackson, a fifth-year graduate transfer from LSU, the Lady Bears became NCAA champions two weeks later.

Of the teams in the men’s preseason AP Top 25, 11 schools added at least one graduate transfer to their roster for the upcoming season, while seven of the women’s preseason AP Top 25 added at least one graduate transfer.

Top teams bring in more grad transfers than they lose

Number of graduate transfers who in the 2019 offseason joined or left programs ranked in the preseason Associated Press Top 25

Men’s Grad Transfers Women’s Grad Transfers
Rk School In Out School In Out
1 Michigan State 0 0 Oregon 1 0
2 Kentucky 1 1 Baylor 2 0
3 Kansas 1 0 Stanford 0 1
4 Duke 0 0 Maryland 0 1
5 Louisville 1 0 Connecticut 1 0
6 Florida 1 1 Texas A&M 0 0
7 Maryland 0 0 Oregon State 0 0
8 Gonzaga 1 0 South Carolina 0 1
9 UNC 2 0 Louisville 0 0
10 Villanova 0 1 Mississppi State 0 0
11 Virginia 0 0 UCLA 0 0
12 Seton Hall 0 0 Florida State 0 0
13 Texas Tech 2 0 Kentucky 0 1
14 Memphis 0 0 NC State 0 0
15 Oregon 2 0 Texas 1 0
16 Baylor 0 0 Notre Dame 2 0
17 Utah State 0 0 Michigan State 0 0
18 Ohio State 0 0 DePaul 0 0
19 Xavier 2 0 Miami 0 0
20 St. Mary’s 0 0 Arizona State 1 0
21 Arizona 1 0 Syracuse 3 1
22 LSU 0 1 Arkansas 0 0
23 Purdue 1 0 Minnesota 0 0
24 Auburn 0 0 Indiana 0 0
25 VCU 0 0 Michigan 0 0
Total 15 4 Total 11 5

Sources: 247sports, ESPN

Despite the number of top programs adding graduate transfers to their rosters, these students still represented just 3 percent of players in the men’s game and 1 percent in the women’s game in 2018. And though Johnson and Leslie heard their names called on draft day, they are relatively alone: Johnson was the only grad transfer drafted this year in the NBA, and Leslie was one of three taken in the WNBA. The trend may be for more students to take advantage of the grad transfer loophole, but if coaches are worried that students are just using it to jump to the pros, they likely shouldn’t be.

We’ve Been Waiting For This Andrew Wiggins

Minnesota Timberwolves swingman Andrew Wiggins has been one of the most confounding NBA players of the late 2010s. Despite being pegged as a superstar out of high school, going first in the 2014 draft and winning Rookie of the Year honors in 2015, Wiggins was largely seen as a disappointment over his first five pro seasons — particularly given the max contract extension he signed in 2017. After arguably the worst season of his career in 2018-19, it appeared that Wiggins’s chance at superstardom — which once seemed guaranteed — had basically passed him by.

But something has clicked with Wiggins in this, his sixth NBA season. Over the Wolves’ first 11 games, he has come out like a man on a mission to reclaim his former promise. He’s averaging nearly 26 points per game (including scoring 30 or more in four of his past five contests) to go with career-best numbers in assists per game, rebounds per game and true shooting percentage. Still just 24 years old, is Wiggins figuring things out? Can he finally deliver on his long-dormant potential?

Certainly, this is the best Wiggins has played in a very long time. To help ballpark a player’s performance in any given game, I created a Game Score-like box score estimate of points added above replacement via RAPTOR — our new player-value metric — which I’m calling the RAPTOR Game Score, or RAGS.2 Over the past five games, Wiggins has averaged a RAGS of 10.7 points above replacement, which is the most he’s had in any five-game stretch since February 2017, nearly three full seasons ago:

For a little perspective, at no point last year did Wiggins average a five-game RAGS any better than 5.0. So this has already been a remarkable start to the season for him.

The seasonlong numbers bear out Wiggins’s rapid improvement as well. Going into the year, our projection system thought Wiggins would be worth -0.3 points above average per 100 possessions. While that was better than how many other advanced stats regarded him — in Box Plus/Minus, for instance, Wiggins was worth -2.9 points per 100 last season relative to average — it was still quite a bit lower than the career-best +1.9 RAPTOR level (including +5.0 on offense) Wiggins has actually played at to start the season.

One of the key ingredients in Wiggins’s resurgence has been a dramatic uptick in scoring efficiency despite a concurrent increase in usage rate, which doesn’t usually happen. A year after letting his true shooting percentage dip below 50 percent for the first time in his career, Wiggins is now up to a career-high 56.3 mark, to go with a usage of 28.3 percent — 3.9 points higher than last year — as well. He has connected for career-best shooting percentages on both 2-pointers (53.2) and 3-pointers (36.1), despite being assisted on the lowest share of made baskets in his career to date, per data from Cleaning the Glass.

This mix of volume and efficiency is rare. Wiggins is one of only 17 players this season with a true shooting percentage of at least 56 percent and a usage rate of at least 28 percent — joining an elite offensive list with names like James Harden, Giannis Antetokounmpo and Damian Lillard on it. And it bears mentioning that, at 6.3 percent, Wiggins also has the lowest turnover rate of anybody on that list.

Some of Wiggins’s improved performance owes to better shot selection — a longtime issue for a player whose repertoire always more resembled midrange gunslingers of yesteryear, such as Kobe Bryant, than modern Moreyball purveyors like Harden. As recently as 2017-18, Wiggins took 47 percent of his shots from midrange and only 24 percent from 3-point range. (Harden, by comparison, takes about 20 percent of his shots from midrange and half of them from three.) Over the past two seasons, Wiggins has steadily improved those ratios; he’s now taking only 35 percent of shots from midrange and 29 percent from three, all while continuing to get to the rim at a solid clip.

Since he’s starting to shoot more from higher-value areas of the court, it makes sense that Wiggins would be a more efficient offensive player this season. But he has also benefited from some trends that may prove unsustainable over the course of the season. According to tracking data from Second Spectrum, Wiggins has taken shots with an expected effective field-goal percentage of 47.7 percent so far this season. That’s still well below the league median (among players with at least 50 shot attempts) of 51.7 percent, but Wiggins has made up for it by shooting an effective field-goal percentage 5.8 points better than we’d expect from shot quality alone. Plenty of star players thrive by making tough shots — Kevin Durant comes to mind — so maybe Wiggins can keep that up … but we’ve also found that shot-making is a highly variable stat from game to game and even season to season. For instance, last year Wiggins shot an effective field-goal percentage 1.4 points below expected after accounting for shot difficulty, according to Second Spectrum.

In truth, Wiggins probably won’t be this good going forward, though he was also never as bad as he appeared to be last season. His Predictive RAPTOR (what we call “PREDATOR”), which is a version that regresses down statistical components known to be noisy in a small sample, has been +0.4 to start the 2019-20 season, somewhat lower than the +1.9 standard RAPTOR that gives players credit for performances even if they might be driven more by luck than skill. Last season, by contrast, Wiggins’s PREDATOR (-1.1) was higher than his regular RAPTOR (-2.0), indicating that luck was working against him.

Wiggins is playing better — and luckier — this season

Andrew Wiggins’s performance by season according to two versions of FiveThirtyEight’s RAPTOR

Standard RAPTOR Predictive RAPTOR
Season Offense Defense Total Offense Defense Total Luck/100
2014-15 -0.4 -2.1 -2.4 -0.7 -1.9 -2.6 +0.18
2015-16 +0.5 -0.8 -0.2 +0.0 -0.7 -0.7 +0.45
2016-17 +1.4 -0.8 +0.7 +1.5 -0.4 +1.1 -0.37
2017-18 -0.8 +0.2 -0.6 -0.4 +0.4 +0.0 -0.57
2018-19 -1.1 -0.9 -2.0 -0.7 -0.4 -1.1 -0.88
2019-20 +5.0 -3.1 +1.9 +2.5 -2.1 +0.4 +1.52

Standard RAPTOR attempts to credit players for every event that happens on the court (and their involvement).

Predictive RAPTOR is a version that regresses down statistics that are driven more by luck and gives more weight to those which contain less random variance.

Source: NBA Advanced Stats

After stripping away the luckier aspects of Wiggins’s season to date, he still looks like a markedly better offensive player so far than at any other point in his career. This also makes sense: Our projections have consistently compared Wiggins to current Spurs veterans DeMar DeRozan and Rudy Gay3 when they were the same age, and both of them easily set career-best RAPTOR marks (to that point in their careers) in their age-24 seasons — just like Wiggins is doing now.

But of course, as we’ve written about before, there’s always a catch with Wiggins. Even in a breakout season, his offensive PREDATOR ranks only 53rd among players with at least 150 minutes played, putting Wiggins at roughly the same level as Will Barton and Buddy Hield. Why? Despite the improved scoring, his shot selection inherently limits his ceiling, and even with an improved assist rate, he still doesn’t pass enough (or draw enough fouls) to enhance his offensive value much otherwise. And then there’s defense, where Wiggins still ranks as decidedly subpar. According to defensive PREDATOR, he sits 188th out of 252 players with at least 150 minutes this season — and again, that’s after filtering out the inherent bad luck in things like his -11.4 on-versus-off defensive plus/minus per 100 possessions.

At his best, Wiggins is showing that he can be a valuable scorer and an offensive complement to teammate Karl-Anthony Towns, who is also having an amazing season (+9.9 RAPTOR) so far. Wiggins will provide some really nice highlights and could even be part of a playoff run for Minnesota. (We give the Wolves a 79 percent chance of getting back to the postseason for the second time in three years — or the third time in 17 years, depending on how you look at it.) But Wiggins is also a very particular kind of polarizing player, about whom the eye test and many different kinds of numbers can never quite agree. That debate didn’t really matter when he was playing poorly by all accounts, but it’s probably going to resurface now that Wiggins is hot again and finally realizing some of his old potential.

Check out our latest NBA predictions.

How Low Can You Roll?

Welcome to The Riddler. Every week, I offer up problems related to the things we hold dear around here: math, logic and probability. Two puzzles are presented each week: the Riddler Express for those of you who want something bite-size and the Riddler Classic for those of you in the slow-puzzle movement. Submit a correct answer for either,4 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

Inspired by Catriona Shearer (if you don’t know who she is, seriously, check out her puzzles), this week’s Riddler Express is a geometric conundrum:

Upside-down semicircle inscribed inside a right side-up semicircle. The smaller semicircle is dark gray, while the larger one is light gray.

The picture above shows two semicircles. The lighter region (inside the larger semicircle but outside the smaller one) has an area of 7. What’s the area of the darker region?

Submit your answer

Riddler Classic

From Ricky Jacobson comes a puzzle of seeing how low you can roll:

You are given a fair, unweighted 10-sided die with sides labeled 0 to 9 and a sheet of paper to record your score. (If the very notion of a fair 10-sided die bothers you, and you need to know what sort of three-dimensional solid it is, then forget it — you have a random number generator that gives you an integer value from 0 to 9 with equal probability. Your loss — the die was a collector’s item.)

To start the game, you roll the die. Your current “score” is the number shown, divided by 10. For example, if you were to roll a 7, then your score would be 0.7. Then, you keep rolling the die over and over again. Each time you roll, if the digit shown by the die is less than or equal to the last digit of your score, then that roll becomes the new last digit of your score. Otherwise you just go ahead and roll again. The game ends when you roll a zero.

For example, suppose you roll the following: 6, 2, 5, 1, 8, 1, 0. After your first roll, your score would be 0.6, After the second, it’s 0.62. You ignore the third roll, since 5 is greater than the current last digit, 2. After the fourth roll, your score is 0.621. You ignore the fifth roll, since 8 is greater than the current last digit, 1. After the sixth roll, your score is 0.6211. And after the seventh roll, the game is over — 0.6211 is your final score.

What will be your average final score in this game?

Submit your answer

Solution to last week’s Riddler Express

Congratulations to 👏 Harry Elworthy 👏 of San Francisco, California, winner of last week’s Riddler Express.

Last week, you had to generate the largest number your could using exactly four threes. Specifically, you were allowed to add, subtract, multiply, divide, exponentiate or write them side-by-side. For example, if you had three nines instead of four threes, the biggest number you could make is 999, which equals 9387,420,489.

Many solvers submitted a similar answer — 3333. Remember, when you’re evaluating exponents, order of operations (PEMDAS, anyone?) dictates that you start from the topmost exponent and work your way back down. So 3333 equals 3327, or 37,625,597,484,987. That’s a big number!

But solver Erin Seligsohn looked closer at the expression 3327. Sure, that’s big, but it could be made even bigger if that topmost exponent, 27, were replaced by 33. Then you’d have 3333, which equals 35,559,060,566,555,523. Sure enough, that’s the largest number you can make with four threes and the operations you were allowed to use. As solver Kyle Pekosh points out, this number “…has 2,652,345,952,577,569 (about 2.65 quadrillion) digits. If you were to write one digit of that number every second, a decade wouldn’t quite cut it.”

I was delighted by how much of Riddler Nation went beyond the list of allowed operations stated in the problem. Many proposed solutions included factorials, while others divided by zero (e.g., 33/(3−3)), which … don’t get me started.

Solver Josiah Kollmeyer went so far as to invoke an operation known as tetration. What is tetration? I thought you’d never ask. Just as multiplication is repeated addition, and exponentiation is repeated multiplication, tetration is repeated exponentiation. For example, 2↑↑5 (where the ‘↑↑’ is the sign for tetration) is 22222, an exponential stack of five twos. And so 3↑↑3 equals 3↑↑3↑↑3, which is only about 8 trillion (something you can at least type into your calculator). But what about 3↑↑(3↑↑3)? That’s 33..., with a total of 33, or 27, threes in the stack. That’s an unconscionably large number — and that was just 3↑↑(3↑↑3). If you were to evaluate 3↑↑(3↑↑(3↑↑3)) … your head, and calculator, would just explode. Fortunately for your head, tetration was not in the approved list of operations for this puzzle.

Solution to last week’s Riddler Classic

Congratulations to 👏 Dave Paschal 👏 of Reno, Nevada, winner of last week’s Riddler Classic.

Last week, you were in a store with three kinds of candy were being sold: Almond Soys, Butterflingers and Candy Kernels. You wanted to buy at least one candy and at most 100, but you didn’t care precisely how many you got of each or how many you got overall. So you might have bought one of each, or you might have bought 30 Almond Soys, six Butterflingers and 64 Candy Kernels. As long as you had somewhere between one and 100 candies, you left the store happy. But as a member of Riddler Nation, you couldn’t help but wonder: How many distinct ways were there for you to make your candy purchase?

There are too many ways to write them all out. But there is an elegant approach that requires us to think beyond the candies. Suppose you want to buy exactly 100 candies, and instead of using a bag to hold them, you line them up on the counter one at a time. Oh, and suppose you also happen to have two dividers in your pocket.

You start by purchasing Almond Soys and lining them up, until you put your first divider down on the counter. Then you buy Butterflingers, one at a time, until you put the second divider down. Finally, you buy Candy Kernels until you have 100 candies in total.

By changing the positions of the two dividers relative to the rest of the candy, you can specify how many of each type you’re buying. So in order to compute how many ways we can buy 100 candies, we need to know how many ways we can position two dividers within a sequence of 102 objects (the 100 candies plus the two dividers). This can be done using combinatorics, a branch of discrete mathematics.

Now that’s just for the case of 100 candies, whereas the problem said you bought at most 100 candies — you could have purchased, 99, 98, 97, and so on, all the way down to one, for a grand total of 100 different cases. Indeed, many solvers broke the problem down into these 100 cases. They solved each case and and tallied up all the ways to buy candy with the aid of a computer, like the team of Ben Finkel, Cullen McAndrews, Zack Smith and Matt Thachet. Solver Aditya Radhakrishnan, meanwhile, impressively found a way to do this using one line of code in the programming language R: sum(unlist(lapply(seq(1, 100), function (x) { factorial(x + 2)/factorial(2)/factorial(x) }))).

Solver Gareth McCaughan took the idea of dividers one step further, imagining that there was a fourth candy that you could purchase, which, in an apparent reference to Douglas Adams, he named “Dingo’s Kidneys.” In Gareth’s approach, if you were to ever buy any fewer than 100 candies, then the difference would be made up of these Dingo’s Kidneys. For example, if you bought 30 Almond Soys, 30 Butterflingers and 30 Candy Kernels, then you’d imagine having bought 10 Dingo’s Kidneys, so that the total number of candies remains 100.

Then, imagine you have three dividers and exactly 100 candies (as opposed to anywhere from one to 100 candies). All we have to do is choose where the three dividers will be in a sequence of 103 objects (100 candy slots and three dividers). Again, we can use combinatorics to calculate the answer in a single shot — it’s 103 choose 3, or (103·102·101)/(3·2·1). That equals 176,851, which was not the answer.

Huh? Oh right, the original riddle said you had to buy at least one candy. Out of our 176,851 scenarios, one of them was the purchase of zero real candies and 100 Dingo’s Kidneys. Excluding that one case reveals the correct answer to be 176,850.

Dingo’s Kidneys do sound vile … but I’m sure they’re mostly harmless.

Want more riddles?

Well, aren’t you lucky? There’s a whole book full of the best puzzles from this column and some never-before-seen head-scratchers. It’s called “The Riddler,” and it’s in stores now!

Want to submit a riddle?

Email Zach Wissner-Gross at [email protected].

Significant Digits for Friday, Nov. 15, 2019

You’re reading Significant Digits, a daily digest of the numbers tucked inside the news. Today’s number is more than 233,000, for the number of students who have experienced gun violence since Columbine; two students were killed and three wounded in a shooting on Thursday at Saugus High School in Santa Clarita, California.


600 dentists

Los Algodones, in Mexico, isn’t very big. Just 14 miles from Yuma, Arizona, it has fewer than 5,000 permanent residents. But Los Algodones has approximately 600 dentists, and they do huge business each year serving thousands of Americans and Canadians looking to save 40 to 60 percent on dental services. American clients like 76-year-old Eugene Richardson and his wife, who drove 2,000 miles from Isle, Minnesota, to get extensive work done. The Richardsons, like 23 percent of Americans overall, don’t have dental insurance. [HuffPost]


5,000 kids charged as adults

A new report at Reveal News found that nearly 5,000 Mississippi children were charged as adults in the last 25 years, and that 75 percent of those kids were black. Among the girls who are also in the Mississippi court system, 60 percent are black. Analysis by reporters Ko Bragg and Melissa Lewis show that most of the charges involve drug use, burglary, larceny and armed robbery, but black children “serve significantly longer sentences than white children in the adult system.” And they trace this disparity to laws from the late 1800s [Reveal News]


50.2 percent

Through Week 10 of this season, NFL teams are going for it on 14.5 percent of all fourth downs, the highest rate in more than two decades. Which makes a ton of sense — the strategy worked last year. Running or passing on fourth down resulted in a first down (or touchdown) 59.4 percent of the time in games played during the 2018 season; as FiveThirtyEight’s Ty Schalter points out, this is the highest success rate since 1998. However, this year the success rate of fourth-down conversion attempts has fallen to 50.2 percent, which is below the 51.7 percent average of the last 10 seasons. [FiveThirtyEight]


800 million jobs

Automation, algorithms, and artificial intelligence have already reduced the amount of human labor in specialty manufacturing, warehouse parcel delivery and resume screening. But a new report from analysts at Bank of America Merrill Lynch estimates the rise of automation could make up to 800 million jobs — nearly half of all jobs worldwide — obsolete by 2035. [Yahoo Finance]


5,136 votes

On Thursday, Kentucky Gov. Matt Bevin conceded to Attorney General (now Gov.-elect) Andy Beshear after a re-canvass of votes confirmed the margin of victory was still 5,136 votes out of more than 1.4 million cast in last week’s election for governor. Bevin, who had received the support of President Trump, had raised unspecified allegations of voter fraud but finally acknowledged Beshear’s victory in a brief speech. “We’re going to have a change in the governorship based on the vote of the people,” he said from a podium in front of the governor’s office. [New York Times]


40,000 transcribers

Freelance workers for Rev, a popular on-demand transcription service, recently alerted users that the company was significantly slashing their pay. Other freelancers are warning users that all of the company’s 40,000 transcribers have access to a database of customer information, including full names, business titles and audio and video files for jobs that have not yet been “claimed.” Rev advertises a “strict customer confidentiality policy” and more than 100,000 customers on its website, including major companies such as Google, Amazon and NBC. [OneZero]


SigDigs: Nov. 15, 2019

Significant Digits for Thursday, Nov. 14, 2019

You’re reading Significant Digits, a daily digest of the numbers tucked inside the news. Today’s number is two, for the number of tails on Narwhal the puppy, including the very cute one on his head.


1,567 tear gas canisters

The Associated Press reports that Hong Kong police officers have escalated their response to protestors. Officials say that police fired 1,567 tear gas canisters, 1,312 rubber bullets and 380 beanbag rounds on Tuesday alone. The protests, which are now in their fifth month, have shut down major subway routes, blocked streets, and resulted in spontaneous rallies in the Hong Kong’s busy Central business district. [Associated Press]


6 feet of water

The mayor of Venice cited climate change as the reason behind severe flooding in the famed Italian city on Wednesday, with water levels peaking at six feet and causing “grave damage” to St. Mark’s Basilica. This is the second-highest water level since official records began in 1923, and Venice is flooding increasingly often. As of Wednesday, there were two deaths due to the floods, including a man who was electrocuted after trying to start a pump in his home. [BBC News]


$22,000 worth of hidden cocaine destroyed

Wild boars can cause a lot of damage, including to drug deals, apparently. The animals stumbled upon cocaine hidden in an Italian forest and worth approximately $22,000. They then dug the drugs up, destroying them in the process. Police learned of the incident after wiretapping members of the gang as part of a larger investigation and heard someone complaining about what the animals had done. Four suspects were arrested, but no one knows what happened to the boars after their discovery. [Newsweek]


80.5 percent field goal success rate

After decades of great conversion rates, NFL kickers are struggling this year. Michael Salfino notes this year’s field-goal success rate of 80.5 percent “is at its lowest level since 2003.” There are 12 teams that currently have field-goal success rates at or below 75 percent, which hasn’t happened since 2003. [FiveThirtyEight]


4,000 wild black bears

Sometimes living in a mountain town means nature comes to your street in the form of a 200-pound hungry bear drawn to the smells of garbage and bird seed. Asheville, North Carolina’s human population has grown nearly 40 percent in the past two decades, but long-term conservation efforts have also increased the wild black bear population to more than 4,000 in the surrounding mountain region. Researchers plan to teach residents how to secure garbage in bear-proof containers, clean barbecue grills and restrict the usage of bird feeders. [Wall Street Journal]


$43 million in lost scallops

There’s been another reported case of mass seafood deaths after a major Chinese seafood supplier announced it had found more than 80 percent of scallops at a major farm had died due to unidentified natural causes. The owners of the farm said that it supplies more than 50,000 tons of scallops a year, a big portion of the over 100,000 tons that are traded globally. The dead scallops were valued at $43 million dollars. That’s an expensive, smelly mess to clean up. [Bloomberg]


SigDigs: Nov. 14, 2019

How Seriously Should We Take Michael Bloomberg’s Potential 2020 Run?

Like any self-respecting election nerd, I keep a spreadsheet of prospective 2020 presidential candidates, which includes a column called “Odds They Run.” When a candidate declares, I move it to 100 percent; when a candidate passes on the race, I move it to 1 percent.

The reason I move it to 1 percent and not 0 percent is people like Michael Bloomberg. Bloomberg announced in March that he would not run for president, but on Friday, he is expected to file paperwork to become a candidate in the Democratic primary in Alabama, as today is the state’s filing deadline. This move doesn’t mean Bloomberg is definitely running, however. At this point he’s just keeping his options open, although Bloomberg advisers told The New York Times on Thursday that he would decide for sure whether to run within a matter of days, not weeks.

Even so, Bloomberg’s move is a big surprise given just how late it is in the electoral calendar. (Since 1976, the latest an eventual nominee has launched his or her presidential campaign was August of the year before the election.) Now there are fewer than three months before the Iowa caucuses, and if Bloomberg does end up running, he’ll have to scramble to make the debate stage, let alone get himself in a position to win any states.

The reason, though, why Bloomberg is considering a last-minute bid is that he is reportedly worried about the way the Democratic primary is unfolding, as one adviser told the Times. Back in March, Bloomberg said he believed that it was essential that the Democratic nominee be able to defeat President Trump, and last month it was reported that he would reconsider his decision not to run if former Vice President Joe Biden continued to struggle. Presumably, Bloomberg has now changed his mind after seeing Sen. Elizabeth Warren — whose ideas, especially the wealth tax, he has lambasted as socialism — gain ground in the polls and Biden struggle with fundraising.

But there is arguably very little appetite among Democratic voters — donors may be a different story — for yet another presidential candidate. In October, a YouGov/HuffPost poll found that 83 percent of Democratic or Democratic-leaning voters were either enthusiastic or satisfied with their presidential choices. And it looks like there is even less appetite for Bloomberg specifically. According to last week’s Fox News poll, just 6 percent of likely Democratic primary voters said they would definitely vote for Bloomberg should he enter the race. And a hypothetical Harvard-Harris Poll of Hillary Clinton, John Kerry and Bloomberg mixed in with the rest of the Democratic field gave Bloomberg the same 6 percent of the vote.

And those polls would probably qualify as good news for Bloomberg, given that he was generally registering around 2 or 3 percent in national primary polls before first taking his name out of consideration in March (which is also when pollsters largely stopped asking about him).

In a field this crowded, entering the race in the high single digits wouldn’t even necessarily be a bad thing, but the problem is that it might be harder for Bloomberg to build on that support than it would be for other candidates. In an average of polls from January and early February, I found that 62 percent of Democrats knew enough about Bloomberg to form an opinion (which was pretty high), but his net favorability (favorable rating minus unfavorable rating) was only +11 (which was pretty low). As you can see in the chart below, Bloomberg was a real outlier — for as well known as he is, we would have expected him to be much better-liked, with a net favorability of about +35, not +11.

And history suggests Bloomberg’s low favorability ratings would be a major obstacle to winning the nomination. Our past research indicates that people who win presidential primaries tend to either be (a) already well known and well liked or (b) relative unknowns to start off the campaign. Only one nominee since at least 1980 has been in Bloomberg’s position (well known but not well liked), and that’s Trump himself.

Assuming Bloomberg does decide to run, where would he fit into the race? Well, as the former Republican-turned-independent mayor of New York City (he reregistered as a Democrat last year), he would be running as a moderate alternative in the Democratic primary. He favors liberal but not progressive solutions to issues like health care and climate change, and he can be downright libertarian on fiscal issues, like the regulation of banks and, of course, taxes. (It’s little wonder that he has denounced Warren for years.) That said, he has become a progressive leader on the issue of gun control, founding and investing millions in the group Everytown for Gun Safety, which has emerged as a powerful political counterweight to the National Rifle Association. Unfortunately for Bloomberg, though, candidates from the more moderate end of the party (other than Biden) have struggled to gain traction in the polls, so there’s a real question of how much support he can reasonably expect to attract. His candidacy may depend on eating into Biden’s support (which, ironically, could make it likelier that someone like Warren wins the nomination).

Thanks to his successful financial services and media company, Bloomberg will at least have money at his disposal (he has a net worth of $52 billion) to help him overcome his late start. It’s not hard to imagine Bloomberg doing what fellow billionaire-turned-presidential-candidate Tom Steyer has done — use his own money to saturate the early states with advertising and augment his state-level polling numbers just enough to qualify for the debates. However, as we noted when Steyer entered the race, wealthy self-funders don’t actually have an electoral advantage. For example, in 2018, we found that self-funders1 in congressional and gubernatorial races won Democratic primaries at about the same rate as non-self-funders did. And getting 4 percent in four individual polls (the polling threshold for the December debate) is a far cry from being in legitimate contention to win the nomination.

Instead, it seems likelier that Bloomberg will affect the race primarily because of the effect he will have on other candidates. For example, he could cause Biden to slip in the polls by chipping away at his moderate base or throw cold water on the ascent of South Bend, Indiana, Mayor Pete Buttigieg. He could also turn voters away from Warren or Sen. Bernie Sanders by becoming the designated attack dog against progressive policies. Or he could fail to make much of an impact at all, leaving the current state of the race unchanged. We’re getting ahead of ourselves, though; it’s still possible he won’t run at all. Either way, there’s a long way to go before Bloomberg is a factor in this race.

Championships Aren’t Won On Paper. But What If They Were?

You probably missed this as baseball’s postseason was coming to an end last week — but congratulations are in order to the Houston Astros. Why? Because Houston finished the 2019 MLB season with the No. 1 Elo rating in Major League Baseball, of course.

The Astros were No. 1 in 2019 … on paper

Top 10 MLB teams in 2019 according to FiveThirtyEight’s Elo rating*

Reg. Season
Team Wins Losses Playoffs Elo Rating
1 Astros 107 55 Lost WS 1595
2 Dodgers 106 56 Lost LDS 1590
3 Nationals 93 69 Won WS 1589
4 Yankees 103 59 Lost LCS 1585
5 Athletics 97 65 Lost WC 1570
6 Rays 96 66 Lost LDS 1554
7 Cardinals 91 71 Lost LCS 1543
8 Braves 97 65 Lost LDS 1541
9 Indians 93 69 None 1538
10 Twins 101 61 Lost LDS 1537

*Using the version of Elo found in our “Complete History of MLB” interactive, which does not adjust for the quality of pitchers.

I mean, sure, the Washington Nationals just beat the Astros to win the World Series in seven games. But still, the Astros were your official Elo champs for the 2019 season. (Somehow I doubt the Astros will throw a parade or put up a banner for the honor.)

Because Elo takes a long view of the entire season, being the best in it is a pretty good proxy for being the best team in the league “on paper.” And it’s actually not uncommon for Elo’s Paper Champion and the team that wins the World Series not to be one and the same. Including Houston this year, it’s happened 28 times — or in a whopping 52 percent of seasons — since the late 1960s, when MLB expanded to a division-based playoff format. Simply put, baseball is a sport in which the best team doesn’t always win. (Or even if it does, maybe we don’t always know who the best team is anyway.)

Take a tour through MLB’s Hall of (Paper) Champions

Actual World Series champions and end-of-season MLB Elo champions* (for years where they were not the same team), 1966-2019

Paper Champ Actual Champ Paper Champ Actual Champ
Year Team Elo Team Elo Year Team Elo Team Elo
2019 HOU 1595 WAS 1589 1995 CLE 1596 ATL 1580
2017 CLE 1596 HOU 1572 1993 ATL 1588 TOR 1565
2015 TOR 1562 KC 1561 1992 MIL 1558 TOR 1555
2014 BAL 1559 SF 1542 1990 OAK 1567 CIN 1544
2012 TB 1566 SF 1561 1988 NYM 1569 LAD 1555
2011 NYY 1575 STL 1555 1987 TOR 1562 MIN 1521
2010 PHI 1570 SF 1563 1985 NYY 1571 KC 1544
2008 BOS 1567 PHI 1564 1982 MIL 1555 STL 1552
2006 NYY 1551 STL 1531 1980 BAL 1577 PHI 1545
2003 NYY 1567 FLA 1547 1974 LAD 1569 OAK 1559
2001 OAK 1596 ARI 1567 1973 BAL 1569 OAK 1556
2000 SF 1559 NYY 1542 1972 PIT 1560 OAK 1557
1997 ATL 1572 FLA 1538 1971 BAL 1591 PIT 1568
1996 CLE 1568 NYY 1547 1969 BAL 1576 NYM 1567

*Using the version of Elo found in our “Complete History of MLB” interactive, which does not adjust for the quality of pitchers.

Ratings include results from all regular-season and postseason games.

Other sports have their own Paper Champs. (Although none happen anywhere near as frequently as in MLB.) I went back to the start of the Super Bowl era in 19662 and looked at the other sports for which we keep Elo — the NFL, NBA, college football, and men’s and women’s college basketball. Using the versions of our Elo that contain no adjustments for trades or players being in and out of the lineup,3 I found each case where the champion at the end of the season4 was not the team that finished atop the Elo leaderboard. Across all of these sports, these Paper Champs come up more frequently than you might think:

Since 1966, all but seven seasons5 (2013, 2009, 2005, 1998, 1989, 1979 and 1967) contained at least one Paper Champion across these five sports. Some years featured a lot more: In 2011, for instance, there were four Paper Champs — the Yankees in MLB, the Patriots in the NFL, and Ohio State (men’s) and UConn (women’s) in college basketball. (The only champs that actually led in Elo were the Dallas Mavericks in the NBA and Alabama in college football.)

In general, there is about a 50-50 chance that a given baseball season would produce a Paper Champ and somewhere between a 20 and 30 percent probability each of the other sports will as well.

How each sport’s Paper Championship rate compares

Frequency of Elo “Paper Champions” (and rate of the real champion being decided head-to-head) by sport, 1966-2019

Sport
MLB NFL NBA CFB MBB WBB*
No. of Paper Champs 28 11 15 14 14 4
Share of seasons 51.9% 20.4% 27.8% 25.9% 25.9% 22.2%
% of Paper Champs lost H2H** 46.4% 90.9% 60.0% 14.3% 78.6% 25.0%

* Women’s NCAA basketball data only goes back to 2001-02.

** This is the share of seasons with a Paper Champ that saw the actual champ beat them head-to-head in the postseason (or, in college football, the entire season).

Source: Sports-Reference.com

What does all of this mean? Well, it could be that Elo is broken. Even though it is calibrated to be the best predictor for a team’s next game — given its recent form, long-term expectations, wins and losses, scoring margin, opponent quality and game locations — maybe there are certain aspects of each sport’s postseason that aren’t captured by the algorithm. (This is the “Billy Beane’s Shit Doesn’t Work In The Playoffs” theory.)

A fundamental challenge of forecasting is the balancing act between considering a large amount of information — some of which may be of less relevance than others — and a more specific one that is more relevant, but also more prone to factors such as random variance in a small sample. Our Elo models attempt to straddle this divide, but it’s impossible to find the perfect mix of information that works in every single case.

Then again, maybe the real issue is that playoff systems are too small of a sample to determine the best team. Perhaps the best we can do is be content believing the champion was simply one of the top teams in a given season, nothing more.

Still another way of reconciling Paper Champs to postseason reality, though, is to consider that most of these actual champions vanquished their on-paper rivals head-to-head along the way. When there was a Paper Champ in baseball, for instance, 46 percent of the time that team lost directly to the eventual champion in a postseason game or series. (See: Nationals over Astros.) In the NBA, that number was 60 percent; in men’s college basketball, 79 percent; and in the NFL, a whopping 91 percent. Although Elo still wasn’t convinced that the matter was settled afterward, the Paper Champ at least had a chance to make its case on the field or court.

And in most of the cases where things weren’t settled head-to-head, you only have to zoom out a little to find a path of head-to-head superiority between the actual champ and the paper one. Like in the 2017 NCAA Women’s Tournament, when UConn finished as Paper Champ … but lost in the national semis to Mississippi State, which then lost to South Carolina in the title game. Almost every disagreement between Elo and the official championship can be settled either directly head-to-head or in this manner — with the exceptions of a few pre-wild-card MLB seasons (in which the Paper Champ didn’t even make the postseason at all) and a number of older college football campaigns that underscored just how broken the sport’s pre-playoff system truly was.

In 2007, famously one of the weirdest college football seasons ever, USC was Elo’s choice, while LSU prevailed in the BCS. To find a head-to-head path that put LSU over USC, even if you open up the possibilities to include the regular season,6 you needed to follow a trail of four games: LSU beat Ohio State, which beat Washington, which beat Stanford, which beat … USC.

But at least the BCS existed by then. Before it came along, the 1970s and ’80s often required even more ludicrous daisy-chaining of head-to-head results to reconcile the championship. In 1976, the path from actual champ Pitt to Paper Champ USC required a string of five games. And in 1983, the chain went like this: Actual champ Miami beat Notre Dame, which beat Boston College, which beat Clemson, which tied Georgia, which beat Texas, which beat Auburn, Elo’s Paper Champion. No wonder college football fans clamored for a proper playoff (even if the one they have now could also probably stand to be expanded).

But even with the perfect playoff system, you can never really avoid Paper Champs. Random variance and matchups — plus a million other factors — will always cause teams to play better or worse than they look on paper. And would we really want it any other way? If we look at who would have benefited most over the past half-century if Elo perfectly aligned with actual championships, the rich would mostly have gotten richer:

Who has beaten their on-paper odds most (and least) often?

MLB, NFL, NBA, college basketball (women’s* and men’s) and college football teams with the biggest positive — and negative — differentials between their actual championships, 1966-2019

Biggest gainers No. of championships
Team Sport Actual Elo Diff
Notre Dame Fighting Irish CFB 4 0 +4
Los Angeles Lakers NBA 11 7 +4
New York Giants NFL 4 1 +3
St. Louis Cardinals MLB 4 1 +3
Miami/Florida Marlins MLB 2 0 +2
Houston Rockets NBA 2 0 +2
Kansas City Royals MLB 2 0 +2
LSU Tigers CFB 2 0 +2
Clemson Tigers CFB 3 1 +2
Miami Heat NBA 3 1 +2
San Francisco Giants MLB 3 1 +2
Connecticut Huskies MBB 4 2 +2
Ohio State Buckeyes CFB 4 2 +2
Boston Celtics NBA 9 7 +2
Biggest losers No. of championships
Team Sport Actual Elo Diff
Baltimore Orioles MLB 3 8 -5
Cleveland Indians MLB 0 3 -3
Oklahoma Sooners CFB 4 7 -3
San Antonio Spurs NBA 5 8 -3
Connecticut Huskies WBB 9 12 -3
Milwaukee Brewers MLB 0 2 -2
Philadelphia 76ers NBA 2 4 -2
Kentucky Wildcats MBB 4 6 -2
North Carolina Tar Heels MBB 5 7 -2
New England Patriots NFL 6 8 -2
New York Yankees MLB 7 9 -2
Alabama Crimson Tide CFB 9 11 -2

*Women’s NCAA basketball data only goes back to 2001-02.

Source: Sports-Reference.com

Although I feel bad for the Orioles and Indians, who would have won multiple extra championships if Elo had been perfectly predictive, other teams that have won plenty of real titles — Alabama football, UConn women’s basketball, the Patriots, the Yankees, the Spurs, etc. — “should” have even more under Elo.

Oftentimes, it’s the unpredictability of sports that make them great — just ask the Nationals. So we’re sorry, Astros. Although the first-ever Elo Championship pennant would be something to behold.

Icing The Play-Caller Really Doesn’t Work

With four seconds left until halftime, trailing the San Francisco 49ers in a 14-7 game, the Arizona Cardinals defense had forced a fourth and goal at the 1-yard line. In what turned out to be a pivotal moment in the game, Niners quarterback Jimmy Garoppolo handed off to Jeff Wilson Jr. for a run up the middle that was stuffed for no gain. But what should have been cause for celebration for the Cardinals defense quickly turned into disappointment. The refs waived off the result; Arizona head coach Kliff Kingsbury had called a timeout before the play began.

San Francisco made Kingsbury pay on the next play, with Garoppolo tossing an easy touchdown to Emmanuel Sanders as time expired. After the game (which the Cardinals ended up losing 28-25), Kingsbury explained his reasoning behind the timeout, saying that the stoppage was to get a better look at what the 49ers might be planning offensively and that he “wanted them to hopefully burn their best play.”

Unfamiliar with the concept of burning a play, I reached out to a handful of coaches and experts to understand what Kingsbury might have meant. I was told that teams will often carry just three or four 2-point conversion plays into a game because of the rarity of the situation. Since the inventory of available plays is limited, a coach will usually call his favorite one first. Once a team lines up, some of the surprise is lost, so calling a timeout could force the play-caller to abandon the best play in favor of one the defense hasn’t yet seen.

One obvious problem with this explanation is that goal-line stands aren’t 2-point conversions. A play run on fourth and 1 at the goal line has some similarities with a 2-point conversion attempt — they’re both near the end zone, after all — but I was told that teams won’t necessarily limit themselves to their 2-point playsheets in goal-to-go situations. And if that was true of the 49ers, a timeout wouldn’t exhaust play inventory in the way Kingsbury might have assumed. For the gambit to work, it seems that a coach would need to have fairly detailed information about how an opponent organizes and prepares its game plan.

This made me wonder: Is there any evidence to suggest that icing the play-caller is a successful tactic? Do defenses tend to fare better following a timeout on fourth and goal at the 1-yard line? To find out, using data from Armchair Analysis, I looked at every play run on that down and distance from 2000 to 2018 in both the regular season and playoffs. I separated timeouts called by the offense from those called by the defense and compared the touchdown rate on plays with timeouts to plays with no timeouts called. The results were surprising.

On goal-line stands after the defense calls a timeout, offenses score a touchdown about 17 percentage points more often than when no timeout is called, on average. Defenses have called timeouts in these situations just 46 times in the past 18 years, so there is still uncertainty about the true difference in touchdown rates, but there’s also no evidence that the touchdown rate after those timeouts is lower. In other words, the benefits of icing the play-caller appear to not exist — and the call may backfire more often than not, just as we saw with Kingsbury against the 49ers.

Perhaps this result shouldn’t be surprising. We’ve known for a while that icing the kicker is likely ineffective. In their book “Scorecasting,” Tobias Moskowitz and L. Jon Wertheim found that when they adjusted for distance, field goals attempted in high-pressure situations weren’t affected by defensive timeouts. If athletes aren’t affected, why would we think coaches would be rattled by a little gamesmanship? A play-caller’s job is to choose plays, and a timeout probably just gives him more time than usual to do it.

Check out our latest NFL predictions.

Are Some Democratic Voters Reluctant To Support A Gay Candidate?

There’s nothing like a national election to illuminate the complex and slippery nature of bias at work in the country today. Just ask Pete Buttigieg. Always something of an underdog in the Democratic primary, Buttigieg has started to poll well in Iowa and New Hampshire relative to his national numbers and has proved to be a formidable fundraiser. But as his profile has risen, murmurs about how his sexual orientation might affect his bid have gotten louder and louder.

There are plenty of reasons, of course, why Buttigieg might struggle to gain traction among more voters. His lack of statewide or national political experience is one potential stumbling block. Voters of all races may also balk because he has faced criticism for his handling of the predominantly white police force in South Bend, where a white officer recently shot and killed a black man, and for implementing economic policies that some feel ignore or harm communities of color. And another scapegoat has emerged: Last month, a leaked memo described the results of a focus group conducted by Buttigieg’s own campaign in July, which found that some black voters in South Carolina were uncomfortable with his sexual orientation.

It’s hard to know how much that discomfort truly matters — even a number of the skeptical focus group voters were still open to supporting Buttigieg — and to the extent that it exists, it’s certainly not confined to one group. But regardless of the reasons behind his depressed support, Buttigieg’s candidacy is a case study in the dilemma facing gay and lesbian candidates running at all levels of office today. It’s remarkable, in one sense, how little Buttigieg’s sexual orientation has come up in the primary so far, considering that only 10 years ago, the election of a lesbian woman as Houston’s mayor was enough to make national headlines. Voters’ willingness to support gay and lesbian candidates is at an all-time high, and multiple studies by political scientists have suggested that Democrats are especially unlikely to discriminate against candidates like Buttigieg. “If anything, there are some subgroups of Democrats who would be more likely to vote for a gay candidate,” said Gabriele Magni, a political science professor at Loyola Marymount University.

Stop there, and you’d have a pretty rosy electoral prognosis for Buttigieg — focus group skepticism notwithstanding. But it also isn’t the full story. Some Democrats haven’t moved as quickly to the left as others on gay rights issues. And a substantial chunk of Republicans are still comfortable saying they wouldn’t support a gay candidate. As ever, it’s difficult to know what actually keeps a voter for pulling the lever for a particular candidate, but Buttigieg’s sexuality could be a sticking point for some. Experts like Magni said Buttigieg might find it tough to draw support from the most conservative or religious corners of the Democratic primary electorate, not to mention Republicans in the general election. And in a primary driven by voters’ concerns about how electable the candidates are, the perception that a significant slice of voters would never support a gay candidate might be an even bigger hurdle than the reluctant voters themselves.

Just a few election cycles ago, a debate about the electoral impact of a gay candidate’s sexual orientation would have had a clear answer — because being gay was a dealbreaker for almost half the country. As recently as 2007, only 55 percent of Americans said they would vote for a gay or lesbian candidate for president, which is only slightly higher than the share who currently say they would vote for a socialist. But many voters’ qualms about the prospect of a gay or lesbian president evaporated over the following decade, and 76 percent of Americans — including a majority of Republicans — now say they wouldn’t have a problem supporting a gay candidate for president. That’s still not the near-uniform level of hypothetical support the same polls show for a female or black candidate, but it’s also not obviously disqualifying. After all, only 63 percent of Americans say they’d vote for a candidate over the age of 70, which describes the three top-polling candidates in the Democratic primary.

There are plenty of signs, too, that a Democratic primary is particularly friendly terrain for a gay candidate. Political scientists have found in studies and interviews with candidates that gay and lesbian candidates overwhelmingly run as Democrats, in part because Democratic voters don’t seem to penalize candidates for their sexual orientation. A recent experimental study co-authored by Magni found that voters who identify as very liberal and nonreligious were more likely to support a gay candidate over a straight candidate.

The impulse to size up the electoral landscape and run where their support is strongest can partially help explain why gay and lesbian candidates often don’t find their sexuality to be a serious barrier. “When you talk to gay and lesbian candidates, they’ll generally tell you their sexual orientation didn’t matter much in their race, and that’s in part a function of the fact that they tend to run in more liberal areas, like cities,” said Donald Haider-Markel, a political science professor at the University of Kansas and the author of “Out and Running: Gay and Lesbian Candidates, Elections, and Policy Representation.”

But there are still pockets of the Democratic electorate where voters’ views of gay people aren’t as liberal. And that poses a few potential problems for Buttigieg, who has to run a national campaign. A significant chunk of his base is composed of white college-educated Democrats; this is also a subset of voters where his sexual orientation is highly unlikely to be a roadblock, given that several decades of data from the General Social Survey shows that people in this group are especially likely to say that homosexual relationships are never wrong.

But as my colleague Nathaniel Rakich wrote recently, Buttigieg has some fierce competition from Elizabeth Warren for white college-educated voters. And while the groups with whom he might be hoping to expand his support — like religious voters or whites with lower levels of education — are certainly not uniformly opposed to gay candidates, they are groups where his sexual orientation might be more of an issue. People who attend church frequently are much less likely than non-churchgoers to believe same-sex marriage should be legal, according to the Pew Research Center. Likewise, lower levels of education tend to come with lower levels of support for gay marriage.

Voters’ feelings about gay candidates could show up in more nuanced ways as well. The specter of electability, for example, could turn out to be a bigger roadblock for Buttigieg than outright hostility toward gay people. For instance, a recent Politico/Morning Consult poll found that voters were basically split on whether the country was ready for a gay or lesbian president, and only 26 percent said that their neighbors were ready.

To be clear, several experts told me these electability concerns don’t have a lot of evidence to support them, although that may be partially because there hasn’t been a lot of research on how gay candidates perform in real-life elections, and candidates may also avoid contests — like Republican primaries — where they’re all but destined to lose. But discomfort with gay marriage or homosexual relationships won’t necessarily stop voters from ultimately supporting a gay candidate. And Haider-Markel pointed out that the people with the strongest prejudices against gay people are also highly unlikely to vote for any Democrat, which means that in a general election, Buttigieg’s sexuality would probably matter less than the “D” next to his name. Dislodging gut-level intuitions about electability can be tricky business for a candidate, though. That’s particularly true when significant chunks of the electorate — including almost 40 percent of Republicans — are still perfectly comfortable telling a pollster they wouldn’t vote for a gay candidate. It’s hard not to assume that a neighbor’s stubborn opposition to gay marriage will shape their vote in some way — even though in reality, the forces that influence our choice of candidate are far more complex.

This complexity makes it nearly impossible to say for certain whether it’s Buttigieg’s sexual orientation — rather than his age, or his political inexperience, or his policy positions, or some ineffable combination of factors — that has kept him from rising further in the polls. And that will also make it hard to assess, when all the ballots are cast and the Democratic nominee is chosen, just how much Buttigieg’s electoral chances were affected by his sexuality.

But it also means that even if some voters are being held back by Buttigieg’s sexual orientation now, other parts of his biography, like his military service or Christian faith, could still change the way they think about him. The good news for Buttigieg is that there are months to go before the primaries begin, and he has plenty of cash to spend on introducing himself to voters who might currently know next to nothing about him. “At a very basic level, Buttigieg could reduce some bias just by getting voters to see him as a gay man who was also in Afghanistan and goes to church on Sunday,” Magni said. “Sexual orientation is less likely to play a role in vote choice when people move past the stereotypes they have in their mind about who gay people are supposed to be.”