The 2019 Orioles Are One Of The Most Anonymous Teams In MLB History

The 2018 Baltimore Orioles were so bad that we questioned whether they belonged in the major leagues at all. They were our runaway pick for worst pro team of the year, going far beyond the many wannabe Astros and Cubs who’ve jumped on the tanking fad in recent seasons.

Amazingly, things might get even worse this year. Since the middle of last season, Baltimore has traded away established veterans Manny Machado, Jonathan Schoop, Zach Britton, Kevin Gausman, Darren O’Day and Brad Brach, and watched as others such as Tim Beckham, Caleb Joseph and Adam Jones departed in free agency as well. Now there are only three remaining members of the Orioles’ lineup with even two years of MLB service time heading into 2019: Jonathan Villar, Trey Mancini — both average players at best — and Chris Davis, who had arguably the worst individual season in MLB history in 2018 when he hit .168 (in 470 at-bats!) with a .539 on-base plus slugging and -2.9 wins above replacement (WAR).4 Davis’ untradeable contract means the Orioles are stuck with him, so they’ll pencil his name in on opening day no matter how bad he was last season.

And those are the guys in the starting lineup who can be remotely labeled as household names. The rest is filled out with either youngish players who are past “prospect” status or journeymen plucked off the scrap heap. Taken as a whole, the 2019 Orioles’ roster basically recalls this scene from “Major League”:

(No word on whether Baltimore owner Peter Angelos secretly built this team of cast-offs so he can move the team to Miami.)

There have been a few teams who went into a season with less apparent talent than Baltimore — but not many. Using Baseball-Reference.com, we gathered data for each American League team’s opening day lineup since 1973 (to include every team who used the designated hitter full-time) and calculated those players’ established WAR track records going into the season.5 The track records for these Orioles — to the extent they have track records at all — place the team at or near the low-water mark at each position relative to all other AL opening day starting lineups since 1973:

Only three teams in our sample — the 1977 and 1982 Toronto Blue Jays and the 1980 Oakland Athletics — had lower established WAR levels for their starting lineups on Opening Day than the Orioles will have this season.

Baltimore’s place among the worst opening day lineups

Among American League teams since 1973, the four worst opening day lineups according to the sum of players’ established wins above replacement (WAR) levels

1977 Blue Jays 1980 Athletics 1982 Blue Jays 2019 Orioles
Pos Player WAR Player WAR Player WAR Player WAR
C Cerone -0.2 Heath -0.3 Whitt +0.4 Sisco +0.1
1B Ault +0.0 Newman +0.7 Upshaw -0.4 Davis -1.0
2B Garcia -0.6 Picciolo -0.8 Garcia -0.2 Villar +1.8
3B McKay -0.5 Klutts -0.1 Mulliniks -0.3 Nunez +0.9
SS Torres +0.3 Guerrero -0.3 Griffin -1.1 Martin +0.0
LF Scott +0.0 Henderson -0.4 Woods +0.6 Mancini +0.6
CF Woods -0.1 Murphy +0.7 Moseby +0.5 Mullins -0.1
RF Bowling -0.1 Armas +0.3 Barfield +0.7 Hays +0.0
DH Velez +0.7 Essian +1.6 Mayberry +1.8 Santander -0.2
Total -0.5 +1.4 +1.8 +2.1

Established level is calculated as a weighted average of WAR from the previous three seasons.

Sources: Baseball-Reference.com, FanGraphs

One of those teams, the ’77 Jays, was an expansion club that won just 54 games. But the two others finished around .500, meaning there are limitations to predicting off a lineup’s previous MLB track records. Of course pitchers can come to the rescue, as they did for Toronto in 1982 (led by the underappreciated Dave Stieb). Also, young players can emerge in dramatic breakout fashion: Rickey Henderson had done little as a rookie for the 1979 A’s before erupting for 8.3 WAR in 1980, for instance.

But it’s tough to find the next Stieb or Henderson waiting in the wings to save Baltimore this year. De facto staff ace Dylan Bundy had a 5.45 earned run average last season, while the top prospect in the Orioles system, outfielder Yusniel Diaz, is starting the year in the minors and probably won’t be a full-time contributor until 2020. Among those actually in this lineup on opening day, outfielders Cedric Mullins and Austin Hays and catcher Chance Cisco probably have the best breakout potential. In fact, FanGraphs currently projects Mullins for a team-best 1.8 WAR, thanks to a combination of power and speed that could make him one of the few Orioles worth watching this season.

New Orioles general manager Mike Elias told BaltimoreBaseball.com earlier this month that he wants the rebuild to go quickly: “This team finished last last year with a bad record,” he said. “I want to get out of that phase as quickly as possible, and so every decision that we’re going to make is going to be towards accelerating our advancement to be a playoff-caliber team again. I see no reason to stretch that out, drag it out beyond what we have to.”

For now, though, this Baltimore lineup looks like it will battle the Miami Marlins for the saddest collection of mediocre veterans and anonymous prospects in the game. In each case, you’ve probably never heard of half of these guys, and the ones you do know are way past their prime. (Or never had a prime.) We’ll just have to see if this ragtag group can rally together and win the pennant anyway — or more realistically, rally to avoid 115 losses this time around.

How The 2020 Candidates Do With The Voters Who Know Them Best

In politics, the people who really know you are your friends and neighbors. At this stage in the election cycle, many Democratic presidential candidates have low name recognition nationally, which means that their popularity at home might be one way to help us understand whether they can appeal to a larger, national audience. Or if they should consider not running.

To figure out how politicians’ constituents feel about them, we looked at two measures. The first is how popular they were among Democratic voters in their home states, which might be an indicator of their ability to attract support in their party’s presidential primary. The second is how popular they were among all voters in their states compared with the states’ partisan tendencies, which might give us a sense of how effectively the candidates can appeal to the broader general electorate.

When it comes to appealing to both the party and the broader public, Bernie Sanders and Amy Klobuchar get strong marks in their home states. While this doesn’t necessarily mean either will be the Democratic nominee in 2020 — Vermont and Minnesota are not the same as the U.S., of course — it does offer evidence of their potential, particularly for the relatively unknown Klobuchar. Let’s start with approval ratings among Democrats in the table below, where these two senators lead the way.

How Democrats rate candidates from their state

Presidential candidates’ approval ratings among registered voters who identify as Democrats in the candidates’ home states in the final three months of 2018

Among Democrats
Candidate Home State Approval Disapproval Net Approval
Bernie Sanders VT 91.5% 6.2% +85.3
Amy Klobuchar MN 85.9 4.6 +81.3
Elizabeth Warren MA 79.8 10.6 +69.2
Cory Booker NJ 71.1 10.8 +60.3
Jay Inslee WA 71.6 12.5 +59.1
Kamala Harris CA 68.0 10.2 +57.8
John Hickenlooper CO 69.7 14.0 +55.7
Kirsten Gillibrand NY 66.0 11.6 +54.4

Among candidates who have declared they are running and who are included in Morning Consult’s approval polls of senators and governors.

Source: Morning Consult

In the table above, 2020 hopefuls are ranked by their net job approval rating (approval rating minus disapproval rating) among Democrats in their state. The data was collected by Morning Consult in the last quarter of 2018; every quarter, the pollster reaches out to hundreds of thousands of people to create its rankings of “America’s most and least popular” senators and governors. Of course, not all the declared Democratic presidential candidates were either a senator or a governor in the closing months of last year, which means that a few hopefuls have been left out, including former San Antonio Mayor Julian Castro and U.S. Rep. Tulsi Gabbard of Hawaii.

Sanders and Klobuchar scored these high marks among Democrats in their home states while also being the best-known — based on the share of people who were able to form an opinion of them (approval rating plus disapproval rating). Notably, despite being so well-known, both Sanders and Klobuchar had disapproval ratings below 10 percent among Democrats, unlike the other six contenders. Sanders is already well-known nationally from his 2016 presidential bid, but that he remains beloved by Vermont Democrats — even as he has technically served as an independent — suggests that he hasn’t lost ground despite continuing to turn his gaze toward national politics. As for Klobuchar, she doesn’t have Sanders’s national profile, and only 2 percent or 3 percent of Democratic voters named her as their pick in the latest polls of the primaries. Yet, her net approval rating among Minnesota Democrats of +81.3 percentage points is nearly as high as Sanders’s (she easily won all three of her Senate races even though Minnesota is not nearly as blue as Vermont). So we shouldn’t underestimate her ability to make inroads among Democrats around the country once she becomes better-known.

The remaining candidates — except for Sen. Elizabeth Warren, whom we’ll talk about in a minute — had net approval ratings between +54 and +60 points. That suggests that they’re relatively popular among their party bases but not as overwhelmingly popular as Sanders and Klobuchar. Part of that may be because most of them simply aren’t as well-known among their own constituents. Each of these candidates will be working to raise their profiles nationally in the coming months, but support at home could benefit at least one of them in a relatively early primary contest: California has more delegates available than any other state, and Sen. Kamala Harris is very popular there, and already leading in at least one poll.

While home-state popularity within the party may indicate strength in the primaries, winning the general election requires broader appeal. So we also looked at candidates’ approval ratings among all voters and how those numbers compare to the partisan lean of their states.1 This shows us how much more or less popular a candidate is than we might expect based on the political makeup of their state.

Along with former Colorado Gov. John Hickenlooper, Klobuchar and Sanders again led the way, as you can see in the table below. Because Minnesota is a purple state, its electorate includes a higher percentage of Republicans and independents than heavily Democratic Vermont’s does, so Klobuchar has more non-Democrats available to win over than someone like Sanders, which may make it easier for her to top this particular ranking. But the fact that her overall net approval rating far exceeds Minnesota’s slight Democratic lean shows that she is successfully appealing to voters outside her base. The same could be said of Hickenlooper. Democratic candidates who are able to appeal to independents and Republicans in purple states might be better able to win over the same types of voters around the country, having had to win some of those voters to get elected in the first place.

Who’s getting more home-state support than their party?

Presidential candidates’ net approval rating among registered voters in their home state in the last quarter of 2018 vs. the partisan lean of that state

candidate Home
Candidate state Partisan lean Net Approval at home home net approval vs. lean
Amy Klobuchar Minnesota D+2.1 +31.6 +29.5
John Hickenlooper Colorado D+1.5 +18.5 +17.0
Bernie Sanders Vermont D+24.1 +35.9 +11.8
Cory Booker New Jersey D+13.3 +14.3 +1.0
Jay Inslee Washington D+11.6 +11.7 +0.1
Kirsten Gillibrand New York D+22.0 +18.8 -3.2
Kamala Harris California D+23.7 +14.0 -9.7
Elizabeth Warren Massachusetts D+29.4 +15.4 -14.0

FiveThirtyEight’s partisan lean metric is the average difference between how a state votes and how the country votes overall, where 2016 presidential election results are weighted 50 percent, 2012 presidential election results are weighted 25 percent and results from elections for the state legislature are weighted 25 percent.
Among candidates who have declared they are running and who are included in Morning Consult’s approval polls of senators and governors.

Source: Morning Consult

Hickenlooper’s situation is particularly interesting because even though he ranked lower in net approval rating among home-state Democratic voters, his net approval rating among Colorado voters overall was strong, especially relative to the partisan lean of Colorado, which is a highly competitive state. Of the eight Democratic presidential candidates we’re looking at, Hickenlooper had the best net rating (-17.1 points) among Republicans in his home state. This might not bolster Hickenlooper’s appeal to the left wing of the Democratic Party nationally, but he could perhaps use those numbers to argue that he has the potential to broadly appeal to a general electorate and even chip away at the small share of Republicans who may be willing to oppose President Trump.

Warren stands out here. Although the senator had a strong net approval rating among Democrats in her home state in the first table above, her net approval rating among Massachusetts voters overall is weak relative to how Democratic her state leans. (She also underperformed her state’s partisan lean in her re-election victory last year.) That’s because her net approval ratings among home-state independents (-0.3 points) and Republicans (-61.4 points) were the worst among the eight candidates we’re examining here. That could be a sign that Warren will encounter problems when trying to appeal to the broader electorate. Maybe it’s not surprising then that in a recent Gallup poll of U.S. adults, Warren had the highest unfavorable rating of the seven Democratic presidential candidates whom respondents were asked about.

Although home-state approval ratings may not end up proving predictive of how the 2020 Democratic primary and general elections turn out, they may offer us some insight into the national candidacies of people who aren’t well-known across the country — like Klobuchar — or provide clues about the potential strengths or weaknesses of candidates who are more recognizable — like Warren. If the folks who know you best really like you or don’t like you as much as we’d expect, that might be a clue for how voters nationally will receive you.


From ABC News:
Sen. Amy Klobuchar launches 2020 presidential campaign


Check out all the polls we’ve been collecting ahead of the 2020 elections.

How Many Numbers Contain The Numbers Of Their Numbers?

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,3 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 Daz Voss, numbers of numbers with numbers of their numbers:

The number 21322314 acts as its own inventory. That is, it contains two 1s, three 2s, two 3s and one 4. Another example is 22 — it contains two 2s.

These numbers consist of alternating tallies and numerals. First comes the tally, then the numeral being tallied, then another tally, and so on.

How many numbers of this kind exist?

(Assume the numerals have to be tallied in increasing order, so you can’t create new numbers simply by rearranging: 21321423, for example, doesn’t count.)

Submit your answer

Riddler Classic

From Michael Kragh Pedersen, some shapely game theory:

You are playing a game against a single opponent. In front of you is an empty eight-by-eight grid and a pile of the 12 different pentominoes, one of each.

Taking turns, you and your opponent select a pentomino from the pile, rotate it however you like (but not flip it over) and place it anywhere within the grid. By rule, the pieces can’t overlap or extend outside the grid. The person to place the last possible piece wins.

Assume you go first. What is the optimal strategy? How does this game end with perfect play?

Submit your answer

Solution to last week’s Riddler Express

Congratulations to 👏 Ricki Heicklen 👏 of Teaneck, New Jersey, winner of last week’s Riddler Express!

Last week brought us to the virtual Wild West of the video game “Red Dead Redemption 2.” In that game, there is a quest where the main character is meant to collect 12 unique cigarette cards from each of 12 different sets. The easiest way to collect these cards is to buy packs of cigarettes from the general store at $5 a pop, each pack containing one random card from one of the sets. How much would we expect our main character to spend to complete all 12 sets?

It will cost nearly $4,000 — quite a lot in those days.

This is an example of the classic coupon collector’s problem. There are 144 different cards the character wants to collect, each of which he’ll have a 1/144 chance of receiving each time he buys a pack. So how long will this take? The main problem for this main character — and the problem we need to quantify — is that sometimes, or even oftentimes, he’ll buy a pack containing a card he already has. This becomes more and more likely the more cards he has collected.

Let’s go card by card. The first card will be collected in one purchase for sure — the character doesn’t have any cards yet and doesn’t care which one he gets. In his subsequent purchases, the second card will be collected with probability 143/144, the third card with probability 142/144, and so on. It gets less and less likely we get a new card the more cards we already have. Once we have all but one card, each purchase only gives us a 1/144 — or 0.7 percent — chance at completing our quest.

We can rewrite the expected number of purchases this will take as the following:

144 * (1/1 + 1/2 + 1/3 + … + 1/144)

That equals about 799.27 expected purchases to collect all 144 cards. At $5 a purchase, that’s about $3,996.

Collecting cigarette cards … in this economy?!

Solution to last week’s Riddler Classic

Congratulations to 👏 Michael Branicky 👏 of Lawrence, Kansas, winner of last week’s Riddler Classic!

Last week you were home alone playing a solitaire game of “Bananagrams.” You spread the game’s 144 lettered tiles on the table in front of you and wondered, “What grid of words can I create that uses all of these tiles in the fewest possible words?”

Wonder no longer: You can use them all in just nine words.

Solver Michael Branicky found the 13-word candidate solution pictured below — bonus points to Michael for the actual Bananagram tiles. It features the lovely, 28-letter ethylenediaminetetraacetates, a chemical apparently used to dissolve limescale.

Solver Laurent Lessard was able to do a bit better by modeling the puzzle as something called a mixed-integer program. He found the nine-word grid below. And, though this grid is not the only nine-word grid — at the very least, you can shift the words below around to overlap on different points — it is the one made with the fewest words possible. It features those classic pieces of kitchen-table “Bananagrams” vocabulary: keratoconjunctivitides, paraformaldehyde, magnetofluiddynamics and oxyphenbutazones. (Notably, “oxyphenbutazone” also features in the theoretically highest possible scoring “Scrabble” play.)

I also asked what completed grid used the most words. Michael is our winner for how he tackled that question, offering this grid that uses 109 words — many of which are those “Scrabble” classics re or od.

Want more riddles?

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

Want to submit a riddle?

Email me at [email protected]

Significant Digits for Friday, March 8, 2019

You’re reading Significant Digits, a daily digest of the numbers tucked inside the news. For even more facts, figures and discussion, check out our live FiveThirtyEight Politics podcast in New York City on March 20.


2 times as often

Philadelphia has become the first major U.S. city to ban cashless stores, which have become a mini-retail fad in recent years. Stores say it saves them time; the city says it locks out poorer residents. The poorest Americans are nearly twice as likely to use cash as the richest ones. [Wall Street Journal]


48 percent of cities

Marijuana has become legal in more places, but sports leagues haven’t done much to change their drug policies. Except the NHL. Hockey players are essentially no longer being punished for positive tests of THC, perhaps because 48 percent of NHL cities now have laws allowing recreational marijuana. [ESPN]


29 percent of top recruits

You and I were never going to make it to the NBA. But one would think top high school recruits had a good shot. Yet only 29 percent of top-100 high school recruits between 1998 and 2013 were drafted in the NBA, and fewer still went on to have halfway decent careers in the league. I guess survival of the fittest still applies to the fittest. [The Pudding]


170 people charged

As we wait for special counsel Robert Mueller to bring his investigation to a close — at least that’s what the rumors keep telling us is coming — this is a good moment to put Mueller’s probe in context. My colleagues looked at all the special or independent counsel investigations since Watergate began, which have charged 170 people with crimes. Mueller’s probe has been responsible for 34 of those charges. [FiveThirtyEight]


1 fewer Democratic candidate

For a while there it seemed like no Democrat would pass up the opportunity to run for president. But on Thursday, Ohio Sen. Sherrod Brown announced that he would not run for the Democratic nomination in 2020. He’s not alone — several other people have said recently that they’re eschewing a campaign. My FiveThirtyEight politics colleagues discussed what that says about the field. [FiveThirtyEight]


47 months

Paul Manafort, President Trump’s former campaign manager, was sentenced to 47 months in prison by a U.S. district court judge on Thursday. Manafort still has another sentencing hearing, in which he could receive an additional 10 years of prison time. [The Washington Post]


From ABC News:
News headlines today: Mar. 8, 2019



Love digits? Find even more in FiveThirtyEight’s book of math and logic puzzles, “The Riddler.”

If you see a significant digit in the wild, please send it to @ollie, who is on vacation, just to taunt him.


PSG’s Collapse Completes A Week Of Champions League Mayhem

We write to you with elbows tucked, stroopwafels out — and a new understanding of what’s possible in soccer. Just half of the second-leg games in the Champions League’s round of 16 are complete, and already two titans of the sport are gone thanks to a youthful rebellion and instant replay. The prediction models are as surprised as the rest of us.

First, to Real Madrid’s dismantling Tuesday. Coming into the second leg of its tie against Ajax, Madrid led 2-1 and had a 75 percent chance of moving on in a tournament the club has won four of the past five years. Ajax was technically better than Madrid in the first leg, and our model gave the Dutch team some respect as a result. But it’s one thing to be Madrid’s equal and another to beat it 4-1 at the Bernabeu, exposing seemingly each of Los Blancos’ flaws since they lost Cristiano Ronaldo to Juventus at the end of last season. Led by homegrown phenoms Frenkie de Jong and Matthijs de Ligt, Ajax played the platonic ideal of free-flowing soccer, pirouetting through the midfield and capitalizing on Madrid’s missing captain, Sergio Ramos, who purposely got himself booked at the end of the first leg so he could be fresh (and cardless) for the quarterfinals. Instead of living to fight another day, Ramos serves as a cautionary tale.

Ajax’s win was surprising; Manchester United’s was almost inconceivable. Heading into its second leg against Paris Saint-Germain, Man U had just a 3 percent chance of moving on — and its play on Wednesday showed why. United lost the possession game 72 percent to 28 percent and trailed substantially in shots. This did not appear to be a team that could overcome a 2-0 deficit from the first leg. Yet PSG gifted Man U two goals on defensive errors and could manage only one first-half goal of its own, putting United just one goal from winning on away goals. But Man U couldn’t gain much possession, let alone break through for a goal. But then the replay gods took pity. In the 90th minute, United’s Diogo Dalot fired the ball in the general direction of PSG’s goal, and PSG’s Presnel Kimpembe leapt to block it once it had crossed into the box. His elbow came with him, and as it separated from his chest, it knocked into the ball. The referees initially called a corner but then went to video review. The (controversial) judgment changed the call to a handball. A penalty shot was awarded, and Marcus Rashford converted it for a tie-breaking third road goal. The final whistle blew a few minutes later, and Manchester United had somehow pulled off a miracle.

Two other games happened, we’ve been told. Porto had its own late drama Wednesday against Roma, including another penalty awarded on video review; the Portuguese squad wasn’t favored to move on heading into the match. And on Tuesday, Tottenham finished off Borussia Dortmund as calmly as Harry Kane finishes his penalty kicks.

All of this leaves some havoc in our projections, and we still have four games to go in this round. Buckle up, and please keep your elbows inside the vehicle at all times.

Check out our latest soccer predictions.

How Cable News Reacted To The Cohen Hearing

Not everyone has time to watch C-SPAN for five-and-a-half hours in the middle of the week. Not even to watch President Trump’s former lawyer and fixer Michael Cohen call Trump a “racist,” “con man” and “cheat,“ as happened on Wednesday. And not even to watch Cohen be forcefully questioned by Republicans in response.

As such, we rely on the news media to watch for us. But the media is not a monolith. How an outlet condenses a big event like the Cohen hearings can shade how its audience interprets the events. And when it came to cable news, the networks differed in their coverage of the hearing’s aftermath, as you might expect. But an analysis of how the words used by each network differed is a window into how they’re framing the threats to Trump’s presidency. MSNBC, for example, appeared particularly focused on the legal implications of the hearing — on Robert Mueller and prosecutors. CNN was heavy on issues of credibility, money and payments, and the claim by Cohen that Trump is a “racist.” And Fox News was especially focused on other news altogether, namely what was happening thousands of miles away, where Trump was sitting down with Kim Jong Un.

Certain specific words gave the Cohen hearing these flavors on each of the three cable networks. Using data from the Internet Archive’s Television News Archive and processed by the GDELT Project, we analyzed the coverage of Cohen on CNN, Fox News and MSNBC from 5 p.m. to midnight the day of the testimony.1 To suss out any differences in the networks’ coverage, we first looked at when “Cohen” was spoken and which other words were said within the same 15-second window. (That’s the size of the clips we can access from the data sources.) Then we looked at the 200 most-used Cohen-adjacent words across the three networks and isolated the 15 words that were most particular to each network. (By most particular, we mean the words that were used relatively more often in a network’s Cohen coverage.)2 You’ll see those words plotted on the chart below; we arranged each word by what percentage of clips that used that word came from each network. For example, of all the Cohen-related clips mentioning the word “summit,” 80 percent were on Fox News, 15 percent were on MSNBC and 5 percent were on CNN.

The words close to each network’s corner of the coverage triangle are the ones most specifically associated with its coverage. For CNN, “certainly,” “credibility” and “racist” stood out. Fox News was notable for its use of the word “summit” — presumably in reference to Trump’s meeting with North Korean leader Kim Jong Un, which happened around the same time as the Cohen hearing. And MSNBC’s coverage was distinguished by its talk about “prosecutors” and “Mueller.” (Words in the center, such as “news,” were used a lot but not especially favored by any network in particular.)

And the qualitative flavor of the coverage varied widely as well. CNN talking head Chris Cillizza baldly declared “winners” and “losers” from the hearing. The former included the performance during the hearing of U.S. Rep. Alexandria Ocasio-Cortez of New York — “and man, did she nail it.” Ocasio-Cortez’s interrogation of Cohen was praised elsewhere for being “well thought out.” The latter included Mark Meadows, the chairman of the House Freedom Caucus, who was “out for blood,” revealing little in his questions beyond his contempt for Cohen.

On the other hand, other coverage suggested that the hearing was merely a tool for Trump’s opponents and that given Cohen’s history of lying, the whole thing was something of a farce. The day after the hearing, the morning show Fox & Friends, for example, went meta, declaring that the media “misses the mark.” “This is what you get when you have partisan political operatives masquerading as journalists,” said Ned Ryun, a Republican strategist and the show’s guest. “They can barely control their glee.” He went on to call it “theater of the absurd” and a “total clown show.”

Fox & Friends then stepped hard on the FiveThirtyEight brand, comparing data on the amount of time the cable news networks had spent on Cohen versus the U.S.-North Korea summit during the run-up to the hearing, lamenting the fact that the other networks had given far more airtime to the former. “There’s a reason they call us fair and balanced,” a host said. (FiveThirtyEight has not independently verified those numbers.) The summit fell apart early, and no deal was reached.

There will be more political events in the weeks and months to come. The cable networks’ coverage surely won’t march in lockstep on those, either. We’ll be watching.


From ABC News:


How John Hickenlooper Could Win The 2020 Democratic Nomination

Former Colorado Governor John Hickenlooper announced on Monday that he is running for president, saying that he “can bring people together to produce the progressive change Washington has failed to deliver.”

Since being laid off from his job as a geologist in 1986, Hickenlooper’s luck has only improved. He and three friends founded a craft brewery in Denver’s LoDo neighborhood, helping bring development to a now-thriving part of the city. In 2003, Hickenlooper used that success as a springboard to be elected mayor, and then won re-election in 2007 with 87 percent of the vote. He then steered Colorado through disasters, tragedy and a recession for two terms as governor, leaving office earlier this year with an enviable legacy and a solid approval rating to boot. Now, he’s hoping to channel that success into becoming president of the United States. But in a field of Democrats who are both better known and have more obvious constituencies, is this where Hickenlooper’s climb stops?

Hickenlooper’s campaign is basically starting from scratch. In early surveys of the Democratic field — which mostly reflect name recognition at this point — he is polling between 0 and 1 percent.5 Only 22 percent of Democratic respondents even have an opinion of him, according to an average of national favorability polls since the beginning of the year. And he’s not on the radar of many Democratic activists in early states, either.

What Hickenlooper does have going for him is that he may be more skilled than most at getting his name out there. He’s run some of the best political ads in recent memory, including endearing spots about parking meters and his humble wardrobe, which helped him stand out in a wide-open field (sound familiar?) during his first run for mayor. And later ads in which he went skydiving and took a shower — fully clothed — were nothing if not memorable. His quirky personality was his secret political weapon in Colorado, but it’s unclear how it’ll shake out on a national stage, where his demographics — older, white, male — may pigeonhole him as a retread.

But even Hickenlooper himself admits that his moderate image is likely to be a problem in a Democratic presidential primary. The party’s influential left wing will probably view him with suspicion over his opposition to anti-fracking efforts, sympathies with the oil and gas industry (which once employed him) and wishy-washy comments over one of his signature progressive achievements, a gun-control package passed in the wake of the Aurora theater shooting. (Hickenlooper’s reported flirtation with running for president on a “unity ticket” with Republican Ohio Gov. John Kasich probably won’t help, either.)

Although his tenure as governor was marked by economic success and liberal wins on social issues, his style of getting things done was undeniably bipartisan (perhaps out of necessity, given that Republicans controlled one chamber of the legislature during six of his eight years in office). Indeed, the only time Hickenlooper’s popularity as governor faltered was when he swerved left in 2013, signing those gun-control bills and staying the execution of convicted murderer Nathan Dunlap. He pivoted back to the center for 2014 and became Colorado’s only statewide Democrat to survive that year’s Republican wave. As he was preparing to leave office in late 2018, his approval/disapproval spread stood at 49 percent to 30 percent, giving him a +18 PARG — Popularity Above Replacement Governor. That’s nerd-speak for “a very high approval rating in a politically divided state.” Clearly, Hickenlooper is a consensus-builder, but it’s hard to imagine his motto of “there’s no profit margin in making enemies” resonating with the current mood of the Democratic electorate.

As FiveThirtyEight editor-in-chief Nate Silver has written, Hickenlooper follows in the tradition of other politicians from the American West — candidates who wear cowboy boots and stand amid the great outdoors in ads espousing a nonideological, anti-politician message. That sort of branding has obvious appeal in the Mountain West, but Hickenlooper will have to expand his support elsewhere to win the nomination. In the 2016 primaries, the mountain states6 accounted for just 7 percent of pledged Democratic delegates, a function of their small size and Republican lean. Not to mention that Democrats have never nominated a presidential or vice-presidential candidate from west of the Central time zone in their 191-year history as a party.

Even more troubling, it’s hard to point to a clear base for Hickenlooper — at least one big enough to propel him meaningfully in a national primary. Contrary to his folksy image, the former big-city mayor doesn’t have a great track record of appealing to rural areas. As governor, his administration’s renewable-energy and gun policies alienated some rural counties so much that they symbolically voted to secede from Colorado. And he doesn’t do very well in our five-corners formulation of thinking about the Democratic primary field either:

Perhaps his penchant for viral videos will make him a favorite among millennials; the craft beer lover and banjo player already has a touch of hipster cred. A smarter strategy might be leaning into being the “cannabis candidate.” As governor of the first state where marijuana sales became legal (in 2014), Hickenlooper oversaw the law’s implementation and has nurtured a thriving cannabis industry. But while he hasn’t been shy about touting the benefits of recreational pot, activists may not be willing to forget that he initially opposed legalization in 2012.

Hickenlooper’s campaign might also plausibly try to leverage his background in geology; science, technology, engineering and mathematics professionals are increasingly political engaged and even have their own PAC. But so far, science as a policy issue doesn’t seem to be resonating with voters. When we researched the win rates of various types of candidates in 2018 Democratic primary races, we found that candidates with a STEM background had won just 23 percent of their primaries to that point, compared with 33 percent of non-STEM professionals.

Perhaps the strongest card in Hickenlooper’s hand is his status as a former governor; historically, they have better track records than members of Congress at being nominated for and elected president. But that is no guarantee of future success, and Hickenlooper starts the campaign a clear underdog. Once again, the self-made man will have to lift himself up from nothing to prevail.


From ABC News:


LeBron And The Lakers Have Hit A Low Point

On the one hand, the Los Angeles Lakers’ loss to the Milwaukee Bucks on Friday night wasn’t the most surprising thing. After all, the Bucks — who staged a late run to earn the victory in Los Angeles — own the NBA’s best record and have a leading MVP candidate in Giannis Antetokounmpo.

On the other hand, the Lakers surrendering a 15-2 run — and the lead — over the final three minutes of play may have put the team’s back against the wall in an entirely new way.

With the defeat, LeBron James and the Lakers find themselves staring at just a 14 percent playoff probability in FiveThirtyEight’s NBA projection model, the lowest mark they’ve had all season, and a damning scenario given that there are only 20 games left in the campaign. That 14 percent figure is an enormous drop-off from even a week ago, when the club held 25 percent odds to get in. (Three weeks ago, the Lakers’ number was 41 percent.)

But a number of realities are setting in now. The Lakers are 4 games behind the Los Angeles Clippers for the seventh seed and 3.5 games back of the San Antonio Spurs, who own the head-to-head tiebreaker (meaning their lead is more like 4 games, since the Lakers would miss out on the postseason if they were to finish with the same record as San Antonio). Perhaps the most disheartening thing, aside from having a lot of ground to make up, is the fact that the other teams vying for the last two spots have much easier remaining schedules.

Our projections surmise that it will ultimately take about 44 victories to earn a spot in the Western Conference playoffs. In order to reach that win total, the Lakers would need to finish 14-6 against the NBA’s 10th-toughest remaining schedule — one that has 10 home games and 10 road ones. They still have to play the Denver Nuggets, Boston Celtics, Bucks again, Golden State Warriors and Oklahoma City Thunder, among others.

By contrast, the indestructible Spurs need to go only 10-9 to finish with 44 wins. They have an easier-than-average slate the rest of the way, with 11 of their last 19 games in San Antonio. The Clippers have it even better, needing a 9-9 finish to get to 44 victories, with 12 of their last 18 contests at home. (The young, fun Sacramento Kings are positioned in about the same spot as the Lakers in the standings, needing a 13-7 finish to reach 44 wins. But their remaining schedule is the third-easiest in the NBA, giving them some hope in an uphill battle.)

James has faced late-season pressure to lift his team out of the doldrums each of the past few seasons. But this scenario with the Lakers stands apart, both because of how much time he missed with injury (one that now looks as if it will cost the team a playoff spot), and because of how the young supporting cast struggled to hold the rope during his absence, going 6-11. It’s one thing to coast into the postseason, the way James’s Miami and Cleveland clubs often did. But James himself hasn’t missed the playoffs in 14 years, not since the 2004-05 season.

If there’s a bright side, it’s that the Lakers finally look engaged. They held Antetokounmpo to just 16 points, one of his lowest-scoring outputs in a dominant season. Youngster Brandon Ingram has showcased his scoring ability lately and was unstoppable Friday, finishing with 31 points.

But the time to celebrate moral victories for this team has run out, unfortunately. A sixth-straight season of missing the playoffs — especially now, after adding one of the league’s all-time greats — would be disastrous. And after Friday’s loss, the Lakers are staring directly at that possibility.

Meet The NCAA’s 5-Foot-9 Scoring Machine

No state has a glossier tradition in men’s college basketball than North Carolina. It has produced 13 national championships.1 It is home to the sport’s fiercest rivalry: Duke vs. North Carolina. It has seen more than a hundred All-Americans and dozens of Hall of Famers cut their teeth on its courts, including Tim Duncan, Vince Carter and Michael Jordan.

You might think that the state’s all-time leading scorer is one of those all-time greats. But he’s actually a current player. He’s not a Tar Heel or a Blue Devil or even a Demon Deacon. He’s a Fighting Camel in Buies Creek, North Carolina. Far from the spotlight, Chris Clemons, a senior at Campbell University in the Big South Conference, is making college-basketball history in a state that’s long on it.

As a high schooler, Clemons was an all-state athlete in Raleigh, smack dab in the backyard of the ACC. When the power conferences overlooked him, Campbell didn’t. In fact, it sometimes brought the entire coaching staff to watch him hoop. “I told him when I was sitting in his living room that I thought he’d be the all-time leading scorer in Campbell history,” Campbell head coach Kevin McGeehan told me.

Only five players in NCAA history have scored more than Clemons has. And his 3,106 points have not come easy: Clemons is 5-foot-9, at least 4 inches shorter than each of his teammates. But that disadvantage wasn’t a problem for McGeehan. “I’ve never thought twice about his height,” he said.

Chris Clemons is the shortest of the scoring greats

The top 25 NCAA men’s basketball scorers and their height

Player School Years Points Height
Pete Maravich LSU 1967-70 3,667 6’5″
Freeman Williams Portland State 1974-78 3,249 6’4″
Lionel Simmons La Salle 1986-90 3,217 6’7″
Alphonso Ford Mississippi Valley State 1989-93 3,165 6’1″
Doug McDermott Creighton 2010-14 3,150 6’8″
Chris Clemons Campbell 2015-19 3,106 5’9″
Harry Kelly Texas Southern 1979-83 3,066 6’7″
Keydren Clark St. Peter’s 2002-06 3,058 5’9″
Hersey Hawkins Bradley 1984-88 3,008 6’3″
Mike Daum South Dakota St. 2015-19 3,006 6’9″
Oscar Robertson Cincinnati 1957-60 2,973 6’5″
Danny Manning Kansas 1984-88 2,951 6’10”
Alfredrick Hughes Loyola Chicago 1981-85 2,914 6’5″
Elvin Hayes Houston 1965-68 2,884 6’9″
Tyler Hansbrough North Carolina 2005-09 2,872 6’9″
Larry Bird Indiana State 1976-79 2,850 6’9″
Otis Birdsong Houston 1973-77 2,832 6’3″
Kevin Bradshaw U.S. International* 1984-91 2,804 6’6″
Allan Houston Tennessee 1989-93 2,801 6’6″
J.J. Redick Duke 2002-06 2,769 6’4″
Hank Gathers Loyola Marymount* 1985-90 2,723 6’7″
Tyler Haws BYU 2010-15 2,720 6’5″
Reggie Lewis Northeastern 1983-87 2,709 6’7″
Daren Queenan Lehigh 1984-88 2,703 6’5″
Byron Larkin Xavier 1984-88 2,696 6’3″

*Transferred from another school

Source: Sports-Reference.com

With Clemons on the court, the Fighting Camels score 1.12 points per possession, according to Hoop Lens — or at a rate that would be among the best in the country if maintained over an entire season. When he sits, the team scores 0.84 points per possession, or at a rate that would rank around 300th if maintained. Without Clemons, the team’s effective field goal percentage drops more than 10 percentage points, and the team’s turnover percentage spikes almost 7 percentage points. As a pick-and-roll ball handler, he has scored more points this season than everyone else on the team combined, according to data provided by Synergy Sports.

This season, Clemons is scoring 30.1 points per game, tied for the second-highest mark by any player since 1992. Charles Jones in 1996-97 was the last player to average more than 30 points per game over a season, while Central Michigan’s Marcus Keene came close in 2016-17, hitting 29.97. Clemons has scored in double figures in 111 consecutive games.2

He can also take the roof off an arena. Behind a 44-inch vertical3 and sculpted physique, Clemons has made a habit of dunking on defenders. Hard. Of Campbell’s 22 dunks this season, he has eight. “It’s like, ‘You’re 5-9!?’” McGeehan said, bewildered.4

McGeehan hasn’t wasted a second of his point guard’s eligibility. Clemons has played at least 80 percent of team minutes each season on campus. As a senior, he’s taking on 93.1 percent.

Over the past three seasons, Clemons has finished no lower than seventh nationally in KenPom’s percentage of possessions used metric, which measures how many possessions a player used while on court, assigning credit or blame to the player when his actions resulted in an ended possession. He leads the country this season, absorbing 37.5 percent of the team’s possessions. “Obviously,” McGeehan noted, “he’s heavily relied upon.”

Clemons is leaned on heavily

The NCAA men’s basketball players with the highest share of their team’s possessions that season, since 2016-17

Season Player Team Yr. Possession Height Weight
2017-18 Trae Young Oklahoma Fr 38.5% 6’2″ 180 lb.
2016-17 Ronnie Boyce San Francisco Sr 37.9 6’3″ 158
2017-18 Tiwian Kendley Morgan St. Sr 37.8 6’5″ 190
2016-17 Jordan Washington Iona Sr 37.6 6’8″ 235
2018-19 Chris Clemons Campbell Sr 37.5 5’9″ 180
2016-17 Michael Weathers Miami (Ohio) Fr 37.5 6’2″ 161
2018-19 Lamine Diane Cal St. Northridge Fr 37.1 6’7″ 205
2016-17 Chris Clemons Campbell So 37.0 5’9″ 180
2016-17 Marcus Keene Central Michigan Jr 37.0 5’9″ 175
2017-18 Roddy Peters Nicholls St. Sr 36.8 6’4″ 195
2018-19 Ja Morant Murray St. So 36.2 6’3″ 175
2018-19 Jordan Davis Northern Colorado Sr 36.2 6’2″ 185
2018-19 Justin James Wyoming Sr 36.0 6’7″ 190
2017-18 D’Marcus Simonds Georgia St. So 35.8 6’3″ 200
2017-18 Ethan Happ Wisconsin Jr 35.6 6’10” 235
2017-18 Milik Yarbrough Illinois St. Jr 35.5 6’6″ 230
2018-19 Markus Howard Marquette Jr 35.3 5’11” 175
2018-19 Ethan Happ Wisconsin Sr 35.2 6’10” 237
2016-17 Tiwian Kendley Morgan St. Jr 35.1 6’5″ 190
2017-18 Chris Clemons Campbell Jr 34.9 5’9″ 185

Source: KenPom

Should his average hold, Clemons will finish the season with a 38.9 percent usage rate, which would give him three of the top 25 marks produced by players since 2009, the first year for which data is available.5 His career usage rate (35.9 percent) figures to edge former BYU star Jimmer Fredette for the top spot by any player over the past decade.

There have actually been plenty of undersized, high-usage players on college courts. Clemons is just relied upon much more than any of the rest — and has been since he arrived on campus. Since 2008, there have been 450 seasons that saw a player shorter than 5-foot-10 appear in 20 games and play at least 40 percent of team minutes, according to BartTorvik.com. Of that sample, Clemons is in line to finish with the first, third, fourth and 23rd highest usage rates.

Let’s remove the height restriction. Given those same qualifications, since 2008 there have been 522 seasons of a player producing a usage rate exceeding 30 percent. Of that pool, Clemons is on track to post two usage rates that rank in the top 15 and another one that ranks in the top 45.

This season, only one player6 has taken a higher share of team shots than Clemons’s 39.3 percent.7 This would be the fourth consecutive season that Clemons has finished in the top 25 in percentage of team shots taken, a feat of production that has never been matched since KenPom began tracking such metrics.8

The Fighting Camels made their only NCAA Tournament appearance in 1992, five years before Clemons was born. This season, Clemons has Campbell roaring into shape to make a run at a second, provided the team wins its conference tournament.

Which is to say: Clemons has the perennial green light to do everything. “I always had one,” Clemons said. “But I don’t remember it being like this.”

Have coaches ever told him not to shoot? “I haven’t heard that lately,” he said.

Is there anything McGeehan wouldn’t trust Clemons to do? “Probably give me a haircut.”

Clemons’s numbers are more impressive when you consider the system under which he plays. Watching him, you get the sense that he could get to the rim or drain a jump shot around 10 seconds before he actually does. McGeehan runs a hybrid version of Pete Carril’s Princeton Offense that relies on quick ball movement and constant motion. It also typically suffocates tempo, as the Princeton Offense is ostensibly designed to limit huge production by either team. Campbell has ranked outside the top 190 in average offensive possession length each season since 2013. This season, it ranks 265th in adjusted tempo. Campbell averages 68.4 possessions per 40 minutes, which is tied for 238th nationally. North Carolina, for instance, averages nearly eight more possessions per game.

Despite all of his success, Clemons may not ever reach the next level. To be sure, there have been other players under 5-foot-10 to reach the NBA: Muggsy Bogues, Earl Boykins, Calvin Murphy, Spud Webb. There have even been a few who have done it since the turn of the century: Isaiah Thomas, Kay Felder, Nate Robinson. Clemons, who has declared for the NBA draft after each of the past two seasons only to return to campus, is unlikely to join them via that route.

The irony is heavy: One of the greatest scorers of all time, in arguably the state with the richest college basketball tradition, plays just a short drive down the road from perhaps the most-talked-about college athlete of all time, for a team that packed in fewer than 2,000 fans per home game last season. The smallest guy on the court is dealt the largest offensive burden and has most of the team’s dunks. The pint-sized player with the larger-than-life impact.

“When it’s all said and done,” McGeehan said, “I’m probably going to go back and say, ‘Wow, did that really all just happen?’”

You’re Home Alone. You Can Buy Zillions Of Video Game Cigarettes Or Beat Yourself At Bananagrams

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,9 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 Gerald Dorrer, I reckon this here’s a math puzzle:

In the video game “Red Dead Redemption 2,” there is a side quest where the main character is supposed to collect 12 sets of cigarette cards, each consisting of 12 unique cards.

Some cards can be found lying around in the open world, but the easiest way to collect the cards is to buy cigarettes at the store and collect the single random card included in each pack. Suppose Arthur is too lazy to look for cards in the open world and tries to complete the set only by buying packs at the store.

At $5 a pack, how much money do we expect Arthur to spend to complete all 12 sets?

Submit your answer

Riddler Classic

From Chadwick Matlin, the deputy editor of some website called FiveThirtyEight, a puzzle perfect for the polar vortex:

You’re snowed in alone with nothing to do but play a solitaire game of Bananagrams. As you spread its 144 lettered tiles out on the table in front of you, you begin to wonder:

What grid of words can you create that uses all of these tiles in the fewest possible words?

What grid uses all of the tiles in the most possible words?

(To test if a word is allowed, use the ENABLE word list, a variant of which is used in games such as Words With Friends, for the purposes of this problem. The letters in Bananagrams are distributed as shown here.)

Extra credit: How many completed grids are there, period, that use all 144 tiles?

Submit your answer

Solution to the last week’s Riddler Express

Congratulations to 👏 Erin Seligsohn 👏 of Atlanta, winner of last week’s Riddler Express!

Last week, we found ourselves on an island with a curious property: All of its young people told the truth and all of its old people lied. Specifically, there was an age limit L — a positive integer — and all islanders who were younger than age L only told the truth, while islanders who were at least L years old only told lies. We met five of them, and they had this to say.

A: “B is more than 20 years old.”

B: “C is more than 18 years old.”

C: “D is less than 22 years old.”

D: “E is not 17 years old.”

E: “A is more than 21 years old.”

A: “D is more than 16 years old.”

B: “E is less than 20 years old.”

C: “A is 19 years old.”

D: “B is 20 years old.”

E: “C is less than 18 years old.”

What was L, and what could we learn about the ages of the islanders?

L was 19 years old. Islander A was 19 years old, B was 20, C was 18, D was at most 16 and E was at least 20.

The key is to ferret out the liars. Notice first that there are a few direct contradictions in the islanders’ statements: A and D contradict each other over B’s age, B and E contradict each other over C’s age, and E and C contradict each other over A’s age. So at least one member of the three pairs (A, D), (B, E) and (E, C) must be a liar. We also know that D and B aren’t both liars — that would make E both exactly 17 years old and more than 20 years old, which is impossible.

Given those facts, we have a few possibilities we can test. Let’s guess that A, B and E are liars and that C and D are truth-tellers, then check the consistency of their statements. There are two statements about each islander, which we can translate into true facts given our guess about the truthfulness of each of their speakers.

The statements about A check out: A is less than or equal to 21 and A is 19. That means A is 19.

The statements about B also check out: B is less than or equal to 20 and B is 20. That means B is 20.

The same goes for C: C is less than or equal to 18 and C is greater than or equal to 18. That means C is 18.

And D: D is less than 22 and D is less than or equal to 16. That means D is at most 16 years old.

And finally E: E is not 17 and E is greater than or equal to 20. That means E is at least 20.

Because all these statements check out, that means our guess worked! Since A, B and E are liars and C and D tell the truth, that means both of our truth-tellers are at most 18 and all of our liars are 19 or older. Therefore, L must equal 19.

And we’re done! And I don’t know about you, but I’m ready to get off this island.

Solution to last week’s Riddler Classic

Congratulations to 👏 Onufry Wojtaszczyk 👏 of Warsaw, Poland, winner of last week’s Riddler Classic!

Once we got off that island, we headed straight for some rest and relaxation at the card table, where we sat down to play a game of bridge. The game begins with an auction. There are four players, each sitting across from his or her partner. Simply put, the auction goes like this: Beginning with the dealer and orbiting around the table, players can place a bid or pass. Players who’ve passed can re-enter the bidding later. A bid is comprised of a number (one through seven) and a suit. Every legal bid must be higher than the one that came before, meaning either that its number is higher or that its number is the same and its suit is higher (from high to low, the suits go no-trump, spades, hearts, diamonds, clubs). The auction ends if the other three players pass after a bid, or if all four players pass right away.

But the R&R came to an abrupt end with this question: How many different legal bridge auctions are there?

There are 1,574,122,160,956,548,404,565, or more than a thousand billion billion, or more than the estimated number of grains of sand on the planet.

To get there, let’s start with a much smaller number: 35. A bid is made up of one of seven numbers and one of five “suits” (the four suits plus no-trump); multiplying numbers and suits gives us 35 possible bids, each with a unique rank.

From there, let’s build up to our final calculation. We know now that in every auction, the players will make somewhere between one and 35 bids. We will proceed by counting all the auctions with one bid, with two bids, with three bids and so on, and add up all of their possibilities in a big sum.

If an auction comprises \(N\) bids, say, then there are “35 choose \(N\)” choices for what specific bids those could be. And between those bids will be some number of passes. Before the first bid there could be zero, one, two or three passes (four possibilities), and between all subsequent bids there can be zero, one or two passes (three possibilities chosen at \(N-1\) points in the auction). We can account for passes by multiplying by those numbers of possibilities.

That leads us to our final, big sum:

\begin{equation*}1+\sum_{N=1}^{35} {35 \choose N}\cdot 4\cdot 3^{N-1}\end{equation*}

An online calculator can turn this into our numerical final answer. That lone “1+” at the front of the equation is the possibility that the auction ends immediately with all four players passing.

For extra credit, I asked how many different auctions there would be if we incorporated the bridge tactics known as doubling and redoubling. The calculation process is very similar, and the new formula — taking into account that there are now 21 intra-bid combinations of passes, doubles and redoubles, plus seven ways the auction ends with passes, doubles and redoubles — looks like this:

\begin{equation*}1+\sum_{N=1}^{35} {35 \choose N}\cdot 4\cdot 21^{N-1}\cdot 7\end{equation*}

And the final answer is, of course, bigger. Much, much bigger:

128,745,650,347,030,683,120,231,926,111,609,371,363,122,697,557.

But in the grand scale of gnarly game math, the number of possible bridge auctions isn’t all that enormous: It is only about 0.25 percent of the number of possible chess positions.

Want more riddles?

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

Want to submit a riddle?

Email me at [email protected]