A Weird NFL Weekend Leaves Us Even More Confused

gfoster: (Geoff Foster, sports editor): NFL Week 14 was very odd in many respects (and the week isn’t over) with the Raiders, Niners and Giants winning and the Rams, Texans, Steelers and Patriots all losing. It all went a long way to making the playoff picture even murkier. What was your biggest takeaway from the week?

Salfino (Michael Salfino, contributor): How much should we be worried about the Rams? Is every offense entitled to one hiccup like this, or is it part of a pattern that began with the Lions, who did not care about play-action at all and just covered the receivers. The Bears also did not bite. Does Los Angeles have a Plan B?

neil (Neil Paine, senior sports writer): It was definitely startling to see the Rams’ offense be held in check so thoroughly. Jared Goff was terrible. Todd Gurley did next to nothing. The Rams’ offensive expected points added in the game was -23.5. That was 29.6 points worse than their second-worst offensive game of the season … which was Week 13 vs. Detroit. And it was 33.8 points of EPA worse than their low before the Detroit game.

sara.ziegler (Sara Ziegler, assistant sports editor): I know people talk a lot about playing in the cold as a problem, and I usually roll my eyes. But was that a factor here?

neil: The Rams have only played four games where temp was under 50 degrees in the Sean McVay era.

Salfino: I know, that temperature stuff. I just can’t believe these guys from all different parts of the country are suddenly able to only play in warm weather. But that’s going to be the theory of the case now almost certainly.

gfoster: The Rams game at Detroit was definitely curious. They came out of the bye completely flat, Goff played poorly, and most people kinda wrote it off to rust. But now I’m starting to wonder if teams, as Mike alluded to, are starting to figure out this McVay offense.

Salfino: This kind of thing has happened before. The 1994 49ers team that won the Super Bowl and was electrifying on offense laid an egg against the Eagles that year, losing 40-8. They had 189 yards that game.

And on defense, the 1985 Bears famously flubbed a primetime game against the Dolphins late in the year for their only loss that season.

gfoster: That Detroit game was in a dome, Sara.

Salfino: This can’t be Cooper Kupp, can it? My theory of playing the Rams was to just ignore the running game. I figured a team that gave them fits would basically concede Gurley. What really surprised me about the Chicago performance was that Gurley didn’t even get going. They had nothing. The Bears didn’t even accept the slow death of Gurley running.

sara.ziegler: But why didn’t the Rams even try to run Gurley? Only 11 carries?

neil: Good question, Sara. They had been almost exactly balanced (49.7 percent pass/50.3 percent run) on first down, but last night they passed on 73 percent of first downs. Despite averaging 4.6 yards per carry on the first downs where they did run.

Salfino: My feeling watching the game with Gurley is that the Rams wanted to get wide open passes by faking to Gurley like they usually do, and then when that didn’t work, they were behind the down and distance and had to straight-up pass, with typically disastrous results.

neil: Right — Goff had a 23.0 passer rating on 1st down.

Salfino: What happens to a play-action offense when no one buys into the deception? When the defense just ignores it?

sara.ziegler: I guess I would argue that you should just run the ball.

Especially with a guy like Gurley.

Salfino: Exactly right. That’s when they should run. This is probably why McVay was so hard on himself in the post-game.

sara.ziegler: And for good reason! LOL

gfoster: Let’s talk about the ending of the Dolphins-Patriots game, which was probably the highlight of the year. Although, I may be partial to the Chris Carson front flip a couple weeks ago.

neil: Incredible. Just look at the diagram:

Longest game-winning scoring play in NFL history, I believe.

Salfino: Kenyan Drake made the best nonlateral decision ever on a lateral play. Or maybe the worst since anyone other than Gronk easily tackles him.

gfoster: The funniest sequence of that was when Drake was looking to lateral more and then you could see him to say to himself, “Oh wait, I can just run this in from here.”

sara.ziegler: Gronk definitely failing his defensive audition.

Salfino: Why was Gronk even on the field? It was not a Hail Mary situation 69 yards from pay dirt.

neil: Right, clearly Ryan Tannehill doesn’t have the arm for that throw in the air. He’s not Patrick Mahomes.

gfoster: Tannehill can throw 80 yards. You didn’t know that?

sara.ziegler:

Salfino: Did Drake just laugh at the idea of Gronk tackling him?

Gronk looked like I imagine Brady would trying to make that tackle. And ironically, he finally looked like Gronk on offense.

neil: And Bill Belichick left Devin McCourty off the field for that play in favor of Gronkowski. Shades of Malcolm Butler in the Super Bowl. Too cute for his own good.

gfoster:

sara.ziegler: I just need to take a second to thank Drake for giving me 12.2 fantasy points on that play.

🙌

Salfino: Fantasy scoring on that play is hilarious. It goes as a Tannehill touchdown pass. There were no screams about that because who’s playing Tannehill?

sara.ziegler: LOL, good point.

Salfino: How dumb are the Dolphins though for not getting a guy as electric as Drake the ball more on offense? But they’re 7-6. Have to be one of the worst winning teams at this juncture of the season in memory, but they do own wins against the Bears and Patriots.

gfoster: Salfino is not a Frank Gore fan.

Salfino: OMG, Frank Gore. The dude is a survivor, I’ll give him that. He’s going to end up in the Hall of Fame, too. But he’s been just a guy for so long now.

gfoster: Neil wrote about this recently. Is this Patriots team just not intimidating?

Gronk’s “defense” is getting the headlines, but he had a big game offensively Sunday. There’s no doubt the offense is world’s different when he’s healthy — and it’s hard to remember a playoffs when he was remotely healthy.

Salfino: The Patriots’ problem is that they lacked offensive upside. But then Tom Brady really looked like the Brady of old vs. Old Brady. Gronk got rolling. Gordon was hyper-efficient. The running game was trash, but that’s sort of old-school Patriots too. Even the defense struggling seemed normal. But losing that game was not normal at all. If you told me New England was going to lose, I’d figure it was their offense flagging the game.

neil: Although it’s worth noting they forced zero turnovers against a Tannehill Phins offense.

sara.ziegler: That’s sort of the beauty of those lateral plays actually working: It’s a fluke, not a systemic problem for them.

neil: But this is definitely the type of toss-up game they usually find a way to win, not lose.

sara.ziegler: For sure. Which is what made it so fun!

neil: At least for 31 fanbases searching for Patriots schadenfreude.

Salfino: I really thought one of the Patriots and Steelers would emerge in the AFC, and now maybe both of those teams are going to be playing wild-card weekend.

gfoster: There are four AFC teams at 7-6: Ravens, Colts, Dolphins, Titans. Which one of those is going to make the playoffs in your eyes? Or will it be the Browns, Neil?

neil: They’re “in the hunt”!

For the first time in approximately 30 years.

Salfino: I think the sixth team in the AFC will be the Colts. Andrew Luck is totally out of his shoulder issue and throwing the ball downfield. T.Y. Hilton has nearly 600 receiving yards in the past four games. Luck-to-Hilton is maybe the most lethal combination in football right now.

gfoster: Not Luck-to-Ebron?

neil: For what it’s worth, our model thinks the Ravens still have the best chance at 55 percent.

sara.ziegler: I like the Ravens. They were super unlucky to lose to the Chiefs.

I mean, come on.

Salfino: The Ravens did everything right on defense. Tyreek Hill takes standing eight counts with three different injuries. And you look up at the end of the game and Mahomes has nearly four bills and the Hill has 139 receiving yards. Spencer Ware looked very dangerous. How can anyone stop the Chiefs?

neil: The diagram on that one is amazing, too:

(In case you can’t tell, I am addicted to these things.)

Salfino: Only one player could make that throw, and only one receiver could race to make that catch. The probability for Mahomes-to-Hill should have been, like, 50 percent 🙂

neil: Maybe Aaron Rodgers could. But certainly only one QB on a playoff-bound team.

Salfino: Yes, maybe Rodgers.

sara.ziegler: (And he probably doesn’t have a receiver this year who could catch it.)

neil: That might not have even been Mahomes’ most jaw dropping pass of the day!

Salfino: The Ravens blitzed the hell out of Mahomes, and it really seemed to give him fits, but then again you look at the stat sheet and are like, “??????”

gfoster: I would normally say here that the Chargers could hang with the Chiefs and look like the AFC’s Super Bowl-bound team. But that was a somewhat lifeless effort by them Sunday. Against Jeff Driskel and probably the worst defense in the NFL.

sara.ziegler: I guess we’ll find that out on Thursday, when the Chiefs and Chargers play.

Salfino: I thought it was going to be typical Chargers. But maybe typical Chargers is having easily the second-best team in the conference and somehow being the fifth seed. As for the Bengals performance, it fits the “letdown game” theory between Pittsburgh and Kansas City.

gfoster: Although, it should be noted that Mike Badgley hit four of four field goals, including a 59 yarder. That actually ties the total of made Chargers field goals for the last two seasons.

sara.ziegler: LOL

Salfino: How is Hill going to play that game on Thursday? He was like Rocky on the stool at the end of that Ravens game.

The thing about the Chargers that’s weird is their pace of play. They were 29th in plays before last week and then put up another relatively low play count.

Could Mike Tomlin get fired if the Steelers miss the playoffs? Should he?

sara.ziegler: Not in light of the Le’Veon Bell mess, I’m guessing.

The Steelers’ running game was nonexistent, with no James Conner even.

Salfino: Has Bell’s absence even hurt the team? Sunday was the first game you could say they lost because of it, arguably. But the defense could not stop Derek Carr. Think about that sentence in the context of this year.

gfoster: I don’t think this Steelers team is very good. They now have the Patriots and then AT the Saints. They could easily be 7-7-1 going into final week.

sara.ziegler: I’ve been amazed all season at how high the Steelers’ Elo was. But they just kept winning — until the past three games.

neil: Looking at our Elo, though, the Steelers usually pick up steam late in seasons. They haven’t had a swoon like this since 2012.

And that year they could pin it on Charlie Batch starting some games.

Salfino: Explain to me how Roethlisberger goes 25-for-29, the Steelers defense is still near the top of the league in yards allowed per play, and they lose to the Raiders. Tomlin totally blows the end of the game by not calling a timeout, he gets his own lateral play to work, and then the kicker falls down.

sara.ziegler: Field goal kicking has been about average this year, at 84.2 percent made. But this week was a bad one for kickers, who made only 42 field goals out of 56 attempts.

Salfino: Steelers vs. Patriots is the game of the week, but not for the reasons we thought going into the season. These are two desperate teams now. The Steelers are teetering on elimination, and New England can’t win the conference without a bye, IMO.

Do you know how hard it is to pass like Roethlisberger has this year and have a top defense in yards per play and still struggle to win?

gfoster: They haven’t beaten a truly good team this season. Especially now that we can safely call the Panthers a bad team.

And the Jags.

And the Falcons.

The Steelers’ best win was arguably over the Browns!

Salfino: They really should have beaten the Chargers, though. But you’re right.

sara.ziegler: And they would have beaten Oakland, if not for their kicker falling down.

neil: … another really bad team.

Salfino: Are we worried about the Saints? They did not really bounce back at all offensively from last week’s loss to the Cowboys. You have had the feeling all year that Sean Payton was worried about the depth of his receiving corps, and they had nothing other than Michael Thomas on Sunday — and Thomas really had to grind it out. Nothing in that game against a terrible Tampa defense was easy, which is shocking.

gfoster: But on Thanksgiving, we were singing the praises of Brees’s ability to throw TD passes to four different guys who walked in off the street and put on Saints jerseys.

Salfino: I think the Saints defense is underrated now and the Saints offense is overrated.

gfoster: Where are Austin Carr, Keith Kirkwood, Tommylee Lewis and Dan Arnold?

Salfino: Arnold was inactive on Sunday. Crippling loss.

That quartet sounds like a country rock band lineup.

neil: Are we rattling off Saints WRs or members of the 1973-74 Cavaliers?

Salfino: Nice pull, Neil!

gfoster: Brees and the Saints are always bad in Tampa, it seems. Kinda like how Brady always seems to struggle in Miami. What is it about Florida?

(Brady was good Sunday, though.)

Salfino: The Saints really pulled that game out of the fire and it was huge. Could give them the No. 1 seed, and I think they really need it.

gfoster: Here’s a question: If we could switch the results of one of the narrow losses in the first eight weeks, are we talking about the Giants as a sleeper playoff team?

neil: People are talking about the Giants as a sleeper playoff team NOW.

(At least on WFAN.)

gfoster: They have two wins more impressive than Pittsburgh: at Texans, Bears.

Salfino: Eli Manning is going to resign the Giants into giving him a lifetime contract. He is sucking all hope out of their prospective QB search.

Eli is the Frank Gore of QBs.

neil: Sara and I theorized that Eli paid Kyle Lauletta to keep screwing up.

“Here’s $50, go commit a traffic violation in New Jersey.”

Salfino: Eli’s last four wins were against opposing QBs Mullen, Fitzpatrick, Daniel, Sanchez.

sara.ziegler: My favorite thing in football this season is how bad the Giants are at tanking.

Salfino: Exactly! It’s not that hard.

sara.ziegler: Remember when the Giants basically didn’t play Saquon Barkley in the second half against Philly?

And handed the Eagles that game?

Salfino: Although the Jets would have the No. 1 pick right now if Sam Darnold didn’t mess it all up by winning.

neil: Broadway Sam can’t help himself.

gfoster: A lot of teams are struggling to tank. These guys need to watch more NBA. The Niners won’t quit.

Salfino: What about Josh Allen’s insane running. This is not a sustainable QB model, or am I wrong?

neil: Did I read that Allen broke a Mike Vick rushing record Sunday?

😳

Salfino: No QB ever had two consecutive games of 99+ yards rushing (in the modern era anyway), and now he has three.

neil:

gfoster: He seems like he’s looking to run. You watch a QB like Watson, Rodgers, Mahomes — they are so reluctant to do it. Always have eyes down the field until they physically cross the line of scrimmage.

Salfino: My theory with Allen is that his running is so effective and reliable that it’s hurting his development as a passer.

gfoster: He also has the worst supporting cast of possibly any QB ever.

Salfino: Darnold escapes to throw and not run, too. Especially on his touchdown pass Sunday, which was an incredible play.

sara.ziegler: Allen is like Bizarro Mahomes.

Salfino: Sara mentioned the Eagles. Is their window closing? They’re going to have to rebuild their defense, which is bereft of impact players. Their skill players are mediocre. Alshon Jeffery is not a No. 1 receiver, remotely. Suddenly their team-building seems suspect.

sara.ziegler: They had seemed to be playing better since that walloping by the Saints. But maybe not so much.

neil: Will we look back at 2017 as a weird, one-off year in general?

Eagles take advantage of a strange hiccup in the general arc of the game?

Salfino: Especially winning with Foles.

gfoster: One more NFC East thought: The Amari Cooper trade was probably the most derided front office move of the year. I myself openly laughed at it. Now…

sara.ziegler:

Salfino: The Cooper trade is going to go down in history as the best in-season deal. He was the missing piece. Everyone else is now in the role that they are suited to be in. Dak Prescott was explosive Sunday. I can’t believe I’m saying it.

The bug for the Cowboys was supposed to be their decision-making by Jerry Jones and his family, and that’s turned out to be their strength. Seriously, name the team that’s drafted better recently than the Cowboys. Now add Cooper to this. God, I hate myself for saying this.

gfoster: The issue with that trade was more relative to the wide receiver market. Golden Tate, Josh Gordon and Demaryius Thomas were all traded for a far smaller return. The difference is that prior to the deal, I would have lumped Cooper in with that tier of receiver, but it’s possible that his problem was even more Derek Carr than I thought.

Salfino: My view is that Cooper is an explosive player who can take the top off the defense. Those other guys, including Gordon now, cannot.

gfoster: OK, before we go, I want to get in one college football question off the big news of the weekend: Did Heisman voters get this one right?

Salfino: Oklahoma somehow pulled a Favre to Rodgers in college football.

sara.ziegler: I’m still kinda shocked that Kyler Murray won.

It really came down to performances in the conference championships.

neil: It’s funny because the Tua-is-unstoppable narrative had been in place for like two months (or more). Yet the signs were there that Kyler was a legitimate threat to him. He had a better NCAA passing efficiency, better QBR. Betting odds even favored Kyler on Saturday AM.

Salfino: My take is that Alabama would be great/No. 1 in the country with pretty much a typical college QB, but Oklahoma absolutely must have Murray to rank where they are. So Murray is the MVP of college football for sure.

gfoster: Here’s the flip side of that: Tagovailoa barely played in the fourth quarter most of the season, while Murray was needing to score 40 to 50 points every week to beat any team.

Salfino: I’ve heard the draft people concede the Murray would be a first-round pick. Now the question I guess is how high? There are no rules anymore after Baker.

sara.ziegler: Unless he’s an Oakland A by that point…

gfoster: And, likewise, Mahomes coming out of the same Big 12 nonsense football that looks like two 11-year-olds playing Madden.

Salfino: If Murray is a top-five pick, the financial calculus changes dramatically. He may never get another MLB contract. Look at ninth-overall picks in history, for example.

neil: Yes it does change the formula we looked at here.

gfoster: All right, last question:

Quickly give me the Week 14 Super Bowl prediction and No. 1 overall pick prediction, since we have a legitimate race this year.

neil: I guess I shouldn’t stick with my October pick of the Vikings…

sara.ziegler: LOL

They’re not out of it…

neil: Although who knows! We’ll see tonight.

sara.ziegler: I still think it will be the Chiefs out of the AFC, but who knows from the NFC.

Even after last night, I just can’t get behind the Bears.

gfoster: Sara, just pick the Vikings, it’s fine. We will forget about it.

sara.ziegler: I can’t. The moment I do, it’s over for them.

Salfino: I think the Raiders are the favorite for the No. 1 pick because San Francisco is going to face backups against the Rams in Week 17. I picked Saints-Steelers in the preseason. I will stick with the Saints — figuring that Payton figures it out. AFC has to be the Chiefs now. They do have a pass rush. They absolutely cannot lose Hill or Kelce though. (And obviously Mahomes.)

Betting on Andy Reid in January though and the Chiefs in January at home, yikes.

sara.ziegler: I’ll take a long shot for the No. 1 pick with the Jaguars. That’s a team that looks like it does not care.

neil: I will speak for the model and pick the Saints coming out of the NFC. We have them at 26 percent to win it all. AFC favorite is Chiefs, though I share Mike’s concerns there, despite my crush on Mahomes.

And I’ll go with the Cardinals at No. 1, our model’s pick for the worst record.

Forecasting the race to the bottom

Fewest projected 2018 wins according to FiveThirtyEight’s NFL Elo model

Wins
Team Current Projected remaining Total
32 Cardinals 3 0.6 3.6
31 49ers 3 0.7 3.7
30 Raiders 3 0.8 3.8
29 Jets 4 0.9 4.9
28 Bills 4 1.2 5.2

Simulated ties included as half-wins.

gfoster: I’m saying Saints over Chargers. Cardinals with the first pick (they have at Seattle and Rams still).

And if the Chargers get drilled by the Chiefs on Thursday, I will sneak into WordPress and delete all of this.

Check out our latest NFL predictions.

The Little, Mathematically Determined House On The Prairie

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,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

From Josh Vandenham, precipitation permutations:

Louie walks to and from work every day. In his city, there is a 50 percent chance of rain each morning and an independent 40 percent chance each evening. His habit is to bring (and use) an umbrella if it’s raining when he leaves the house or office, but to leave them all behind if not. Louie owns three umbrellas.

On Sunday night, two are with him at home and one is at his office. Assuming it never starts raining during his walk to his home or office, what is the probability that he makes it through the work week without getting wet?

Submit your answer

Riddler Classic

From Randi Goldman and Zach Wissner-Gross, little misanthropic houses on the prairie:

Antisocial settlers are building houses on a prairie that’s a perfect circle with a radius of 1 mile. Each settler wants to live as far apart from his or her nearest neighbor as possible. To accomplish that, the settlers will overcome their antisocial behavior and work together so that the average distance between each settler and his or her nearest neighbor is as large as possible.

At first, there were slated to be seven settlers. Arranging that was easy enough: One will build his house in the center of the circle, while the other six will form a regular hexagon along its circumference. Every settler will be exactly 1 mile from his nearest neighbor, so the average distance is 1 mile.

However, at the last minute, one settler cancels his move to the prairie altogether (he’s really antisocial). That leaves six settlers. Does that mean the settlers can live further away from each other than they would have if there were seven settlers? Where will the six settlers ultimately build their houses, and what’s the maximum average distance between nearest neighbors?

Submit your answer

Solution to the previous Riddler Express

Congratulations to 👏 Brenda Holloway 👏 of Manchester, Connecticut, winner of last week’s Riddler Express!

Last week, Alice and Bob were facing off in a footrace. The route was a straight path out followed by the same straight path back. But Alice literally wanted to face off — she wanted to run faster than Bob such as to maximize the amount of time they spent facing each other after she made the U-turn but before the two passed each other going in opposite directions. Assuming the turnaround time is negligible and the two each commit to running the race at a constant speed, how much faster than Bob should Alice run?

If Bob runs at speed \(B\), Alice should run at speed \((1+\sqrt{2})B\), or about 2.4 times his speed.

This (obviously) is a maximization problem, meaning we’re trying to find the value of a variable that maximizes some larger expression. That means we need a mathematical expression to maximize. Let’s try to piece that together.

Suppose the distance to the U-turn — that is, the halfway point of the race — is \(D\). (It won’t matter exactly what that distance is.) Suppose Alice’s speed, which we’re trying to calculate, is \(A\). The time it takes Alice to get there is \(D/A\). In that amount of time, Bob will have run a distance of \(B\cdot (D/A)\). So the distance between the two runners at the moment Alice turns around is \(D-B\cdot (D/A)\).

Now the two are running toward each other, so we must combine their speeds. They will, combined, cover that distance and pass each other after a time of \((D-B\cdot (D/A))/(A+B)\). That’s the amount of time we want to maximize.

A handy mathematical way to maximize things is to take derivatives. Specifically, we want to take the derivative of the expression above with respect to \(A\), set it equal to zero, and solve that for \(A\). That will give us a set of extremes, be they maxima or minima. The derivative of the expression that we’ll set to zero looks like:

\begin{equation*}\frac{D(-A^2+2AB+B^2)}{A^2(A+B)^2}=0\end{equation*}

That equation equals zero for two values of \(A\): \((1+\sqrt{2})B\) and \((1-\sqrt{2})B\). The latter of these doesn’t make sense for the case of our footrace (it’s negative), so we know the first must be the value we’re after. If Bob runs at speed \(B\), Alice should run at speed \((1+\sqrt{2})B\), or about 2.4 times his speed.

Solution to the previous Riddler Classic

Congratulations to 👏 Peter Ingraham 👏 of Brooklyn, New York, winner of last week’s Riddler Classic!

Last week, after facing each other down for so long in that footrace, Alice and Bob fell in love, got married and wanted to have kids. Potentially lots of kids. They figured out that the amount of work involved in having a child is equal to 1 divided by the age of that child. So if a child is 0, the work is infinite. If the child is 1, the work is 1; if a child is 2, the work is ½; and so on. They decided that they only wanted to have another kid if and when the total work involved for all of their other kids was equal to 1 or less. Ignoring real-world technicalities such as death, twins and the inability to decide precisely when one has a child: Will Alice and Bob have an infinite number of children? Can we predict, with an equation, when they will have their Nth child?

First, yes, Alice and Bob would have infinity children if they could. But they would have them very, very slowly. After about 10,000 kids, they’ll have to wait more than 10 years to have the next one, and the gap between children keeps increasing forever. Solver Scott Wu plotted how long they will wait to have their next child, based on how many children they’ve had so far, for their first 50,000 children. That looks like this:

Our winner this week, Peter, provided the following proof-by-contradiction of why the number of children the couple will eventually have is infinite: Suppose for the sake of argument that there was some point where Alice and Bob stopped having more kids, and that at this point they have \(k\) kids and \(n\) years have passed. Let’s examine the amount of work required at time \(2n\). All the kids are now age \(n\) or more. As an upper bound, each kid is worth at most \(1/n\) work, the amount required for the youngest child. Therefore the total work is at most \(k\cdot (1/n) = k/n\). But the number \(k\) is always less than \(n\): The gaps start at 1 year and always get longer. So Alice and Bob should have had another kid by this time since their total work \(k/n\) is less than 1. This is a contradiction. Therefore they never stop having kids.

There does not appear to be an exact, elegant formula for when Alice and Bob have their Nth child. But the Nth child does come at approximately N log N, a fact that we could ascertain by generating lots of simulated child data and running a regression. So, for example, the 1,000th child would be born approximately 1,000 log 1,000 — or 6,908 — years after the first child. Solver Laurent Lessard plotted the birthdates by the ordinal childbirths:

“Meet my brother, Bob Smith CMXLVII. Yeah, he’s my big brother, can’t you tell? He was born about 5,000 years ago.”

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! Consider your holiday shopping done.

Want to submit a riddle?

Email me at [email protected]

Trump Is Only Popular In Rural Areas

Welcome to Pollapalooza, our weekly polling roundup.

Poll(s) of the week

The 2018 midterm election confirmed America’s urban-rural divide; Democrats excelled in cities, Republicans dominated in the country and the suburbs were the tiebreaker that handed Democrats the House. Will the 2020 election play out the same way? This week, we got two polls of President Trump’s approval rating that suggest it might.

First, a Selzer & Co. (one of our favorite pollsters) national poll conducted Nov. 24-27 for Grinnell College found that Trump had a 43 percent approval rating and a 45 percent disapproval rating among all adults. However, his support isn’t distributed equally across different types of communities. He’s enormously popular among residents of rural areas, with a 61 percent approval rating and a 26 percent disapproval rating. In small towns, that breakdown is 44 percent approve vs. 42 percent disapprove. But in suburban areas, only 41 percent of residents approve of the job that Trump is doing as president, while 50 percent disapprove. Trump’s approval rating is lowest among urbanites — 31 percent approve of him while 59 percent disapprove.

We saw similar geographic trends in an Investor’s Business Daily/TIPP poll that was conducted from Nov. 26 to Dec. 2. Trump again got the highest marks from residents of rural areas — a 62 percent approval rating and a 35 percent disapproval rating. And yet again, his standing took a nosedive among suburbanites and urbanites. In suburban areas, Trump’s approval rating was 32 percent, and his disapproval rating was 60 percent. In urban areas, his approval rating was 27 percent, and his disapproval rating was 67 percent. (The IBD/TIPP poll didn’t include “small town” as an option for respondents.) Overall, Trump’s approval/disapproval spread was much lower in the IBD/TIPP poll (39 percent approve, 55 percent disapprove) than it was in the Selzer poll, which explains why the IBD/TIPP poll is worse for Trump in all three geographic categories as well.

Here are the results of the polls side by side:

Trump is more popular in rural areas

Presidential net approval rating among adults by density type

Category Selzer Poll IBD/TIPP Poll
Urban -28 -40
Suburban -9 -28
Small town* +2
Rural +35 +27
Overall -2 -16

* The Investor’s Business Daily/TIPP poll did not include a breakdown for “small town.”

Sources: Selzer & Co., Grinnell College, Investor’s Business Daily

 

This is perhaps stating the obvious, but Trump would do well to improve his standing among suburban and urban voters before 2020. Less than 20 percent of the U.S. population lives in rural areas. Granted, not all rural voters will cast their ballot for the president, nor will all urban and suburban voters back whoever is the Democratic nominee. But elections are winner-take-all contests waged within discrete geographic areas — states or districts. According to the Congressional Density Index from CityLab, a news website covering urban issues, just 70 congressional districts are “pure rural,” and an additional 114 are a “rural-suburban mix.” CityLab is still in the process of making similar assessments for states, but David Montgomery, a journalist for CityLab, told FiveThirtyEight that 11 states could be classified as mostly rural, while an additional 17 could be classified as a mix of rural areas and suburbs. The former are worth a combined 53 electoral votes, while the latter are worth a combined 138; 270 are needed to win a presidential election.

None of this means that Trump lacks a path to electoral victory. It’s still early in the 2020 campaign; approval ratings may change, and a person’s feelings about the president aren’t the only determinant of his or her vote. But those numbers aren’t great for Republicans even if institutions like the Electoral College give disproportionate influence to rural areas. Without urban and suburban areas, they’ll find it difficult to cobble together a sustainable majority.

Other polling nuggets

  • A YouGov poll found that only 26 percent of Americans either “strongly” or “somewhat” approve of the way Trump is handling the fallout from the killing of journalist Jamal Khashoggi. There was a sharp party divide in responses: 57 percent of Republicans approved, while only 6 percent of Democrats did.
  • 59 percent of Americans today either “strongly” or “somewhat” approve of the way George H.W. Bush handled his presidency, from 1989 to 1993, according to YouGov. That includes 53 percent of Democrats and 77 percent of Republicans. You can see for yourself how Bush’s approval rating changed over the course of his presidency using our presidential approval rating tracker.
  • A Gallup poll taken in France earlier this year, before the riots in December over increased living costs, found that President Emmanuel Macron’s job approval had declined by double digits among the three poorest of France’s five income groups but remained virtually unchanged among the two richest groups.
  • Americans still prefer to watch the news instead of reading or listening to it, according to a study by the Pew Research Center: 47 percent of respondents said they preferred to watch the news, 34 percent said they preferred to read the news, and 19 percent said they preferred listening to it. These habits remained mostly unchanged from 2016. Among those who would rather watch the news, 75 percent said they preferred doing so on the television, while 20 percent said they preferred to watch online.
  • A poll conducted by The Research Moms, a group of researchers at Edison Research, found that only 15 percent of moms said they split parenting responsibilities evenly with another parent. Forty-three percent said they handle the majority of the parenting responsibilities, and 41 percent said they handle all the parenting responsibilities. The Research Moms surveyed mothers in the U.S. age 18 to 64.
  • According to a Quinnipiac poll of New York City voters, 57 percent of New Yorkers approve of Amazon locating one of its new headquarters in Queens, while 26 percent disapprove. But when respondents were asked about the $3 billion in tax breaks that Amazon will get from the city and state government as part of its deal to relocate, support was roughly split, with 10 percent unsure.
  • Support for Amazon is even higher in Virginia, where a second headquarters is planned. A poll by Christopher Newport University asked registered voters in the state whether they approve or disapprove of the announcement that Virginia would provide $573 million in incentives for an Amazon headquarters; 68 percent said they approve of the announcement, while 30 percent said they disapprove.
  • The masses have spoken, determining that “Die Hard” is not a Christmas movie. That’s according to a Morning Consult/Hollywood Reporter poll, which found that only a quarter of American adults considered the Bruce Willis vehicle a Christmas movie; 62 percent said it was not, and 13 percent said they didn’t know or had no opinion on this important matter.
  • Armenians go to the polls Sunday to participate in snap parliamentary elections, the first since the “non-violent velvet revolution” protests in April and May that resulted in the resignation of Prime Minister Serzh Sargsyan. An in-person survey of 1,100 voters conducted this week by an affiliate of the Gallup International Association in Armenia found that more than 69 percent of Armenians who plan to participate in elections said they would vote for the My Step alliance, led by current acting Prime Minister Nikol Pashinyan, followed by about 6 percent who said they would vote for the Prosperous Armenia Party. The Republican Party of Armenia, which had served as the ruling party, polled at less than 2 percent.

 

Trump approval

According to FiveThirtyEight’s presidential approval tracker, 42.1 percent approve of the job Trump is doing as president, while 52.3 percent disapprove (a net approval rating of -10.2 points). At this time last week, 42.4 percent approved and 52.6 percent disapproved (a net approval rating of -10.2 points). One month ago today, Trump had an approval rating of 41.8 percent and a disapproval rating of 52.8 percent, for a net approval rating of -11.0 points.

 

Did The Packers Squander Aaron Rodgers?

The dismissal of Green Bay coach Mike McCarthy — who was let go after the Packers’ stunning home loss to the Arizona Cardinals on Sunday — wasn’t exactly a shock. Perennially tabbed as a Super Bowl contender out of the NFC, McCarthy’s team had gone just 11-16-1 over the past two seasons, including a disappointing 8-9-1 in games that featured future Hall of Fame quarterback Aaron Rodgers as Green Bay’s primary passer.1 It was time for a change along the sidelines that Vince Lombardi once roamed.

Things weren’t always so bleak on the frozen tundra. The McCarthy era had its high points, particularly early on — when he and Rodgers appeared to have Green Bay positioned on the cusp of a potential dynasty. But between postseason near-misses, roster changes, injuries and coaching miscues, McCarthy’s Packers never fulfilled that promise. Instead, it’s fair to wonder whether Green Bay squandered the prime of one of the most talented QBs in NFL history.

The Packers team that McCarthy inherited in 2006 from Mike Sherman2 was one in transition — and that meant navigating some heavy-duty Brett Favre melodrama in his first two seasons at Green Bay’s helm. However, McCarthy quickly found that he had an all-time great on his hands in Rodgers, who, when he took over the starting job at age 25, was just entering his best years as a passer. The McCarthy-Rodgers marriage sputtered to a 6-10 finish in its first season but yielded great results shortly thereafter: an 11-5 playoff campaign in Year 2, then a Super Bowl crown in Year 3 and a 15-1 regular season (with Rodgers winning MVP) in Year 4. The sky seemed to be the limit for McCarthy and his star QB.

Since the end of the 2011 regular season, however, the Packers have gone just 5-6 in the playoffs; by comparison, Tom Brady and the postseason Patriots are 13-5 over the same span. Green Bay’s record includes a crushing home defeat against the New York Giants two weeks after that 15-1 season ended and another loss in which they watched helplessly as ex-49er Colin Kaepernick destroyed their defense in 2012 — still one of the greatest individual QB games in playoff history. The Packers’ postseason circumstances have not always been ideal: For instance, that Giants game was actually the only time since 2011 that Green Bay lost in the playoffs while favored — meaning the rest of the losses were as underdogs. But at the same time, the Pack have also had clear chances to return to the Super Bowl, and they came up short in each of them.

All told, it remains mystifying that a quarterback of Rodgers’s stature hasn’t won more frequently. If we run a simple logit regression between a QB’s Yards Above Backup in a season and whether his team made the Super Bowl,3 we’d expect Rodgers to have made 1.86 Super Bowls in his career through 2017 — roughly twice as many as he’s actually been to. (Meanwhile, other contemporary QBs — such as Ben Roethlisberger, Eli Manning and even Brady — have gone to more than twice as many Super Bowls as we’d expect from their individual stats.)

Rodgers hasn’t won as much as he should have

Top 10 NFL starting quarterbacks by Yards Above Backup QB, 1990-2017, with their actual and expected Super Bowl appearances

Super Bowls Made
Quarterback Years Starting YARDS Above Backup Actual Expected Diff.
1 Peyton Manning 17 21,585 4 4.15 -0.15
2 Tom Brady 16 19,735 8 3.63 +4.37
3 Drew Brees 16 17,250 1 2.89 -1.89
4 Brett Favre 19 13,047 2 1.86 +0.14
5 Aaron Rodgers 10 10,988 1 1.86 -0.86
6 Ben Roethlisberger 14 10,945 3 1.35 +1.65
7 Philip Rivers 12 10,721 0 1.54 -1.54
8 Steve Young 8 10,022 1 1.65 -0.65
9 Matt Ryan 10 8,251 1 1.14 -0.14
10 Tony Romo 9 8,192 0 1.11 -1.11

1990 was the first season of the NFL’s current playoff format. Expected Super Bowls are based on a season-by-season logit regression between a QB’s Yards Above Backup and whether his team made the Super Bowl.

Source: Pro-Football-Reference.com

Over time, it became more and more difficult for the Packers to come within striking distance of the Super Bowl. In 2015, Rodgers slumped to career-worst numbers without top wideout Jordy Nelson, though the team as a whole was still good enough to get to the divisional playoffs before losing. In 2016, it was more of the same when Rodgers mused that Green Bay could still “run the table” — sparking an eight-game winning streak that saw the QB return to vintage form and left the Packers a win away from the Super Bowl.4 By then Rodgers was 34 years old, so a sense of urgency was setting in when 2017 came and went without a playoff berth — even though that could be written off as the byproduct of Rodgers missing nine starts.

The 2018 season was always going to be the real crossroads for McCarthy. With a healthy Rodgers leading the way, the Pack could always count on contending in the past, so this year’s expectations were no different. But Rodgers’s numbers have been merely good, not great. Brett Hundley isn’t around anymore to take any blame. And unlike in 2015, when Green Bay was talented enough to survive despite a downturn in its QB’s individual stats, there has been no answer from the team’s supporting cast this time around. It all came crashing down around McCarthy in the loss to Arizona as 13½-point favorites, Green Bay’s single most disappointing defeat since the merger according to Pro-Football-Reference’s point-spread data.

We can visualize the Packers’ decline over McCarthy’s final few years at the helm using FiveThirtyEight’s Elo ratings. Specifically, I’ve been tinkering with an experimental version of Elo that keeps a separate adjustment for the primary QB in each game, similar to how we treat starting pitchers in our MLB ratings.5 Using this, we can trace how a team’s performance rises and falls independent of its QB — which is useful in cases like 2017, when Rodgers was hurt and Hundley started nine games. (For instance, by season’s end, the Packers would have projected to be a 1529 Elo team with Rodgers starting — compared to a 1427 team with Hundley. And remember, 1500 is average.)

At the beginning of 2015, the Packers had an effective Elo of 1622, which included a 73-point boost from having Rodgers at QB and a 49-point boost from his teammates. By the end of the year, Green Bay’s effective Elo was still in the same neighborhood (1597), despite Rodgers’s adjustment actually dropping to negative 11, because the rest of the team carried a larger share of the weight (+108). Meanwhile, at the peak of the Packers’ run-the-table surge in 2016, the team’s 1657 effective Elo arose out of a 61-point boost from Rodgers and 97 additional points (relative to league average) from the rest of the team.

But fast-forward to now, and it’s clear how much the Packers have crumbled around Rodgers. His own adjustment is 16 points of Elo above an average QB, the lowest it’s been since Week 10 of the 2016 season. But he’s still expected to be above average; his supporting cast, by contrast, has fallen to a negative-67 score relative to the average team. That’s the worst they have been in Rodgers’s entire NFL career, and it isn’t especially close. Keeping QB play constant, the Packers’ Elo has dropped by a total of 139 points since the end of the 2016 season, which is essentially the difference in current Elo ratings of the 11-1 Los Angeles Rams and the 6-6 Carolina Panthers.

The reasons for the slide are varied, but many can be traced back to a series of poor drafts under former general manager Ted Thompson, who was replaced by current front-office chief Brian Gutekunst in January. As Sports Illustrated’s Kalyn Kahler pointed out last week, only three of Green Bay’s 17 draftees from 2014 and 2015 remain on the current roster. While no team can avoid dry spells in the NFL draft if given enough time, the Packers also — largely by design — did little in the way of enlisting outside help as a backup plan.6 Combine those infrastructural problems with criticisms of McCarthy’s offensive scheme (criticisms of a perceived lack of innovation that Rodgers apparently shared), plus legitimate complaints about Rodgers’s own decline in performance, and a season like this was bound to happen to Green Bay sooner or later.

Even so, it came contrary to preseason predictions. Going into the schedule, you might have penciled in this week’s matchup against the Atlanta Falcons as a marquee game with playoff implications. Instead, it will be the third-worst game of the week, according to our combination of matchup quality (i.e., the harmonic mean of the teams’ Elo ratings in each game) and game importance (how likely it is to swing every team’s odds of making the playoffs):7

The best matchups of Week 14

Week 14 games by the highest average Elo rating (using the harmonic mean) plus the total potential swing for all NFL teams’ playoff chances based on the result, according to FiveThirtyEight’s NFL predictions

Playoff % Playoff %
Team A Current Avg. Chg* Team B Current Avg. Chg* Total Change Game Quality
BAL 65.0% +/-15.1 KC 100.0% +/-0.0 31.8 1628
PHI 28.3 24.2 DAL 81.1 19.4 51.0 1578
MIN 59.7 19.9 SEA 87.7 11.8 41.4 1572
CHI 94.4 4.3 LAR 100.0 0.0 10.8 1615
IND 9.8 10.1 HOU 98.2 2.4 23.6 1515
MIA 6.6 7.4 NE 99.7 0.5 16.2 1537
WSH 26.4 11.7 NYG 0.1 0.1 24.9 1435
TB 1.7 2.0 NO 100.0 0.0 6.9 1570
TEN 18.9 10.6 JAX 0.1 0.1 21.9 1478
PIT 83.8 11.0 OAK 0.0 0.0 23.7 1453
CAR 17.8 9.9 CLE 0.3 0.4 22.0 1454
DEN 20.9 11.1 SF 0.0 0.0 24.7 1427
LAC 96.0 3.6 CIN 0.9 1.3 9.1 1504
GB 1.4 1.3 ATL 1.1 1.2 5.5 1469
DET 0.3 0.3 ARI 0.0 0.0 4.2 1412
BUF 0.0 0.0 NYJ 0.0 0.0 2.4 1377

Game quality is the harmonic mean of the Elo ratings for the two teams in a given matchup. Total Change adds up the potential swing in playoff odds for every team in the league (not just the two teams listed).

*Average change is weighted by the likelihood of a win or loss. (Ties are excluded.)

Source: ESPN.com

Although the Packers hadn’t replaced a coach at midseason since 1953, Sunday’s loss forced their hand. Now they’ll need to figure out who’s next, from a candidate list that includes big names among both pro coordinators (Josh McDaniels) and up-and-coming college coaches (Lincoln Riley). They’ll also need to hope Rodgers’s issues were more related to McCarthy’s offense and less to his getting older and less productive — basically, that the next Packer coach will be more Mike Shanahan to Rodgers’s John Elway than Jimmy Johnson to his Dan Marino. So while the Packers may not have much on the line over the rest of their games, this promises to be the most interesting offseason Green Bay has had since Favre was retiring and unretiring more than a decade ago.

FiveThirtyEight vs. the readers

Make sure to check out FiveThirtyEight’s Elo ratings using our NFL prediction interactive, which simulates the rest of the season 100,000 times and tracks how often each team should make the playoffs and win the Super Bowl. And did you know you can also pick against the Elo algorithm in our prediction game? Maybe you can also climb up our giant leaderboard (or, if you’re like me, fall down it with each passing week).

Here are the games in which Elo made its best — and worst — predictions against the reader picks last week:

Elo’s dumbest (and smartest) picks of Week 13

Average difference between points won by readers and by Elo in Week 13 matchups in FiveThirtyEight’s NFL prediction game

OUR PREDICTION (ELO) READERS’ PREDICTION
PICK WIN PROB. PICK WIN PROB. Result READERS’ NET PTS
CIN 52% DEN 59% DEN 24, CIN 10 +8.5
PIT 62 PIT 54 LAC 33, PIT 30 +6.7
LAR 68 LAR 79 LAR 30, DET 16 +4.1
ATL 53 ATL 50 BAL 26, ATL 16 +1.5
KC 83 KC 89 KC 40, OAK 33 +0.4
SEA 83 SEA 83 SEA 43, SF 16 -1.5
MIA 58 MIA 57 MIA 21, BUF 17 -2.3
NE 67 NE 65 NE 24, MIN 10 -3.8
PHI 69 PHI 66 PHI 28, WSH 13 -4.1
TEN 78 TEN 72 TEN 26, NYJ 22 -4.4
CHI 69 CHI 72 NYG 30, CHI 27 -6.1
HOU 77 HOU 69 HOU 29, CLE 13 -6.6
CAR 59 CAR 64 TB 24, CAR 17 -7.8
GB 73 GB 79 ARI 20, GB 17 -10.5
IND 51 IND 62 JAX 6, IND 0 -14.0
NO 64 NO 74 DAL 13, NO 10 -15.3

Home teams are in bold.

The scoring system is nonlinear, so readers’ average points don’t necessarily match the number of points that would be given to the average reader prediction.

After a series of narrow wins at midseason, the algorithm handed the readers their worst loss (-55.2 points on average) since Week 3. Some of the blame can go to the subject of this column — the Green Bay Packers, whose loss not only cost Mike McCarthy his job but also cost users 10.5 points on average. But readers were also burned by the Jaguars’ win over the Colts and the Cowboys’ upset victory over the Saints. Add it up, and Elo has beaten the average reader 12 times in 13 weeks this season.

But congrats to Mike Edelstein, who led all users in Week 13 with 137.0 points, and to one of my favorite leaderboard names, Greg Chili Van Hollebeke, who maintained his No. 1 ranking on the season with 1,002.1 points. Thanks to everyone who has been playing — and if you haven’t, be sure to get in on the action! You can make picks now and still try your luck against Elo, even if you haven’t played yet.

Check out our latest NFL predictions.

Is There Any Stopping Another Alabama-Clemson Title Game?

gfoster (Geoff Foster, sports editor): After weeks of crunching every possible playoff scenario, we finally got our two matchups for the College Football Playoff: Clemson vs. Notre Dame and Alabama vs. Oklahoma. We have to wait until Dec. 29 to see those games. But in the meantime, we have some 37 bowl games to distract us from our families over the holidays.

Let’s start with the big two. Were you surprised by the playoff selections? I think the committee avoided all of the doomsday scenarios as the conference championships played to form.

neil (Neil Paine, senior sports writer): Yeah, they mostly got out of the woods compared with some of the scenarios we talked about here. Only thing that would have helped them more would be if either Oklahoma or Ohio State lost, but that didn’t happen.

To your question, I wasn’t too shocked about the picks. Much was made of Georgia potentially making it, but it seemed very unlikely that they’d take a two-loss nonconference champ over a pair of one-loss conference champs — even if UGA was probably better talent-wise. I was really only slightly surprised they took Oklahoma over the Buckeyes. If you look at the power ratings like Football Power Index or Simple Rating System, or something like ESPN’s Strength of Record, Ohio State was the superior team. But the committee probably held OSU’s strength of schedule against it — as well as that bad loss to Purdue and the near-loss against Maryland.

sara.ziegler (Sara Ziegler, general editor): It helped for the Sooners that, in the Big 12 title game, they beat the only team to have beaten them.

neil: Yes — you definitely heard the phrase “they beat every team on the schedule” thrown around.

sara.ziegler: And I feel like the committee members had Ohio State’s near-loss to Maryland on their minds.

(I know I did.)

Josh Planos (Josh Planos, contributor): I wasn’t all that surprised either. Three were shoe-ins, and if the playoff format has taught us anything it’s that most of college football’s elite programs hog all the playoff spots, and the committee will do everything in its power to eschew controversy. Ohio State certainly wasn’t lacking in that department this season.

gfoster: Oddly, I think blowing out Michigan in such ugly fashion actually hurt Ohio State, because most people seemed to write off that game (rightly) as UM being overrated rather than Ohio State beating a team ranked fourth in the country.

neil: And yet the Wolverines were allowed to (easily) be Notre Dame’s most signature win… 🤔🤔🤔

sara.ziegler: Notre Dame getting its special dispensation, as per usual.

neil: It was funny during the selection show to hear the note about how Vegas would have the Irish as underdogs against every other team in the playoff conversation (except UCF, I guess).

Josh Planos: During Northwestern’s third-quarter scoring run on Saturday night, while Gus Johnson was firing off catch phrases, you could almost hear the committee scratching out the Buckeyes. Would a 40-point win in the Big Ten championship game even have gotten Ohio State over Georgia?

gfoster: Last year, if it hadn’t lost to Stanford, Notre Dame could have easily finished the season with one loss and would have not have made the playoff. In that spot, not having a conference championship would have really worked against the Irish because they wouldn’t have had another opportunity for a signature win. But this year, we see the advantage for ND. Win all your regular season games, as easy as they may be, and you are in.

But likewise, Northwestern didn’t really give Ohio State much of a resume boost. So … it’s Wisconsin’s fault for being lousy I guess.

sara.ziegler: But of course, Geoff, that’s only the case for ND — not for any other non-Power Five teams. (😢 UCF)

gfoster: UCF needs to boost its strength of schedule if it wants to be taken seriously.

sara.ziegler: For sure. And I don’t think the Knights should have gotten in. But it’s not like Notre Dame’s schedule was off the charts.

gfoster: For all we knock ND, they are playing teams like NC State, USC, Syracuse — all of whom would be one of the hardest games on UCF’s schedule.

neil: How can they improve their SOS, short of joining a better conference? (Or is that basically it?)

I don’t think any real power team wants to play them nonconference. No upside there, only downside.

gfoster: You could get a mid-tier Power-Five team that would take them at home, no? When Notre Dame was playing Michigan in Week 1, UCF had UConn — quite possibly the worst team in FBS.

neil: Defensively, at least.

sara.ziegler: Though that UConn game was a conference game.

Josh Planos: They followed it up by playing South Carolina State, too.

sara.ziegler: They scheduled North Carolina, but that was canceled because of the hurricane.

gfoster: North Carolina is also terrible.

sara.ziegler: And that’s the other problem: You can schedule a mid-tier Power-Five team, but you can’t guarantee they’ll be good.

neil: Or if you’re Notre Dame, you can schedule prestige Power 5 teams and not know if they’ll be good.

sara.ziegler: UCF did schedule and beat Pitt, which was good enough to get trampled by Clemson in the ACC title game.

gfoster: Truth is, maybe UCF does need to move conferences? TCU managed to do that when it was facing similar problem.

sara.ziegler: Or we could solve this with an eight-team playoff!

neil: This.^^

gfoster: Well, yes.

sara.ziegler: Solve it for this year, anyway.

LOL

gfoster: NO ONE is against that.

neil: Except conference and university presidents.

gfoster: As for this year, this is the first time both playoff games have double-digit spreads. Which falls in line with some lopsided lines in the conference championships. Any reason to like the underdogs here?

Or is this destined for Clemson vs. Alabama again?

neil: Maybe if Tua is still hurt? (He won’t be. And they will destroy Oklahoma.)

sara.ziegler: And it’s hard to see Notre Dame doing much against Clemson.

neil: Clemson vs. Bama Part IV is pretty redundant at this point. But at least there’s a chance it doesn’t play out according to chalk. Under the old BCS system, they’d automatically be slotted in at 1-2. (Although that would have been very uncontroversial.)

gfoster: It’s hard to see any team doing much against the Tigers’ defense. Look at the line of Pitt QB Kenny Pickett in the ACC title game: 4 of 16 for … wait for it … 8 YARDS.

That’s 0.5 yards per pass attempt. (538 math skills, folks.)

Josh Planos: Yeah, we don’t know about Tua’s health. Oklahoma’s offense puts up video game numbers, so you’d expect Alabama to need to bring at least something to the table in that regard. Trevor Lawrence is playing the best secondary he’s seen all season. Brian Kelly is probably pretty motivated that his team’s recent struggles were broadcast on a Showtime series.

sara.ziegler: If Tua plays, you gotta think he’ll go to town on the Oklahoma defense.

Will he actually have to play all four quarters? LOL

Josh Planos: Have we ever seen the likely two top vote-getters in the Heisman race square off in the postseason? Is this the best QB battle in terms of single-season QBR that we’ve ever seen? Each is on pace to set the single-season record (though that will change, I’m sure).

gfoster: It might not change for Tua against that Oklahoma defense that gave up 700 yards to West Virginia.

The Sooners also might be without their best receiver: Marquise “Hollywood” Brown, which would be a significant blow to Kyler Murray.

neil: Josh, we almost got 1-2 Heisman QBs in the 2008 title game: Tebow vs. Bradford.

(But Tebow finished 3rd in the voting.)

(Colt McCoy finished 2nd????)

sara.ziegler: Wow

gfoster: If we had an eight-team playoff, my guess is that it would be Alabama-UCF (lol), Clemson-Washington (I’m thinking they must include a token Pac-12 in this new world), Notre Dame-Ohio State, Oklahoma-Georgia.

neil: What would the line be on that Tide-vs.-Knights game?

gfoster: 28.5

Josh Planos: Without McKenzie Milton? 30+

neil: Isn’t that the same line they gave the Buffalo Bills vs. Alabama?

gfoster: The Bills are like the sixth worst team in the NFL now. Shows how misguided those types of stories are.

sara.ziegler: Would a Pac-12 team even make an eight-team playoff this year? The committee had Michigan at No. 7.

Which is kind of amazing — another two-loss Power-Five team above poor UCF.

neil: I would guess an eight-team playoff would have an automatic berth for a Pac-12 champ.

sara.ziegler: There’s obviously no way to do this without some controversy.

neil: Then we can get into those fun March Madness arguments about “at-large” bids.

gfoster: Right … and one token non-Power Five. (Or in this case two, because of ND.)

neil: Notre Dame is Power Five! (According to our tier system.)

gfoster: Let’s talk about the other bowl games. Any others you are particularly excited for?

neil: UGA-Texas should be fun, I think.

Josh Planos: Maybe it’s because I’m a sucker for the nostalgia of Big 8 football, but Missouri vs. Oklahoma State. FPI is really high on the four-loss Tigers (like, higher on the Tigers than UCF and LSU), and each of Oklahoma State’s past five games have been decided by no more than 7 points. If nothing else, there will be a lot of points.

sara.ziegler: Missouri never should have left the Big 12.

neil: I always forget they aren’t in the Big 12 now.

sara.ziegler: Wisconsin vs. Miami in the Pinstripe Bowl is kind of a fun throwback.

gfoster: That should be called the Pinstripe Lack Of Motivation Bowl.

sara.ziegler: Haha

Josh Planos: Fun is an interesting word.

Is this the underachieving bowl? And did any team underachieve more than Wisconsin? All we heard throughout the preseason was that Jonathan Taylor could win the Heisman, they returned the entire offensive line, and Alex Hornibrook was returning for a 12th year of eligibility.

gfoster: In the bowl games, it’s always fun to identify the games where one team is really pumped to be there and the other has zero interest. For instance, Purdue vs. Auburn in the Music City.

You think Auburn is getting up for that?

Josh Planos: If they couldn’t get up to bully UCF last year, they’re not getting up to try and corral Rondale Moore.

neil: Also, the biggest early spread in a lower-tier bowl might be the Famous Idaho Potato Bowl. ESPN has BYU as a 14-point favorite over Western Michigan

I always root for the Directional Michigans though.

gfoster: Sad that Alabama-Oklahoma might be the most lopsided bowl game.

Also a 14-point spread.

sara.ziegler: I’m actually pretty interested in how UCF does against LSU.

LSU is a good proxy for a playoff team, since the Tigers did OK against Bama (at least early on) and pounded Georgia.

gfoster: I actually think LSU will get up for UCF, mainly because of what happened to Auburn and all this chatter.

neil: Although I wish UCF had gotten one of the just-missed-it playoff contenders like UGA or Ohio State, just for experiment’s sake.

sara.ziegler: Yeah, that would have been better.

Though maybe it’s all moot with no Milton.

neil: True. It wouldn’t have settled the debate.

gfoster: LSU is actually still playing that game against Texas A&M. They are in their 134th overtime.

sara.ziegler: 🏈 💤

gfoster: But Neil, didn’t we see kinda see that the year they let Hawaii play UGA?

neil: Hah, yes I was thinking of that exactly. Poor Colt Brennan.

Josh Planos: Should’ve known that a haircut like this didn’t stand a chance.

gfoster: OMG

sara.ziegler: Why … would … you … do … that?

gfoster: He even has the little island that they don’t let anyone on.

neil: The run-n-shoot makes you do crazy things.

gfoster: OK, what’s the worst bowl game.

This answer is two parts.

Worst name and worst game.

neil: Bad Boy Mowers Gasparilla Bowl

Can anything top that?

Short of bringing back the Poulan Weed Eater Bowl?

sara.ziegler: I love the Boca Raton Bowl.

Congrats, teams! You’re going to … Boca Raton!

gfoster: That’s the Cheribundi Boca Raton Bowl, Sara.

Josh Planos: Best bowl experience: The Makers Wanted Bahamas Bowl, FIU vs. Toledo.

Last season, folks could bring beer into the stadium. There were also archery opportunities near the concession stand.

I apologize for not answering the question, Geoff.

neil: I do miss the Popeye’s sponsorship for that one. Gave us an excuse to pick up fried chicken at Times Square and eat it in the office.

sara.ziegler: As if you needed an excuse for that, Neil.

neil: (Sorry again about the spicy tenders, Geoff.)

gfoster: The DXL Frisco Bowl is a rare short name that is terrible.

Jared Birmingham Bowl? It sounds like its named after someone named Jared Birmingham.

neil: I think “Jared Birmingham” is UCF’s backup QB.

Josh Planos: LOL

gfoster: I will say. I’m a big fan of the Cheez-It Bowl

I wish I had a bowl of Cheez-Its right now.

sara.ziegler: We didn’t talk about the best game of the weekend.

neil: Iowa State!

sara.ziegler: A dominant (not at all) win over powerhouse (not at all) Drake!

Josh Planos: I think Washington State could beat Iowa State by 40 points. Or the Cyclones could ride the Matt Campbell relevancy train to a 13-10 win.

sara.ziegler: That’s quite a range.

neil: And nothing in between.

sara.ziegler: LOL

neil: I’m also kinda intrigued by the Peach Bowl: Michigan vs. Florida. Feels like that is a constant matchup in the tier of bowls just below the prestige level.

That has happened in many Citrus Bowls, for instance.

Josh Planos: How. Do. These. Teams. Keep. Playing.

gfoster: Harbaugh’s only bowl win at Michigan was a romp of Florida. And Lloyd Carr’s final win was an upset of Tim Tebow Florida.

neil: And don’t forget about the 2003 Outback Bowl!

sara.ziegler: Who can forget?

neil: Grossman vs. Navarre.

gfoster: I’m excited for West Virginia vs. Syracuse in the Camping World. That feels like a 100-point game. I also am oddly interested in Boca Raton bowl! UAB is an amazing story. They won Conference USA just a few years removed from having their football program eliminated.

It’s at this time where I’d normally ask for predictions. But I imagine no one is picking an upset in the first two playoff games?

So let’s skip to the final predictions.

sara.ziegler: It’s pretty hard to pick against Alabama.

neil: Alabama 27, Clemson 24

sara.ziegler: Clemson has been dominant, obviously, ever since squeaking by Syracuse. But Bama is just too good.

Alabama 30, Clemson 18

gfoster: Clemson 35, Alabama 28

This isn’t (entirely) me being the contrarian. I think the Crimson Tide are kinda vulnerable to an upset. They start slow every game (tied with Citadel at halftime, remember) and it’s going to catch up to them at some point. Clemson defense can keep Tua off the field enough to win.

(Assuming they beat ND, who I think will make a game of it against Clemson.)

Josh Planos: Alabama over Oklahoma 35-14

Clemson over Notre Dame 21-7

Alabama over Clemson 28-14

sara.ziegler: I guess there’s nothing left to do but watch the games!

The Republican Party Has Changed Dramatically Since George H.W. Bush Ran It

George H.W. Bush, whose death at 94 was announced on Friday by his family, was a hugely influential figure in the Republican Party: chairman of the Republican National Committee, vice president, president and father of another GOP president. But the GOP has changed dramatically since it nominated Bush for the presidency in 1988 — a fact reflected in the ex-president’s strained relationship with the GOP’s new standard-bearer, President Trump.

So, I think Bush’s death is another moment to highlight what my colleague Clare Malone described in the summer of 2016 as “The End Of A Republican Party.” Let’s run through some of the big shifts that have occurred within the GOP:

The GOP was once a more moderate party

Ideology is complicated to measure. By some standards, the Republican Party has moved to the left. In a poll conducted last year, 42 percent of Republicans backed same-sex unions; it’s safe to assume that number was far lower during George H.W. Bush’s presidency. In 1992, one of South Carolina’s senators was Republican Strom Thurmond, who ran a 1948 presidential campaign featuring his opposition to civil rights for blacks. Today, one of South Carolina’s senators is Republican Tim Scott, who is African-American.

But by most other measures, the GOP is far more conservative than it used to be. The General Social Survey, for example, shows self-identified Republicans moving far more toward the “extremely conservative” end of its scale (as opposed to “extremely liberal”) over the past several decades.1

Political scientists, using DW-Nominate scores,2 have concluded that the Republicans now in Congress are much further to the right of congressional Republicans in the 1970s and 1980s. And even anecdotally, figures like former House Speaker John Boehner, Ohio Gov. John Kasich and the late Arizona Sen. John McCain — considered solid conservatives in the George H.W. Bush era — found themselves cast as insufficiently right-wing by the party’s base in recent years.

In Bush’s era, Fox News did not exist. Deeply conservative billionaires Charles and David Koch and their allies had not created a huge network of right-wing groups that constitute basically an alternative political party. There was no tea party or House Freedom Caucus. Trump may be personally more conservative than Bush, but even if he weren’t, the forces that push a Republican president to the ideological right are stronger now than they were in the 1980s.

Bush himself famously signed a tax increase to help reduce the federal budget deficit, a move that angered the party’s conservative base. His two GOP successors (George W. Bush and Trump) never even really considered tax hikes, aware of the power of the party’s conservative coalition.

The GOP used to be more in line with the nation demographically

Trump’s rallies include lots of older white people, as the stereotype goes. But that is the Republican Party of today. The country has become older, more diverse and more educated. The GOP, meanwhile, has grown even more disproportionately old. And while its voters grew more diverse along with the country through the 1980s (though at a bit of a lag), that shift stalled in the party in the 1990s. Same with education: The share of non-Hispanic white voters without a college degree fell throughout the 1950s, ’60s, ’70s and ’80s — in the electorate overall and the GOP. But beginning in the 1990s, it stopped falling among Republicans.

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In 1992, according to the Pew Research Center, about 38 percent of registered voters who identified as Republicans were 50 years or older. By 2016, that number had grown to 58 percent. In 1992, 61 percent of Republicans were under 50, compared with 41 percent today.3

Republicans were once competitive on the coasts but weak in the South

The ideological and demographic shifts described above have corresponded with big changes in the GOP’s geographic coalition. In 1988, Bush won California — the sixth straight election in which the Republican presidential candidate carried the Golden State. Bush also won Connecticut, Delaware, Illinois, New Jersey and Vermont. The Republican caucus in the U.S. Senate from that era included two members from New Hampshire, two from Oregon and one from both Delaware and New York. There were zero Senate Republicans from Alabama, Arkansas, Georgia, Louisiana and Tennessee.

All but one of the 16 Senate seats I just described are now held by the opposite party that controlled them in 1992, as the GOP gained ground in the South but lost power on the coasts. Trump, reflecting the growing weakness of Republicans in California, lost there in 2016 by 30 percentage points. Bush lost West Virginia in 1988 by 5 percentage points. Trump’s 42-point win in West Virginia in 2016 was his second-biggest margin in any state.4

In short, today’s Republican Party is centered in the South and almost completely out of power on the West Coast.


Let me not overstate the changes in the Grand Old Party. It still loves to cut taxes, like it did in the George H.W. Bush era. It’s still overwhelmingly white. The majority of its voters are still whites without college degrees. White evangelical Protestants are still about a third of the party. It still deploys negative racial stereotypes about non-white Americans to appeal to white voters. (Remember the Willie Horton ad of the 1988 Bush campaign?)

But the changes highlighted here have dramatically altered the power dynamics in Washington. Presidents Nixon, Reagan and George H.W. Bush always had to share power with Democrats, who controlled the House from 1949 to 1994.5 But since the GOP won the House in 1994, the party has held the chamber for all but four years.6 (They won’t be in control in 2019, of course.)

When George H.W. Bush won 53 percent of the national popular vote in 1988, it was not that remarkable. Richard Nixon had won more than 60 percent in 1972, and Ronald Reagan breached 50 percent in both 1980 and 1984. But 1988 was a watershed moment for the Republican Party — it was about to start a measurable decline in terms of its national standing. In the seven presidential elections since then (including Bush’s 1992 defeat), Republicans have won more votes nationally than Democrats just once (2004).

The tensions between Trump and the Bush family, and between Trump and McCain, speak to this broader narrative. Trump is a different kind of Republican — and he is changing the party in his image in ways they don’t like. But he is also the product of a different Republican Party than the one that the Bushes and McCain ascended in. Trump got the GOP nomination in some ways by embracing what the Republican Party had become, not what the Bushes wished it were.

CORRECTION (Dec. 1, 2018, 1:30 p.m.): An earlier version of this article said President Trump’s largest margin of victory in any state in the 2016 election was in West Virginia. It was in Wyoming.

What Reddit Can Tell Us About NBA Fan Bases

The highs and lows of being an NBA fan aren’t necessarily dependent on wins and losses. When your team has been bad for a while, small victories — like upsetting a better team during the regular season — can be your version of winning a championship. Conversely, when your team is a finals contender every year, some setbacks can feel like signs of the end times.

One way to quantify the ups and downs of a fan base is to look at how active they are: After all, the only fans worse than angry fans are uninterested ones. To help with that, we have Reddit — the discussion-based website with more than a million communities, or subreddits, each devoted to a different subject.

Over the years, basketball fans have flocked to /r/nba, the site’s professional basketball subreddit, to discuss games in progress, seek meaning in the latest trade rumors and debate the legality of surrounding Steph Curry in a moving ring of teammates with locked arms. With more than 11 million comments made from January through October, /r/nba is the third most active subreddit this year, trailing only /r/AskReddit and /r/politics. It’s also far and away the busiest sports subreddit: Through October, /r/nba had received 3 million more comments this year than /r/soccer, the next most active sports-related subreddit.

One noteworthy aspect of /r/nba is that users have the option to publicly display their allegiance to teams through “flairs,” which appear as icons next to usernames in posts and comments. Although flairs are optional, roughly 80 percent of comments made on /r/nba since October 2011 have been made by users with those icons.

We used those flairs on comments made since Oct. 1, 2011,1 to chart the activity of each NBA fan base. To normalize for the growth of Reddit over time, we calculated the total number of daily comments made by a fan base divided by the total number of daily comments made by all users (flaired or not). In addition, we used a 30-day rolling average2 indexed to each team’s highest point in order to make the trends for teams in small markets as clear as the trends for teams in large markets. Indexing allows us to show all 30 teams at once, but cross-team comparisons must be made with caution because the y-axis for each team’s chart is unique.

In other words, each fan base’s activity is relative only to itself. We can infer from the data that 76ers fans on Reddit were more active earlier this year than they had ever been, but we shouldn’t infer that 76ers fans were more active than Warriors fans.

The spikes on the charts often correlate with playoff runs, blockbuster trades or better-than-expected starts to a season. For example, the fever of Atlanta Hawks fandom peaked during their 60-win season in 2014-15, which culminated in an appearance in the Eastern Conference finals. The spike on Minnesota’s chart lines up with the summer the Timberwolves traded for Jimmy Butler. Meanwhile, Sacramento Kings fans were most active during the start of the 2014-15 NBA season, when the team jumped out to a 9-5 record (they finished 29-53).

To understand the landscape of fandom on /r/nba, it’s worth examining in detail the comment activity for fans of the Philadelphia 76ers, the Cleveland Cavaliers, the Los Angeles Lakers and the New Orleans Pelicans.

76ers

No team has had a more linear increase in comment activity over the past seven years than the Philadelphia 76ers. Sixers fans have grown more active on /r/nba each year, in part because their team has gotten incrementally better each year.

But the 76ers also have two players on their roster who are a lightning rod for discussion on /r/nba. Hardly a day goes by without a post on the latest Joel Embiid sound bite or Markelle Fultz lowlight (when he’s not injured or involved in trade rumors). So long as those two players are on the team, Philly fans will have plenty to talk about.

Because the data is available only through Oct. 31, 2018, the chart does not capture the reaction to Philadelphia’s trade for Butler earlier this month. But given that the replays of Butler’s game winners against Charlotte and Brooklyn are two of the most-upvoted, or highest-rated, posts of the season thus far on r/nba, 76ers fans are likely just warming up.

Cavs and Lakers

When LeBron James returned to Cleveland in the summer of 2014, Cavaliers fans instantly became more active on /r/nba. With their teammate squabbles, midseason trades and postseason heroics, the Cavaliers were a constant topic of conversation from 2014 to 2018 — and never more so than during their championship run in 2016.

But when James left to join the Lakers this offseason, he took the conversation with him. We charted the comment activity for fans of the Lakers and Cavaliers together, without indexing the shares so that we could make a cross-team comparison.

The most-upvoted post of all time on /r/nba is the announcement that LeBron would sign with the Lakers. Ever since then, the L.A. fan base, which is one of the largest on /r/nba, has been more active than usual. Meanwhile, comment activity among Cleveland fans is the lowest it’s been since the first post-LeBron era.

Pelicans

The New Orleans Pelicans exist in one of the smallest media markets in the NBA. Their team-specific subreddit, /r/NOLAPelicans, has the second fewest subscribers among all 30 teams (/r/memphisgrizzlies has the fewest). And yet, their fan base is in the middle of one of its most active periods. For every subscriber to /r/NOLAPelicans, Pelicans’ fans made more than 1.2 comments on /r/nba in the month of October, the highest mark of any fan base.

The NBA’s most talkative fans

Number of subscribers to team subreddits and comments made by users with that team’s flair on r/nba, October 2018

Comments from flaired fans on r/nba
team subreddit Subscribers Total Per subscriber
NOP r/NOLAPelicans 6,595 8,131 1.23
TOR r/torontoraptors 52,289 58,935 1.13
MIA r/heat 15,170 16,933 1.12
UTA r/UtahJazz 16,157 17,789 1.10
POR r/ripcity 15,955 17,139 1.07
CHA r/CharlotteHornets 7,918 8,402 1.06
DEN r/denvernuggets 10,394 11,007 1.06
SAS r/NBASpurs 25,673 27,127 1.06
OKC r/Thunder 23,646 24,898 1.05
SAC r/kings 12,846 12,836 1.00
BRK r/GoNets 9,449 9,430 1.00
LAL r/lakers 84,137 82,127 0.98
PHO r/suns 14,302 13,872 0.97
DET r/DetroitPistons 10,695 10,072 0.94
DAL r/Mavericks 19,264 17,858 0.93
MEM r/memphisgrizzlies 5,520 4,838 0.88
MIN r/timberwolves 21,578 18,544 0.86
MIL r/MkeBucks 20,869 17,430 0.84
LAC r/LAClippers 10,418 8,250 0.79
BOS r/bostonceltics 61,601 48,665 0.79
IND r/pacers 9,358 7,102 0.76
PHI r/sixers 44,368 32,824 0.74
WAS r/washingtonwizards 17,926 13,178 0.74
CLE r/clevelandcavs 36,850 25,861 0.70
NYK r/NYKnicks 35,482 24,294 0.68
ATL r/AtlantaHawks 12,110 8,131 0.67
ORL r/OrlandoMagic 10,186 6,799 0.67
HOU r/rockets 35,090 23,237 0.66
CHI r/chicagobulls 41,935 21,248 0.51
GSW r/warriors 113,331 48,315 0.43

Subscribers were counted on Oct. 9. Subscribers to r/warriors are likely inflated because of a feature of Reddit’s mobile app.

Source: Reddit

If Anthony Davis decides to stay in New Orleans beyond his current contract, it’s likely to be in part because of the small but passionate fan base that supports him.

It’s worth pointing out that Golden State Warriors fans rank dead last in comments per subscriber. But this is probably misleading because of the inflated subscriber count of /r/warriors. During the summer of 2017, those who downloaded Reddit’s mobile app were subscribed to the Warriors subreddit by default if they selected the NBA as a category they were interested in. This resulted in a spike in subscribers to /r/warriors and created a gap between them and the rest of the league.

Check out our latest NBA predictions.

Alice And Bob Fall In Love

Welcome back to The Riddler — I hope you had a wonderful Thanksgiving. 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,14 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 Graydon Snider, road race intimidation tactics:

Two runners, Alice and Bob, are participating in a footrace. The route is a straight line out some distance and the same straight line back — the starting point and the finish line are the same. As the starting gun is about to go off, Alice hatches a race plan: Her legs feel good and she wants to run fast enough compared to Bob that after the U-turn, they are staring face-to-face for as long as possible. How much faster than Bob should Alice run to spend the maximum amount of time facing Bob before they pass each other going in opposite directions? Assume that, on the advice of their coaches, they’ve each committed to running at a constant speed the whole time, and that the turn-around time at the halfway point is negligible.

Submit your answer

Riddler Classic

From Dan Johnston, in which Alice and Bob then fall in love:

After their Riddler Express footrace, Alice and Bob fell in love and got married. Now they want lots of kids. However, as you may know, having one child, let alone many, is a lot of work. But Alice and Bob realized children require less of their parents’ time as they grow older. Alice and Bob, romantics that they are, decided to calculate how this relationship worked. They figured out that the work involved in having a child equals one divided by the age of the child in years. (Yes, that means the work is infinite for a child right after they are born. That may be true.)

Anyhow, since having a new child is a lot of work, Alice and Bob don’t want to have another child until the total work required by all their other children is 1 or less. Suppose they have their first child at time T=0. When T=1, their only child is turns 1, so the work involved is 1, and so they have their second child. After roughly another 1.61 years, their children are roughly 1.61 and 2.61, the work required has dropped back down to 1, and so they have their third child. And so on.

(Feel free to ignore twins, deaths, the real-world inability to decide exactly when you have a child, and so on.)

Five questions: Does it make sense for Alice and Bob to have an infinite number of children? Does the time between kids increase as they have more and more kids? What can we say about when they have their Nth child — can we predict it with a formula? Does the size of their brood over time show asymptotic behavior? If so, what are its bounds?

Submit your answer

Solution to the previous Riddler Express

Congratulations to 👏 Chris Sears 👏 of Maysville, Kentucky, winner of the previous Riddler Express!

The World Chess Championship ended this week after 12 straight draws followed by a series of speedier tie-breaking games. That’s fitting, because two weeks ago we posed this question: What are the chances that the better player wins a 12-game match? Specifically, suppose one of the players is better than his opponent to the degree that he wins 20 percent of all games and loses 15 percent of games; the other 65 percent end in draws.15 What are the chances the better player wins a 12-game match? How many games would a match have to be in order to give the better player a 75 percent chance of winning the match outright? A 90 percent chance? A 99 percent chance?

That better player wins a 12-game match about 52 percent of the time. The number of games required for those larger thresholds are, in order, 82, 248 and 773. (Call me crazy, but I’m totally game for a two-year-long World Chess Championship.)

So how do we get there? Solver Dan Swenson suggested this tidy approach. Consider the following expression:

\begin{equation*}\left(0.2x + 0.65 + 0.15x^{-1}\right)^{12}\end{equation*}

The coefficients 0.2, 0.65 and 0.15 are the probabilities of the three outcomes of an individual chess game, and the whole thing is raised to the 12th power because of the 12 games of the match. Expand that expression, multiplying it all out and grouping its like terms. Then, the coefficient on \(x^r\), where the exponent \(r\) is some integer, is the probability that the better player wins the match by \(r\) games. Finally, add the coefficients on all the positive powers of \(x\), which gives the probability that the better player wins the match. The result is about 0.5198, or about 52 percent.

For the second part of the problem, we can use that same approach, and just change the “12” in the exponent of the expression, and then, the same as before, expand it and sum the coefficients on the positive powers of \(x\). The smallest exponent that gives a 75 percent chance or better is 82, the smallest that gives a 90 percent chance or better is 248, and the smallest that gives a 99 percent chance or better is 773.

Solver Chris Sears illustrated how the probability that the better player wins the match increases as the number of games increases.

Somebody get FIDE on the phone!

Solution to the previous Riddler Classic

Congratulations to 👏 Scott Wu 👏 of Baton Rouge, Louisiana, winner of the previous Riddler Classic!

Two weeks ago, we presented you with a sort of Riddlerified version of beer pong. You had an infinite supply of ping-pong balls, each labeled with some number 1 through N. There was also a group of N cups, labeled 1 through N, each of which could hold an unlimited number of ping-pong balls. The game was played in rounds that had two phases: throwing and pruning. During the throwing phase, you draw balls randomly, one at a time, from the supply and toss them at the cups. The phase ends when every cup contains at least one ball. Next comes the pruning phase, in which you go through all the cups and remove any ball whose number does not match the number on its cup. Every ball drawn had a uniformly random number, every ball landed in a uniformly random cup, and every throw landed in some cup. The game was over when, after a round was completed, there were no empty cups.

How many balls would you expect to need to draw and throw to finish this game? How many rounds would you expect to need before finishing this game?

The first question — how many throws would you expect to need — is like a “coupon collector’s problem,” and versions of it have appeared in this column before, for example in a problem about collecting Riddler League football cards. In this case, we’re filling cups instead of collecting coupons.The solution to this problem is well-known and stems from the fact that as we collect coupons — or fill cups — it becomes more and more difficult to collect or fill the ones that remain. The specific solution is quite mathy and involves something called the Euler-Mascheroni constant, but the main takeaway is that the more coupons or cups there are, the more and more time we can expect to spend collecting coupons or throwing balls.

Solver Laurent Lessard illustrated how the expected number of throws increases roughly quadratically as the number of cups (and numbers on the balls) increases:

The second question — how many rounds would you expect to play — turns out to be much trickier. Lessard also illustrated his mathematical approach when playing with three cups. The cups in his diagram are either empty (white), filled with a correct ball (green), or filled with an incorrect ball (red). The arrows and the numbers next to them represent transition probabilities — the chances a cup goes from empty to correctly filled, for example, are 1/N. Our goal, of course, appears in the bottom right of the diagram, where all three cups are correctly filled and the game is over.

From there, Laurent describes how to turn this diagrammatic approach into an answer. He uses a Markov chain, which describes probabilistic sequences of events that depend on the current state of events — such as the balls currently in our cups. And from there he invokes a holy Riddler trinity: absorbing states and limiting distributions and nilpotent matrices.

The result of all this is that while the number of throws required grows roughly quadratically, the number of rounds required grows roughly linearly.

Of course, this is also a problem that admits programmatic, simulation-based approaches. Solver Robert Chin shared Python code for a Monte Carlo simulation, and Michael Branicky shared his code as well. Finally, solver Tim Book shared the results of his computer simulations, which illustrate how the expected number of rounds increases as the number of cups (and the number of numbers on the balls) increases — roughly linearly:

Happy tossing!

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! Consider your holiday shopping done.

Want to submit a riddle?

Email me at [email protected]

CORRECTION (Nov. 30, 2018, 11:50 a.m.): An earlier version of this article misstated the name of one of the solvers. It is Robert Chin, not Robin Chin.

Michael Cohen Is The 33rd Person Mueller Has Charged — And Could Be Among The Most Important

After a quiet period, there was a potential blockbuster development in special counsel Robert Mueller’s investigation into possible coordination between Russia and the 2016 Trump campaign this morning, when the president’s former lawyer, Michael Cohen, made a surprise appearance in a Manhattan courtroom to plead guilty to making false statements to Congress.

According to the formal charging document, Cohen lied about a Trump real-estate deal in Russia — specifically, the “Trump Tower Moscow” project. This doesn’t prove that members of Trump’s 2016 campaign coordinated with Russia. But according to the document, discussions of the Trump Tower Moscow project went on for longer than Cohen had previously indicated, and Trump was aware of the discussions. According to Cohen’s plea deal, he is cooperating with the special counsel investigation.

This brings the total number of people charged in Mueller’s investigation to 33.

In the document describing Cohen’s alleged conduct, Mueller’s team says that contrary to Cohen’s congressional testimony in 2017 that the deal to build Trump Tower Moscow had concluded early in 2016, negotiations around Trump Tower Moscow were still going as late as June 2016, when Trump was the presumptive Republican nominee.

Cohen also testified that he only spoke to Trump about the project on three occasions and didn’t brief the Trump family on it, that he never personally agreed to travel to Russia or considered a Russia trip for Trump in relation to the project, and that he didn’t recall any response from the Russian government to the project — all of which was challenged in the Mueller team’s charging document. According to the document, the “status and progress” of the project was discussed more than three times with then-candidate Trump and that Cohen also talked to family members about the project’s trajectory. It also says that Cohen did agree to go to Russia (although the trip never happened) and even considered a potential trip to Russia by Trump. And the document says that Cohen reached Russian President Vladimir Putin’s personal spokesperson to ask for help reviving the deal.

Speaking to reporters on the White House lawn this morning, Trump repeatedly said that Cohen was lying in the hope of receiving a shorter prison sentence.

This is the first time that the Trump Tower Moscow project has been mentioned in a charge filed by Mueller’s team. The deal ultimately collapsed but has been scrutinized as a possible point of connection between Trump and high-level Russian operatives. According to some reports, Cohen has provided more than 70 hours of testimony to the special counsel, including about contacts between the 2016 Trump campaign and Russia, questions related to obstruction of justice by the president, and Trump’s previous business dealings in Russia.

Mueller hasn’t charged Cohen before, but this is the second time in three months that Cohen has appeared in a Manhattan courtroom to plead guilty to a federal crime. In August, Cohen pleaded guilty to eight federal charges filed by prosecutors in the Southern District of New York, including a violation of campaign finance laws that appeared to implicate Trump.

Now, Cohen’s cooperation could have big implications for the way Mueller’s investigation continues to unfold. Trump submitted written answers to questions from Mueller’s team last week; they reportedly included queries about the Trump Tower Moscow project. If Trump’s responses differ from Cohen’s testimony to Mueller, that could spell trouble for the president.

The Patriots Aren’t Quite Their Usual Dominant Selves This Year

Here’s a surprise: The New England Patriots are 8-3, leading the AFC East, with some of the best odds in the conference of winning the Super Bowl.

Oh, right. I’ve just described basically every Pats season in recent memory. This is the ninth consecutive season that New England has won at least eight of its first 11 games. The team’s current Elo rating of 1641, however, is the lowest it’s been through the same stage of the season since 2009 (and we don’t talk about that season).

So what are we to make of these Patriots, then? After overcoming the typical early season hiccups, is this year’s version ready to build championship momentum down the stretch like normal? Or is there still something a little bit off about a team that was showering its punter (of all players) with praise after an uncharacteristically modest win over the lowly New York Jets last week?

In advance of New England’s showdown Sunday against the Minnesota Vikings, let’s take a look at some of the Patriots’ calling-card metrics to see whether this season is business as usual in Foxboro.

Road warriors?

One of the Pats’ most eye-catching statistics during Bill Belichick’s time as head coach has been their near-invincibility at home, where they’ve won 87 percent of their games this decade. But their road record — winning more than 70 percent of the time away from Gillette Stadium — could be even more remarkable. From 2010 to 2017, the Pats’ winning percentage on the road was about 10.5 percentage points higher than what we’d expect from their home record — the third biggest gap in the NFL (behind the Cowboys and Eagles):

This year, though, New England is a perfect 5-0 at home but only 3-3 on the road — respectable but nowhere near the league’s best. (The Pats have also been outscored by 11 points in away games, against a road schedule that ranks just 28th in average opposing Elo.) And this might come up in the playoffs, unlike so many seasons in which the Pats had home-field advantage through the AFC title game.1 Right now, New England is in line for the AFC’s No. 2 seed behind the Kansas City Chiefs, but only a half-game separates them from the fourth-seeded Steelers.

Owning the turnover battle

Turnover margin is one of the most important factors in determining who wins or loses any football game. Conventional stathead wisdom, though, tells us that outlier turnover seasons — whether avoiding them on offense, forcing them on defense, or both — are unsustainable. While there are some ways a team can influence its tendency to have more takeaways than giveaways, a lot of it also comes down to luck.

Unless, of course, you’re the Patriots. New England perennially dominates this category, ranking first by a mile from 2010 through 2017 with a +116 turnover differential, almost double that of the next-best team. A lot of that is a function of having Tom Brady at QB; he’s tied for the second-lowest interception percentage of any passer in NFL history. But the Pats are also great at avoiding fumbles — only the Falcons had coughed it up fewer times since 2010, and no team had lost fewer fumbles than the Pats. And their defense had forced the second-most turnovers of any team this decade (behind the Giants), ranking second in interceptions and tied for third in fumbles recovered.

Such opportunism has historically paid big dividends for New England, but this year’s squad is still trying to recapture that formula. The Pats are currently +5 in turnovers, which ranks ninth in the league but is nothing special by their standards. Brady has his highest interception rate since 2013 (his seven picks already are only one off of his full-season total from last year), driving a big overall increase in giveaways per game, though the team is being more careful in recent weeks. And while the Pats have forced at least one turnover in all but one game this season, they are tied for eighth-to-last in the league in games with three or more takeaways, six behind the league-leading Bears.

Yards and points

In addition to — and correlated with — their dominant turnover differential, the Patriots have always had another trick up their sleeves in terms of winning extra games. It involves their yards per point (YPP): essentially, how efficiently they turn field position into scores on offense and how inefficiently they force opponents to do the same. By definition, when you have a lower YPP than the opponent, you will win more often because you’re trading field position for points at a more favorable rate than they are.

Like turnover margin, YPP is supposed to be pretty inconsistent from year to year, bouncing around with a team’s luck at picking up key first downs and converting red zone chances, along with the all-important knack for “bending but not breaking” on defense. Yet the Pats dominate this category so thoroughly and so consistently, it might be the single biggest factor in their ongoing success. Not only had they ranked first in both offensive and defensive YPP since 2010, but their net YPP differential of +5.6 was more than double the No. 2 Packers’ +2.5 mark over that span.

(This is one of the big reasons that worries about the Patriots’ defense always need to be tempered. Belichick’s team has traditionally punched above its weight in terms of points allowed, just because it always makes opponents work so hard to turn gains on the field into rewards on the scoreboard.)

This season, the Pats remain among the top net YPP teams, ranking fourth, but they are not quite dominating like usual. They rank just seventh in offensive YPP and sixth on defense, with a net YPP of +2.8, which trails the Bears, Saints and Chiefs. On top of the increase in turnovers per game from above, New England’s efficiency rankings on third down and in the red zone are worse, and the team has slipped in those same “situational” categories on defense. And if you want another cause for the Patriots’ YPP decline, their net starting field position is -2.6 yards per drive this season (meaning the opponent starts 2.6 yards closer to the end zone than the Pats), after a decade in which that number was a league-best +4.6.

In other words, many of the little things that usually add up to that massive YPP advantage for New England aren’t quite working as well so far this year. But the good news for the Pats is that their turnover margin and net YPP tend to improve radically from this point in the season onward, in no small part because Belichick specifically tries to build a tough, physical team that thrives in bad weather. So even in a relative down season by their key indicators, don’t be surprised if the Patriots build them up at least some before season’s end.

Gronk smash!

Tight end Rob Gronkowski has long been the Pats’ not-so-secret weapon on offense, helping the team transition seamlessly from the powerful Randy Moss-Wes Welker offense of a previous era to the version that’s been terrorizing the league for most of this decade.

But the famously fragile Gronk has appeared to show his age and mileage this season more than perhaps ever before. He’s missed three games with various ailments, and when he has played, he’s been limited to just 63.0 yards per game with a career-low 0.25 touchdown catches per contest. Gronkowski’s reduced mobility has hurt his trademark ability to rumble after the catch for spectacular gains, and it’s made him much less of a focal point in the offense than he’s accustomed to being. When on the field, Gronk has seen only 18.7 percent of the targets in the Pats’ passing game, his lowest number since getting 17.7 percent as a rookie.

But Gronk’s influence on the Patriots’ offense remains undeniable. In the eight games the star tight end has played in 2018, Brady’s passer rating is 98.2; in the three he missed, it dropped to 91.6 (league average is 94.9). Even with Gronkowski playing in a more limited physical condition than usual, producing less of a statistical footprint than before, this is confirmation that he’s still one of the biggest engines driving the Patriots’ success. The biggest question might simply be what kind of durability Gronkowski’s banged-up body will have over the rest of the season.

Brady stays ageless … sort of

Along with Belichick, the one constant in New England’s dynasty has been No. 12 under center. Brady has probably been the single most valuable player in the NFL this century, and he’s been crucial in engineering five Super Bowl titles for the Patriots with his consistency, leadership and ability to rally the team back from seemingly insurmountable deficits.

But at 41 — an age at which almost no other QB has ever been productive — there is a near-constant watch for any sign of slippage in Brady’s performance. And he has been a bit less sharp statistically than in years past. His adjusted net yards per attempt index at Pro-Football-Reference.com, which measures passing efficiency relative to the league (where 100 is average), is 111 this year, down from 117 last season and 138 the year before that. It hadn’t been so low since Brady was barely above average (102) in 2013.

Of course, there are reasons for Brady’s decline that go beyond his advanced age, from Gronk’s aforementioned absences to a four-game suspension for top target Julian Edelman at the start of the season and a WR corps in flux early on before adding Josh Gordon and shuffling roles for the likes of Phillip Dorsett and (WR-turned-RB) Cordarrelle Patterson. But Brady has managed to work around weird receiving situations before — and, in fact, his passer rating was better over the season’s first four games (94.0) than it’s been over the four most recent ones (90.8).

Combine that with a ProFootballFocus grade that’s down a bit from last season (though still sixth-best among QBs) and those ubiquitous stats about Brady’s off-target throws (at 22.1 percent, no qualified passer has thrown an errant pass more frequently this year), and it’s fair to ask whether Brady is playing at quite the same level as he did over the past few seasons. Whether because of Brady or the receivers, the Pats are currently tied for eighth in adjusted net yards per attempt — their worst showing since (again) 2013, a season that saw New England fall short in the AFC title game.2


Taken altogether, these numbers reveal a Patriots squad that is not fully playing at the level it’s used to at this stage of the season. And that shows up in big-picture indicators such as Elo or even point differential, where the Pats’ +58 margin is its weakest of the decade through 11 contests. But even so, a lessened version of the Patriots still ranks among the league’s top teams. And as we mentioned above, the Vikings will be a good opponent for Belichick to use as a measuring stick for his roster. According to our combination of matchup quality (i.e., the harmonic mean of the teams’ Elo ratings in each game) and game importance (how likely it is to swing either team’s odds of making the playoffs), this will be the fourth-best game of the week:

The best matchups of Week 13

Week 13 games by ranking of average Elo ratings (using the harmonic mean) plus ranking of total potential swing for the two teams’ playoff chances, according to FiveThirtyEight’s NFL predictions

Playoff % Playoff %
Team A Current Avg. Chg* Team B Current Avg. Chg* Total Change Game Quality
WSH 38.9% +/-19.5 PHI 23.7% +/-11.1 30.6 1525
BAL 46.1 18.0 ATL 4.2 3.1 21.0 1539
DAL 60.3 13.3 NO >99.9 0.1 13.3 1635
MIN 62.8 13.6 NE 99.1 1.0 14.6 1610
LAC 88.2 7.0 PIT 93.9 5.6 12.6 1619
CAR 30.9 14.9 TB 0.7 0.8 15.7 1492
IND 29.5 14.5 JAX 0.1 0.1 14.7 1468
DEN 13.0 10.7 CIN 6.4 5.3 16.0 1453
DET 1.3 1.6 LAR >99.9 <0.1 1.6 1550
SEA 74.9 10.5 SF <0.1 <0.1 10.5 1460
CHI 96.3 3.9 NYG <0.1 <0.1 3.9 1474
HOU 95.7 3.8 CLE 1.3 1.4 5.2 1462
KC 99.9 0.2 OAK <0.1 <0.1 0.2 1479
TEN 20.5 7.2 NYJ <0.1 <0.1 7.2 1417
MIA 5.1 3.9 BUF 1.3 1.3 5.2 1425
GB 6.1 3.0 ARI <0.1 <0.1 3.0 1412

Game quality is the harmonic mean of the Elo ratings for the two teams in a given matchup.

*Average change is weighted by the likelihood of a win or loss. (Ties are excluded.)

Source: ESPN Stats & Information Group

While the game has a lot more at stake for Minnesota, whose spot in the playoffs is still not fully locked in, there is still plenty for the Patriots to play for as well. Not only will this game affect seeding for the postseason (Elo says the Pats currently have a 60 percent chance of securing a first-round playoff bye), but it will also be another telling data point as to whether the Pats can get back to their mega-dominant form of the recent past, or if they’ll be merely good — but mortal — according to their signature metrics.

FiveThirtyEight vs. the readers

If you want to know where your team stands, FiveThirtyEight’s Elo ratings are a good indicator. You can check them out in our NFL prediction interactive, which simulates the rest of the season 100,000 times and tracks how often each team should make the playoffs and win the Super Bowl. Did you know you can also pick against the Elo algorithm in our prediction game? Try it out, and maybe you can climb up our giant leaderboard.

Here are the games in which Elo made its best — and worst — predictions against the reader picks last week:

Elo’s dumbest (and smartest) picks of Week 12

Average difference between points won by readers and by Elo in Week 12 matchups in FiveThirtyEight’s NFL prediction game

OUR PREDICTION (ELO) READERS’ PREDICTION
PICK WIN PROB. PICK WIN PROB. Result READERS’ NET PTS
CIN 75% CIN 63% CLE 35, CIN 20 +13.4
CAR 62 CAR 57 SEA 30, CAR 27 +4.1
CHI 53 CHI 59 CHI 23, DET 16 +2.9
HOU 58 HOU 63 HOU 34, TEN 17 +2.3
NE 77 NE 82 NE 27, NYJ 13 +0.4
PIT 70 PIT 69 DEN 24, PIT 17 +0.0
DAL 65 DAL 67 DAL 31, WSH 23 -0.5
IND 68 IND 70 IND 27, MIA 24 -0.6
NO 81 NO 83 NO 31, ATL 17 -0.9
LAC 84 LAC 85 LAC 45, ARI 10 -0.9
BAL 83 BAL 81 BAL 34, OAK 17 -2.3
TB 63 TB 58 TB 27, SF 9 -5.7
PHI 80 PHI 69 PHI 25, NYG 22 -8.1
MIN 71 MIN 60 MIN 24, GB 17 -10.0
BUF 56 JAX 55 BUF 24, JAX 21 -13.0

Home teams are in bold.

The scoring system is nonlinear, so readers’ average points don’t necessarily match the number of points that would be given to the average reader prediction.

On average, Elo beat our readers by 18.9 points in the game last week, bringing its record to 11 wins and one loss so far this season. Readers had the best pick of Week 12 — rightly pumping the brakes on Cincinnati’s chances of beating the Browns — but they were punished for picking against Elo in the Bills’ upset over the Jaguars, and they didn’t show enough faith in the victorious Vikings, Eagles and Bucs.

Among individual users who did better than average, congrats are in order to Ryan Gnizak, who led all users in Week 12 with 263.5 points, and to Greg Chili Van Hollebeke, who held on to a slim lead for the entire season with 934.5 points. Thanks to everyone who has been playing — and if you haven’t, be sure to get in on the action! You can make picks now and still try your luck against Elo, even if you haven’t played yet.

Check out our latest NFL predictions.