Guest Posts To Ultimate SEO Sites

Just today I received a long email complimenting me on the site (which one I dont know) but they always conclude with a suggested mention of site xyz.com and its article on the matter.  And I appreciate those and do feel that the value of this site increases with the more diverse information.  But its diverse information that adds value, me going into an existing article to add a competing link is not really provided a diverse view of the subject just an additional exit page from this site.

Theres actually a lot of these messages that filter in throughout the week. If I spend the time to add the links where they want it would probably consume half a day read messages, finding article, review their link, add it to mine and now to the next.  Its also questionable how long that material is going to be up before the ir site changes or goes and the link becomes just a redirect and I have no idea what the target is too.

Guest Post Content Submissions

So a way to help increase my site’s content and diversity of opinions I have SEOArticle.org which serves as a place for you to Guest Post.  The site is also content.ultimateseo.org.  If you want to do a guest post on any Ultimate SEO site then this is the way to do it.  With your own content that may include a link or two you can submit it through the content site.  I can then review the content in one place and approve it or not.

When approved it will feed to the Ultimate SEO Network, a content syndication network.

UltimateSEO Guest Posts, Webseoservice.net, SEOArticle.org UltimateSEO.net, MatthewLeffler.com and finally a network of over a 100 other sites.  So its worth it to you and to me because I get diverse content.

Just follow our submission guidelines and once you’re article is reviewed you’re live.

About the Author

Matthew Leffler 

Matthew Leffler, founder of Ultimate SEO

Ultimate SEO Audits And Technical SEO

Guest Posts To Ultimate SEO Sites

The NHL’s 3-on-3 All-Star Game Is A Scorer’s Delight

The NHL’s annual All-Star weekend is upon us, and that means fast skaters, blistering shots, mascot hijinks and even a new trick-shot competition. For the fifth-straight year, it also means a heavy focus on the 3-on-3 format, including a women’s All-Star Game on Friday night and the four-team men’s tournament on Saturday. The NHL seldom spent any time playing at 3-on-3 before 2015-16, when the league changed its overtime rules to play the tie-breaking period with three skaters per side. But ever since, 3-on-3 hockey has become a somewhat regular fact of NHL life:

The increasing popularity of 3-on-3 makes a lot of sense. With fewer skaters on the ice to clog up the same amount of territory, offensive players have a lot more time to think and room to move — and consequently, scoring goes through the roof. According to data from the statistics site Corsica Hockey, 5.97 goals have been scored per 60 minutes of 3-on-3 hockey in the NHL this season, a whopping 140 percent increase over the scoring rate (2.49 goals per 60 minutes) of regular 5-on-5 play. It’s been bad for goalies, who not only face 6.6 more shots per 60 minutes but also find themselves allowing around twice as many goals per shot at 3-on-3 (16.1 percent) than at 5-on-5 (8.2 percent). But it’s good news for both the league, which wants more games decided before the shootout,4 and for those players skilled enough to best exploit the extra space on the ice.

That’s what makes 3-on-3 an especially ideal format for an All-Star Game, which is designed to assemble the best — or at least the most offensively productive — players in the sport. And indeed, the most prolific 3-on-3 scorer of the past three seasons, Edmonton’s Leon Draisaitl, will be representing the Pacific Division this weekend. But the best offensive threats in “normal” hockey aren’t always the top 3-on-3 players. For every current All-Star with at least 3,000 minutes at 5-on-5 and 25 minutes at 3-on-3 since the 2017-18 season, I mapped out their Goals Created (GC) per 60 minutes in each situation, looking for the players whose production increased (or decreased) the most with three skaters on each side.

On the high side, Draisaitl has been one of the biggest gainers at 3-on-3, elevating his Goals Created per 60 minutes by 2.91 at 3-on-3. That change ranked fifth among All-Stars, behind St. Louis’s David Perron (+3.89), Winnipeg’s Mark Scheifele (+3.80), Dallas’s Tyler Seguin (+3.35) and Calgary superpest Matthew Tkachuk (+3.16), just ahead of Columbus defenseman Seth Jones (+2.82). Most of those guys are at least solid offensive producers overall — Draisaitl has actually been historic this season — but Jones is a great outlier toward the left side of the chart. More of an all-around blueliner than a pure offensive defenseman, Jones doesn’t score very many points at 5-on-5. At 3-on-3, however, he has 4 goals and 4 assists in 62.4 minutes of play, an especially huge scoring rate coming from the back line. (No other All-Star defenseman has created more than 2 goals per 60 at 3-on-3 over the past three years.)

Meanwhile, there are some stars who do surprisingly little damage at 3-on-3, despite the extra chance for creativity. Boston Bruins right wing David Pastrnak currently leads the NHL in goals with 37 markers in just 51 games so far this season, one of the best goals-per-game rates in recent history. By any measure, Pastrnak is an elite offensive talent, but he has produced only 2 points (and zero goals) in 50.3 minutes of 3-on-3 action over the past three seasons — good for a Goals Created rate 0.40 per 60 minutes lower than during 5-on-5 play. The only All-Star whose production dropped off more at 3-on-3 was Minnesota’s Eric Staal (-0.84), who didn’t score a single point at 3-on-3 in 33.5 minutes. Pastrnak, Staal and Chris Kreider of the Rangers (-0.26) were the only three All-Stars whose stats somehow got worse at 3-on-3 than at 5-on-5. That’s not supposed to happen to players with All-Star skills, although it’s also a testament to the fact that anything can happen in small samples and weird man-strength situations.

And then there are the goalies, who were already tasked with an impossible job under pre-2016 All-Star conditions, when per-game scoring increased by 236 percent compared with the regular season. Then things got even worse between the pipes when the 3-on-3 format was adopted. So no All-Star netminder has had a better save percentage at 3-on-3 than at 5-on-5 — though some dropped off less than others.

Which All-Star goalies hold up the best at 3-on-3?

Among 2020 NHL All-Star goaltenders, smallest drop-off in save percentage from 5-on-5 to 3-on-3 since the 2017-18 season

5-on-5 3-on-3
Goalie Team Minutes SV% Minutes SV% SV% Diff.
Tristan Jarry Metropolitan 2380.9 0.921 30.0 0.882 -0.039
Connor Hellebuyck Central 7943.8 0.925 89.3 0.885 -0.040
Frederik Andersen Atlantic 7877.4 0.925 92.4 0.883 -0.041
Jacob Markstrom Pacific 7086.5 0.922 118.9 0.880 -0.042
Andrei Vasilevskiy Atlantic 7151.5 0.930 86.4 0.872 -0.057
David Rittich Pacific 4313.8 0.923 72.8 0.854 -0.069
Jordan Binnington Central 3178.3 0.930 40.5 0.850 -0.080
Braden Holtby Metropolitan 6508.1 0.917 76.2 0.778 -0.139

Excludes injured goaltenders who were replaced before final All-Star rosters were set.

Source: Corsica Hockey

Tristan Jarry, who is having a breakout season for the Pittsburgh Penguins, checks in at No. 1, though he has faced a paltry 17 shots at 3-on-3. (For a little context, it generally takes 3,000 shots for a goalie’s long-term performance to stabilize, so we’re dealing with absolutely microscopic samples here.) As alternative options atop the leaderboard, Connor Hellebuyck of the Jets and Toronto’s Frederik Andersen have stared down more attempts at 3-on-3 and stopped almost as many of them relative to their 5-on-5 numbers. But Washington’s Braden Holtby is a huge outlier in the opposite direction. Usually one of the top netminders in the game (although he has struggled this season), Holtby has let in a wretched 22.2 percent of opposing shots during 3-on-3 play, easily making him the worst All-Star goalie in those situations.

The Central Division team, led by Colorado’s Nathan MacKinnon and backstopped by Hellebuyck, probably has the best on-paper combination of offensive talent, goaltending and collective 3-on-3 track record. But whoever wins the All-Star tournament, fans are in store for a lot of fast-paced, high-scoring action — thanks in large part to the fundamental nature of the 3-on-3 format. (Sorry, goalies.)

John Isner Is Usually Cold In January And February

If John Isner could start the year like he finishes it, he would have hundreds of thousands of dollars more in prize money, a new shelf of trophies and a better chance all year of challenging the Big Three of Rafael Nadal, Novak Djokovic and Roger Federer. But despite his best efforts, Isner still hasn’t mastered how to start a season well.

The 6-foot-10 American, who joined the tour in 2007, routinely starts the season playing much worse than he finishes it. From January through February, from 2008 through 2019, Isner has won an average of 58.3 percent of his matches. Curiously, from March to November, his play improves. In the final nine months of the season, he’s won an average of 62.3 percent of his matches.

Isner is aware of his poor starts, and he’s worked to fix the problem. But since he almost always holds serve and rarely breaks serve, his matches often come down to tiebreakers, where one errant shot can cost him the set and therefore the match.

In those tight moments, confidence plays an outsized role. But many players, including Isner, gain confidence primarily from playing and winning matches. Without that experience, which he lacks at the start of every year, he struggles.

“My matches, as everyone knows, can be inherently close, and so playing a lot of matches, that sort of gets me tougher in those big moments, where the match may just come down to a point here or there,” Isner said. “Matches are matches, and you can practice all you want. We all practice so hard, and we practice a lot. But you can’t emulate matchplay out there.

“If I’m playing with not so much confidence, I can very easily lose a lot of close matches, a couple weeks in a row, three, four weeks in a row.”

Isner has had bad streaks like that, such as in 2018, when he went 2-5 in January and February. Three of the five losses were decided in the final set.

Isner began 2020 poorly at the ATP Cup, losing all three matches as the U.S.’s No. 1 player at the inaugural team event. But he rebounded in Auckland, going 2-1 and making the semifinals of the lower-level ATP event.

In Melbourne, he is through to the third round of the Australian Open. In the first round, the 19th-ranked Isner beat unseeded Thiago Monteiro of Brazil in four tiebreakers, and in the second, he dispatched of Chilean qualifier Alejandro Tabilo in straight sets. Isner will now face 15th-seeded Stan Wawrinka, who won the event back in 2014.

Late in his career, Isner has been achieving all sorts of milestones that eluded him earlier. In 2018, for instance, he made his first Grand Slam semifinal at Wimbledon and qualified for the year-end ATP Finals for the first time.

Can this year be the first in which Isner advances beyond the fourth round of the Australian Open? If he does, he might just turn the tide of his terrible season starts.

There Has To Be A Better Way To Pick Presidential Nominees … Right?

Not all democracies are the same. Sure, the bedrock principle — governing according to the will of the people — is consistent, but there are lots of structural decisions that influence who “the people” are and how their diverse wishes are tallied. Those decisions have consequences, influencing who runs for office, how they campaign and how they ultimately govern.

Take our presidential primary system as an example. How we seek to find a consensus within parties in the U.S. is, as we’ve covered in past episodes of The Primaries Project, unique among the world’s democracies. It’s accidental and arguably doesn’t always do a great job of finding a consensus. So is there a better way of doing things?

Throughout the month of January, the FiveThirtyEight Politics podcast has been exploring the system we use to choose presidential candidates in an audio documentary series. In the first installment, we looked at the accidental way our modern primary system came to be. In the second, we explored its consequences. In this, the final installment of the series, we ask how things could be different.

To answer that question, we look at how other democracies choose their candidates and ask political scientists how they would design a candidate selection system from scratch. To hear the episode click play above or subscribe to the FiveThirtyEight politics podcast. You can also learn more about our primary system on the FiveThirtyEight Youtube channel.

The FiveThirtyEight Politics podcast publishes Mondays and Thursdays. Help new listeners discover the show by leaving us a rating and review on iTunes. Have a comment, question or suggestion for “good polling vs. bad polling”? Get in touch by email, on Twitter or in the comments.

Is our primary system democratic?

Election Update: It’s Cherry-Picking Season

As of our most recent Election Update this weekend, I noted that polls since last week’s debate had been — this is a scientific term — “pretty weird.”

Well, it’s time to revise that statement. There have been a whole bunch of new polls over the past couple of days. And they’ve been not just pretty weird … but really weird. As far as I’m concerned, that means it’s exactly the sort of time when it’s helpful to take an average instead of fixating on individual polls.

On balance, national polls have been pretty good for Bernie Sanders — but there’s a lot of variation from survey to survey. On Wednesday morning, for instance, a CNN poll came out showing him leading Joe Biden nationally and having gained 7 percentage points from the previous CNN poll in December. But a couple of hours later, YouGov’s weekly tracking poll came out showing Sanders in third place — or having lost ground since last week’s debate.

If you’ve come looking for confident assertions about which poll is “right” and which one is “wrong” — well, that’s just not how we do things around here. And any such claims would be a bit ridiculous given the inherently high margin of error on primary polls. Primary and caucus polling is always a struggle, so the hope is that by accounting for a wider range of pollsters and methods, an average will prove to be a bit less wrong than any individual poll might be. At a bare minimum, averaging or aggregating polls increases the sample size, which is a relevant factor since primary polls often use considerably smaller sample sizes than general election ones.

What you don’t want to do, of course, is to cherry-pick data. There’s actually been quite a bit of post-debate polling, and if Sanders had actually gained 7 points nationally, as the CNN poll shows, we would have seen more signs of it by now.

It would be an equally big mistake, however, to “throw out” or ignore the CNN poll. CNN has not shown especially strong results for Sanders before, and CNN’s pollster, SSRS, is reasonably highly rated. It’s a sign that a pollster is doing honest work when it’s willing to publish a poll that differs a bit from the consensus. So stick the CNN poll in the average instead … and you’ll see there’s still some good signs for Sanders.

National polls show Sanders and Bloomberg gaining, others flat

In fact, at 20.4 percent, Sanders is in his strongest position in our national polling average since April. State polls have been much more of a mixed bag, with Sanders having fallen slightly in our Iowa and New Hampshire polling averages since the debate — but we’ll cover those in the next section. A bit more about those national polls first.

In addition, to CNN and YouGov, Morning Consult and Monmouth University also released national polls on Wednesday morning. These join earlier post-debate national polls from SurveyUSA and Ipsos. Here’s a table showing those polls for the top six candidates:

Biden leads, Sanders second in post-debate national polls

Where the top six candidates stand in six national polls, after the last debate

Post-debate toplines
Pollster Biden Sanders Warren Buttigieg Bloomberg Klobuchar
CNN/SSRS 24 27 14 11 5 4
Ipsos 19 20 12 6 9 2
Monmouth 30 23 14 9 6 5
Morning Consult 29 24 15 8 10 3
SurveyUSA 32 21 14 9 9 2
YouGov 28 18 21 8 6 4
Simple avg. 27.0 22.2 15.0 8.5 7.5 3.3
FiveThirtyEight avg. 26.6 20.4 15.8 7.6 7.3 3.2

Current FiveThirtyEight national polling average and most recent polls, as of Jan. 22

And here’s a companion table showing the change from the previous pre-debate poll for each pollster. Note that the pre-debate poll wasn’t necessarily that recent; the last time that SurveyUSA had polled the race nationally was in November, for instance.

National polls show gains for Sanders, Bloomberg

How the top six candidates’ standing changed from each pollster’s previous pre-debate poll

Pollster Biden Sanders Warren Buttigieg Bloomberg Klobuchar
CNN/SSRS -2.0 +7.0 -2.0 +3.0 +0.0 +1.0
Ipsos -4.0 +0.0 -3.0 +1.0 +1.0 +1.0
Monmouth +4.0 +2.0 -3.0 -2.0 +4.0 +1.0
Morning Consult +0.0 +1.0 +1.0 +0.0 +2.0 +0.0
SurveyUSA +2.0 +4.0 -1.0 -2.0 +6.0 +0.0
YouGov +1.0 -2.0 +2.0 +1.0 +1.0 +1.0
Raw avg. change +0.2 +2.0 -1.0 +0.2 +2.3 +0.7
FiveThirtyEight avg. change* -0.1 +1.7 -0.3 +0.2 +1.5 +0.1

*Calculated change in FiveThirtyEight’s national polling average since Jan. 14

In the tables, you can see how a simple average of the six most recent polls compares to the much fancier FiveThirtyEight polling average: They’re really pretty darn similar. There are some differences, though. For instance, it’s worth noting that the SurveyUSA poll had about three times (1086) as many respondents as Monmouth (372) and more than twice as many as CNN (500); our averages account for that by giving SurveyUSA more weight. House effects are also something of a factor; for instance, SurveyUSA tends to show good numbers for Biden, Morning Consult tends to show good numbers for Sanders, YouGov tends to show good numbers for Elizabeth Warren; and Ipsos tends to show poor numbers for all candidates except Sanders. Those explain some of the differences between the polls, too.1

But none of this really matters that much. As you can see, the FiveThirtyEight average and a simple average of the six post-debate polls produce highly similar results. Likewise, the average change in the polls is pretty similar regardless of which method you use. Take a simple average of the change in Sanders’s numbers since the last time these six pollsters surveyed the field, and he’s gained 2.0 percentage points since the debate. Similarly, he’s gained 1.7 percentage points in the fancy version of the FiveThirtyEight average, calculated since Jan. 14, the day of the debate. Michael Bloomberg has also gained ground regardless of what method you choose and is close to catching Pete Buttigieg in our national average.

State polls haven’t been great for Sanders, though

There have also been quite a few state polls since the debate, however, and they don’t tell a terribly consistent story with the national polls. They’re actually on the weak side for Sanders. Iowa polls from Neighborhood Research and Media and David Binder Research, both conducted since the debate, have Sanders in 5th and 4th place, respectively in the Hawkeye state. But Sanders does have some decent excuses here; the David Binder poll has generally been one of his worst ones in Iowa, and the Neighborhood Research poll has a small sample size and is from a Republican pollster that hasn’t previously released data in Iowa (as such, it receives a relatively small weight in our model).

Nonetheless, based on recent Iowa polls — both before and after the debate — Iowa would appear to be Biden’s state to lose more than Sanders’s. Biden has been ahead or tied for the lead in 4 of the 5 Iowa polls since the new year, as compared with just two for Sanders. (Although one of the polls that had Sanders ahead was the highly-rated Selzer & Co. poll.) In fact, Sanders has basically slipped into a three-way tie for second place in Iowa with Warren and Buttigieg.

Iowa and N.H. trends have been medicore for Sanders

How the top five candidates’ polling averages in Iowa and New Hampshire changed, before and after the last debate

Iowa New Hampshire
Candidate Jan. 14* Current Change Jan. 14 Current Change
Biden 20.1 21.4 +1.3 21.8 17.9 -3.9
Sanders 19.9 17.8 -2.1 20.1 19.5 -0.6
Warren 14.9 16.4 +1.5 16.0 13.8 -2.2
Buttigieg 17.4 16.6 -0.8 13.1 13.9 +0.8
Klobuchar 6.7 8.4 +1.7 4.7 7.1 +2.4

* Polling average as of the day before the debate.

In New Hampshire, true to the theme, the polling has also been weird. Two new polls conducted wholly or partially since the debate, from Emerson College and Suffolk University, each show Sanders leading there — but having lost ground relative to the previous editions of the same polls. Instead, these polls show growth for candidates outside of the top three, such as Buttigieg and Amy Klobuchar. And in the case of the Suffolk poll, there are also a large number of undecided voters. (Sanders, despite being the leading candidate in the Suffolk poll, has only 16 percent of the vote in the poll.) Sanders leads in our New Hampshire polling average, but no candidate should feel particularly secure in the Granite State.

Wednesday also saw the release of polling in four delegate-rich Midwestern swing states from a consortium of universities there, Baldwin Wallace University, Oakland University and Ohio Northern University. Sanders led in Wisconsin, but Biden held leads in Ohio, Michigan and Pennsylvania.

How much do national polls matter?

Overall, our forecast just hasn’t changed that much since we first released it two weeks ago. Biden has a 43 percent chance of a delegate majority, followed by Sanders at 20 percent, Warren at 14 percent and Buttigieg at 8 percent. The chance of no candidate winning a majority of delegates is 15 percent. Technically, Biden is up very slightly and Sanders is down very slightly since we first launched the forecast, but the differences are small and not worth spending a ton of time worrying about.

If you look carefully, though, you can see how the model tends to privilege state polls over national polls (or at least it does so right now, with Iowa set to vote in less than two weeks). For example, Sanders did gain ground in the model following the release of the Selzer poll of Iowa on Jan. 10, but — despite his standing in national polls improving — he’s given those gains back because of an underwhelming series of polls in Iowa and New Hampshire.

You might be wondering: How does the model even use national polls? There is no national primary, after all. But national polls have several uses in the model. To simplify, here are the three most important ones:

    1. We use national polls to calculate a trendline adjustment, which can influence states that haven’t been polled recently. For instance, if there are no recent polls of Oregon, but Sanders has gained 3 points in national polls since the last time Oregon was polled, the model will assume he’s gained ground in Oregon, too.
    2. National polls are used as a baseline to calibrate the bounce a candidate will potentially receive after winning or finishing strongly in a state. The twist is that the higher a candidate’s standing in national polls, the less her bounce after winning a state (and the more she might decline after losing a state). Empirically, the bounce that a candidate gets after winning a state depends strongly on expectations, and national polls are a good proxy for voter and media expectations. Less abstractly, the fact that Sanders is now seen as a national front-runner means that — as is also the case with Biden in Iowa — he might gain less ground following a win there, and lose more ground nationally if he finishes in anything other than first place or a strong second. Warren or Buttigieg, on the other hand, would probably be seen more as underdogs, and those sort of candidates have historically gotten bigger bounces after winning Iowa.
    3. In various implicit and explicit ways, national polls help the model to predict the outcome of states where there isn’t much polling.

So basically, factors No. 1 and No. 3 tend to help a candidate in the model when their national polls are strong, but factor No. 2 (rising expectations) actually hurts them. To be clear, our experience with the model so far — keep in mind that this is the first time we’ve run a full-fledged primary forecast — is that No. 1 and No. 3 usually outweigh No. 2. In other words, a candidate would usually prefer to gain ground in national polls, as far as the model is concerned. Still, there is some ambiguity about the influence of national polls in the model, especially when there is a lot of recent state polling and so the timeline adjustment (No. 1) doesn’t have as much impact.

Essentially, this leaves us with three plausible interpretations of the post-debate polling:

  • National polls tell the true story of the race, and the slightly quirky set of recent Iowa and New Hampshire polls since the debate are misleading. If so, Sanders should expect to do better in the next set of Iowa and New Hampshire polls. This would be good news for Sanders.
  • All of this is noise, and none of the candidates’ positions have changed much since the debate. This would be neutral news for Sanders.
  • Whether or not Sanders is gaining in national polls, Iowa and New Hampshire have their own dynamics, and Sanders does not appear to be closing strongly there. This would be bad news for Sanders.

Polling since the debate just doesn’t provide a lot of clarity on which one of these stories is correct. So we’ll have to wait and see if the picture clears up by the weekend.

Dominic Thiem Doesn’t Play As Much Anymore, And That’s A Good Thing

It took nearly three years, but Dominic Thiem is finally acting like a top 10 player. His reward has been his best tennis yet.

For years, Thiem played far more often than his peers. In 2016, when he first cracked the top 10, he played 82 matches, 17 percent more than the average of the end-of-season top 10 players. But now, a more mature and confident Thiem is designing his playing schedule more carefully, and his results on court have never been better.

The 26-year-old Austrian won a career-high five titles in 2019, including his first at the Masters 1000 level when he beat Roger Federer in March in Indian Wells. Thiem also made his second French Open final, and he beat Novak Djokovic at the season-ending ATP Finals in what he called “probably the best match I ever played.” He finished the season ranked a career-high No. 4, behind only Federer, Djokovic and Rafael Nadal.

“I think when I was younger, I needed [more matches]. It was tough for me to come to a big event without any matches before. I kind of was looking for getting confidence with some smaller events,” Thiem said in May. “But I’m also way more experienced now, so I can play the bigger events also without the smaller events before.”

As Thiem worked his way up to the top of the game, he had been the ATP Tour’s Ironman. In 2015, for instance, he rarely took a week off, playing in 30 tournaments, 48 percent more than the top 10 average of 20.3. That tied for eighth-most among players who finished the year ranked inside the top 100.

The play-often strategy can make sense. Sometimes players hop from tournament to tournament simply to play more matches and earn a better income. Lower-ranked players often lose earlier in tournaments, freeing up their schedules and allowing them to play more events.

But once Thiem reached No. 7 in June 2016, after his first Grand Slam semifinal at the French Open, he didn’t change his habits to reflect his ranking and conserve his body; he kept acting like a player still on the climb.

In 2016, he played 82 matches, while the top 10 averaged 70.1. In 2017, Thiem competed in 77 tour-level contests, almost 15 percent more than the top 10 average of 67.2; and in 2018, Thiem again played in more matches, 74, than the top 10 average of 64.2.

All the while, he defended the strategy.

“If I would have played less, I wouldn’t be here at the finals,” he said in November 2016 during his first appearance at the season-ending ATP Finals, where only the top eight players compete. “I think also the body and the mind and everything gets used to it.

“I think it’s really normal. I think that if I play the same amount of matches and tournaments next year, I will be used to it much better than this year.”

The aggressive schedule didn’t always hurt him. Most of the highest-ranked players spend the week before a Grand Slam practicing at the tournament site and getting used to the conditions. But in 2018, when Thiem made his first French Open final, he played a full week of matches the week before the tournament in Lyon, about 290 miles south of Paris, and won the ATP 250-level tournament.

“[It] was only four matches. And I feel physically completely fine, and the victory of the tournament helped for sure,” he said at the time. “I’m feeling great with a lot of confidence.”

But Thiem, who turned 26 in September, knows better now. He’s wary of adding any extra toll to his body, and he no longer needs more time on court to bolster his belief.

Last season, Thiem didn’t play the week before any Slam, and in total, he played 68 matches — exactly the top-10 average for 2019 and his lowest tally since 2015, when he played 64.

“I think with the age, I have to take little bit more care of myself,” Thiem said. “My goal is to play less tournaments, for sure, but to play well in every single tournament I play.”

Last year in Australia, even after time off between seasons, Thiem still felt worn down from his 2018 schedule. Suffering from an illness, he retired in the second round.

“I had a long season [in 2018]. It was very tough and long. Maybe I took off a little bit too short. I went almost straight to the offseason, which was again tough. That’s why I got ill in Australia,” he said in March.

Thiem is seeded fifth at the Australian Open, matching his 2018 seeding for his highest yet in Melbourne. He won his first-round match over Adrian Mannarino in straight sets. He is feeling fresh and confident at the start of the two-week tournament, which should help him thrive in the Australian summer — temperatures in past years have regularly climbed above 100 degrees Fahrenheit.

Thiem has never reached past the fourth round of the Australian Open. But the Austrian also has never been more physically prepared for the season’s first Grand Slam than he is right now.

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Election Update: The First Post-Debate Polls Are In! And They’re … Pretty Weird.

The first handful of polling since Tuesday night’s debate is out. But it doesn’t tell a terribly consistent story. Pretty much whichever Democrat you’re rooting for, you can find some polls to be happy about and others that you’d rather ignore. Here’s a quick list of those polls:

  • SurveyUSA has a new national poll that shows Joe Biden leading with 32 percent of the vote, followed by Bernie Sanders at 21 percent, Elizabeth Warren at 14 percent and both Pete Buttigieg and Michael Bloomberg at 9 percent. As compared with their previous national poll in November, Biden is up 2 percentage points, Sanders is up 4, Warren is down 1, Buttigieg is down 2, and Bloomberg is up 6.
  • SurveyUSA also published a new California poll, which has Biden leading there at 30 percent, with Sanders and Warren tied for second at 20 percent and Buttgieg in fourth at 8 percent. Biden and Sanders are both up 2 percentage points since their November California poll, when California senator Kamala Harris was still in the running, while Warren has gained 7 points since polling at 13 percent in November.
  • While SurveyUSA has seemingly good news for Biden, an Ipsos national poll for Reuters does not. Instead, it has Sanders ahead nationally at 20 percent, followed by Biden at 19, Warren at 12, Bloomberg at 9, and Buttigieg at 6. As compared with the Ipsos/Reuters national poll conducted roughly a week before the debate, Sanders is unchanged, but Biden is down 4 percentage points and Warren is down 3 points.
  • I should also mention the Ipsos poll conducted with FiveThirtyEight, which surveyed a single group of voters both before and after the debate. It did not include a traditional horse-race question (i.e., “Who is your first choice?”) so it doesn’t figure directly into our polling averages or primary model. However, it showed strong results for Warren, with her making gains on favorability, perceived electability, and the number of Democrats who said they were considering voting for her.
  • Finally, an Emerson College poll of New Hampshire, conducted partially since the debate, has Sanders ahead there with 23 percent of the vote, followed by Buttigieg at 18 percent, Biden and Warren each at 14 percent, and Amy Klobuchar at 10 percent. As compared with Emerson’s previous poll of New Hampshire, in November, Sanders is actually down 3 points and Buttigieg is down 4 points, while Biden and Warren are unchanged and Klobuchar is up 8 points.

As I said, you can cherry-pick your way to pretty much whatever narrative you like. Biden fan? Both those SurveyUSA numbers look nice. Sanders stan? You’ll probably want to emphasize the Ipsos/Reuters poll. Warren aficionado? The SurveyUSA California poll and the Ipsos/FiveThirtyEight poll look good; the others, not so much.

But some of these polls are also pretty confusing. If you’re Sanders, for instance, should you be happy that the Emerson poll in New Hampshire still has you leading, or unhappy that it has you having lost a few points? Or to abstract the question: Should you pay more attention to the trendline within a poll or to the absolute result?

This question does not have a straightforward answer (other than that both are important to some degree). Ideally, you should be comparing a poll not only against the most recent survey by the same pollster, but really against all previous surveys by the pollster in that state — and for that matter also in other states — to detect whether it generally shows good results or poor results for your candidate. And when evaluating trendlines, you should account for when the previous polls were conducted. For example, any poll conducted in October is likely to have shown good results for Warren, since she was at her peak nationally then. So if a new poll came out today showing Warren having fallen by 2 points in Iowa since October, that might might be comparatively good news for her since you’d have anticipated a steeper decline.

If all this sounds like a lot of work … well, it’s the work that our polling averages and our model are doing for you behind the scenes. Usually our model moves in the direction you might expect intuitively, e.g., Sanders gained ground both in Iowa and in our overall delegate forecast after a Selzer & Co. poll showed him leading the Iowa caucuses.

In the presence of strong house effects, however, the model might move in surprising directions. Just as polls can have house effects in general elections — Rasmussen Reports polls have a notoriously pro-Trump/pro-Republican lean, for example — certain pollsters in the primaries persistently show better or worse results for certain candidates.

And it just so happens that all the pollsters who have released polls since the debate have fairly strong house effects. Emerson College has often shown strong results for Sanders, for instance. And SurveyUSA — both in its California polls and its national polls — has consistently had some of the best numbers for Biden. This is good news for Biden in one sense since SurveyUSA is one of our highest-rated pollsters. But it also means that it isn’t necessarily new news when a SurveyUSA poll comes out showing Biden doing well; such a result will be in line with our model’s expectations. Conversely, Ipsos has consistently shown some of the worst results for Biden, so it doesn’t necessarily move the needle in our model when another Ipsos poll comes out showing Biden doing mediocrely.

To give you a sense of the magnitude that house effects can have, here are the various post-debate polls with and without our model’s house effects adjustment:

House effects can make a big difference

Polls since the January debate, with and without FiveThirtyEight’s adjustments for house effects

SuvreyUSA national poll, Jan. 14-16, 2020
Candidate Raw Adjusted
Biden 32.0% 28.0%
Sanders 21.0 20.1
Warren 14.0 14.7
Bloomberg 9.0 9.0
Buttigieg 9.0 8.5
Ipsos/Reuters national poll, Jan. 15-16, 2020
Candidate Raw Adjusted
Sanders 20.0% 20.2%
Biden 19.0 22.9
Warren 12.0 16.1
Bloomberg 9.0 8.5
Buttiigeg 6.0 7.4
SuvreyUSA California poll, Jan. 14-16, 2020
Candidate Raw Adjusted
Biden 30.0% 26.1%
Sanders 20.0 19.1
Warren 20.0 20.8
Buttigieg 8.0 7.5
Bloomberg 6.0 6.0
Emerson College New Hampshire Poll, Jan. 13-16, 2020
Candidate Raw Adjusted
Sanders 22.9% 18.0%
Buttigieg 17.7 17.7
Biden 14.1 13.4
Warren 13.5 13.4
Klobuchar 10.0 10.3
Yang 6.3 5.0

Only candidates polling at 5 percent or more in each survey are shown

Source: Polls

While the SurveyUSA national poll had Biden at 32 percent and Ipsos had him at 19 percent, the gap is a lot smaller once you account for house effects. The adjustment brings the SurveyUSA poll down to 28 percent and the Ipsos poll up to around 23 percent, a difference that is well within the polls’ sampling error given their respective sample sizes.

To be clear, house effects are not the same thing as statistical bias, which can be evaluated only after a state has conducted its voting. For example, SurveyUSA is implicitly suggesting that Biden is underrated by other pollsters. If they’re wrong about that, SurveyUSA polls will turn out to have had a pro-Biden bias. But if Biden’s results match what SurveyUSA’s polls project, then their polls will have been unbiased and all the other polls will have had an anti-Biden bias. Obviously, we think you should usually trust the polling average — that’s the whole point of averaging or aggregating polls. But especially in the primaries, where turnout is hard to project, it’s also worth paying attention to the differences between polls — and sometimes pollsters with strong house effects (even to the point of being “outliers”) turn out to be correct.


For all that said, polls with strong house effects, because of the additional complications they present, aren’t necessarily ideal for evaluating polling swings following news events such as debates. So while it’s tempting to infer from the polls we have so far that the debate didn’t change things very much — no candidate is consistently seeing their numbers surge or crater — we should wait for a few more polls to confirm that.

In the meantime, our topline forecast is largely unchanged. Biden remains the most likely candidate to win the majority of pledged delegates, with a 41 percent chance, followed by Sanders at 23 percent, Warren at 12 percent and Buttigieg at 9 percent. There is also a 15 percent chance no one wins a majority, a chance that could increase if Bloomberg, who has now almost caught Buttigieg in our national polling average, continues to rise.

The Iowa Caucuses Are In 17 Days. What’s Going On In The Early States?

Welcome to Pollapalooza, our weekly polling roundup.

Poll(s) of the week

Throughout the primary, we’ve been tracking polls in the four early states to better understand how the race differs in each of them, as each state tells its own story about how the 2020 Democratic primary could shake out.

For instance, Iowa is a very tight race between former Vice President Joe Biden, Sens. Bernie Sanders and Elizabeth Warren, and former South Bend, Indiana, Mayor Pete Buttigieg. And New Hampshire is really competitive, too, with less than 10 percentage points separating Sanders and Biden in our forecast. (Warren and Buttigieg aren’t too far off either.) Nevada, on the other hand, appears to be more of a two-way race between Sanders and Biden, while South Carolina remains — at least at this point — in Biden’s column, as our forecast gives him a 3 in 5 shot at winning the most votes there.

Of course, things could once again shift before the voting starts in Iowa, but here’s a deep dive on where things stand as of Friday morning in each of these states.


Biden, Sanders neck and neck In Iowa


Iowa and New Hampshire are really competitive

Remarkably, Biden, Sanders, Warren and Buttigieg are all in the hunt in both Iowa and New Hampshire: No one has more than a 35 percent chance of winning either state, according to our forecast.

And our state polling averages illustrate just how close things are. Sanders and Biden are essentially tied in Iowa, with Buttigieg and Warren trailing, but within about 5 points of the lead. And Sanders holds a slim lead over Biden in New Hampshire, although one might expect the Vermont senator to have an advantage there — New Hampshire tends to favor politicians from neighboring states. In New Hampshire, too, Warren and Buttigieg aren’t too far behind the leaders.

Iowa and New Hampshire are wide-open contests

Democratic presidential candidates’ polling averages in Iowa and New Hampshire, as of Jan. 17 at 9 a.m.

Iowa New Hampshire
Candidate Average Candidate Average
Bernie Sanders 20.1% Bernie Sanders 19.6%
Joe Biden 19.5 Joe Biden 18.5
Pete Buttigieg 16.8 Elizabeth Warren 15.1
Elizabeth Warren 15.2 Pete Buttigieg 13.9
Amy Klobuchar 6.7 Amy Klobuchar 6.7
Andrew Yang 3.5 Tom Steyer 3.8
Tom Steyer 3.1 Tulsi Gabbard 3.7
Michael Bloomberg 2.2 Andrew Yang 3.7
Tulsi Gabbard 1.7 Michael Bloomberg* 2.9
Michael Bennet 0.1 Michael Bennet 0.4
John Delaney 0.1 Deval Patrick 0.3
Deval Patrick 0.0 John Delaney 0.2

*Michael Bloomberg didn’t file to run in New Hampshire, but some pollsters have included him when surveying the state.

Source: Polls

But the dearth of state-level polls in December means that some of the polls feeding our averages may no longer reflect the current state of the race, so let’s take a closer look at some of the bigger polls released in the last 10 days. Monmouth University’s latest Iowa poll underscored how important voters’ ideological leanings could be in that state, as many caucus-goers will end up backing their second-choice candidate — not their first — in the realignment process. (If a candidate does not have enough support, usually at least 15 percent of voters at a caucus site, their supporters are asked to align themselves with another candidate.) Overall, Biden was first in that poll with 24 percent support, but he also had the most support (35 percent) among voters who said they were moderate or conservative — and that group made up a majority of the poll’s respondents. Buttigieg came in second among these voters at 15 percent, followed by Sanders at 13 percent. By contrast, among very liberal voters, Sanders led with 29 percent support with Warren in second at 25 percent, while Buttigieg had 14 percent and Biden had just 11 percent. But among the “somewhat” liberal voters in between, Buttigieg actually led with 22 percent, followed by Sanders at 20 percent. In other words, a different candidate led in each ideological “lane,” with a ton of overlap — in particular, Sanders and Warren — which could play to either Sanders’s or Warren’s detriment.

Similarly, a new poll from RKM Research and Communications on behalf of Franklin Pierce University, the Boston Herald and NBC 10 Boston showed New Hampshire has some of the same ideological divides. Overall, the survey found Biden in first with 26 percent, but once again Biden did best among more moderate (and conservative) voters with 27 percent, compared to 16 percent for both Sanders and Warren. Meanwhile, Sanders led among liberal voters with 29 percent, followed by Biden at 26 percent and Warren at 20 percent. (The pollster didn’t separate “somewhat liberal” and “very liberal” like Monmouth did.)

So as some progressives have feared, it’s possible that Sanders and Warren are hindering each other because of the ideological overlap among their supporters. After all, a new Quinnipiac University national poll found that more than 50 percent of Sanders’s and Warren’s supporters picked the other progressive as their second choice. And in New Hampshire, there’s another way they might be holding one another back: Since they both represent neighboring states, they could be limiting each other’s home-field edge.

Nevada’s close, Biden dominates South Carolina

Nevada and South Carolina look a bit different from the first two states in that their polls are less top-heavy, with fewer contenders in the mix. However, Nevada is still very much up for grabs — Biden and Sanders are separated by about 3 points in our polling average, and our forecast gives them both about a 1 in 3 chance of winning. But Biden still has a notable advantage in South Carolina, where he leads the polls by more than 20 points and has nearly a 60 percent chance of winning, per our forecast.

Nevada is competitive, but South Carolina is Biden’s to lose

Democratic presidential candidates’ polling averages in Nevada and South Carolina, as of Jan. 17 at 9 a.m.

Nevada South Carolina
Candidate Average Candidate Average
Joe Biden 22.7% Joe Biden 35.2%
Bernie Sanders 19.6 Bernie Sanders 14.5
Elizabeth Warren 14.4 Elizabeth Warren 11.0
Tom Steyer 8.2 Tom Steyer 9.8
Pete Buttigieg 6.4 Pete Buttigieg 4.2
Andrew Yang 3.9 Michael Bloomberg* 3.8
Amy Klobuchar 2.8 Andrew Yang 2.3
Michael Bloomberg* 2.2 Amy Klobuchar 1.7
Tulsi Gabbard 1.4 Tulsi Gabbard 1.4
John Delaney 0.4 John Delaney 0.3
Deval Patrick 0.2 Deval Patrick 0.2
Michael Bennet 0.1 Michael Bennet 0.1

*Michael Bloomberg didn’t file to run in Nevada or South Carolina, but some pollsters have included him when surveying those states.

Source: Polls

Part of the reason the dynamics of these two states are so different is that Iowa and New Hampshire still haven’t voted, which could dramatically shift the race, and that Nevada and South Carolina’s electorates are just much more diverse than those of the first two states to vote — and Biden has made the most significant inroads with voters of color, followed by Sanders and Warren.

Nevada is a particularly interesting case because even though Biden tends to lead among nonwhite voters nationally because of his strong support from black voters, nonwhite voters in Nevada are more likely to be Hispanic, and two recent Nevada surveys found Sanders narrowly ahead of or tied with Biden among Hispanics.

First, a Nevada survey from Fox News found Biden in first overall with 23 percent and Sanders in second at 17 percent, but both attracted the same share of Hispanic voters (24 percent); Biden’s edge among white voters (22 percent) might have been what tipped him over the edge, as Sanders attracted only 13 percent of that group. Meanwhile, another Nevada poll by Suffolk University, conducted for USA Today and the Reno Gazette-Journal, found Biden and Sanders essentially tied for first with around 20 percent support and neck and neck at about 20 percent among nonwhite voters. Biden had an ever-so-slight edge over Sanders among white voters (19 percent to 16 percent).

As for South Carolina, Biden has consistently led there due to his strong support among black voters, who made up about 60 percent of the Democratic primary electorate in 2016. A Fox News poll released late last week found Biden’s edge in South Carolina mostly holding as he led the field with 36 percent overall, including 43 percent of black voters. Billionaire Tom Steyer did make noise by coming in second with 15 percent overall, though that may partly be the result of him flooding the airwaves in South Carolina. Still, Steyer got support from 16 percent of African American voters, which could be evidence that some black voters are open to alternatives other than Biden. For now, the Biden firewall in the Palmetto State is holding up, though that could, of course, change if he performs poorly in earlier contests.

With just over two weeks to go before the Iowa caucuses, there’s definitely a traffic jam at the top of the polls in Iowa and New Hampshire that makes those states fairly unpredictable at this point. And whatever happens in those initial contests will likely affect Nevada and South Carolina, too, so stay tuned as we keep a close watch on how the polls move once the voting starts.

Other polling bites

  • Speaking of Biden’s base, a new national poll from The Washington Post and Ipsos found 48 percent of black Democratic registered voters supported Biden. Sanders was second with 20 percent support, and no other candidate reached double-digit support. There was a noticeable age difference in Biden and Sanders’s supporters: Among registered voters 50 years and older, Biden captured more than 50 percent support. He also led Sanders, 41 percent to 16 percent, among voters 35 to 49 years old. Yet among those under the age of 35, Sanders attracted 42 percent support versus Biden’s 30 percent.
  • An LX/Morning Consult poll asked Americans about Iowa and New Hampshire’s role in the nominating process and found that only 41 percent felt that these states represented their views in the presidential primary “very well” or “somewhat well” while 59 percent said “not very well” or “not at all.” Still, 59 percent said they were “very” or “somewhat” satisfied with the tradition of Iowa and New Hampshire kicking things off, compared to 41 percent who were dissatisfied.
  • In the wake of reports that Sanders told Warren in 2018 that he believed a woman couldn’t beat President Trump, YouGov asked Americans about their views on this issue. Fifty-three percent of respondents said that either a man or woman could defeat Trump, while 11 percent said that a man could beat Trump but not a woman. Another 20 percent said that neither a man nor a woman could defeat Trump, including 60 percent of Republicans.
  • In a new poll from Pew Research Center, 48 percent of Americans said that the airstrike that killed Iranian military leader Qassem Soleimani was the right decision while 43 percent said it wasn’t. However, 54 percent felt that the Trump administration’s approach to Iran has increased the chances of a military conflict between the U.S. and Iran and a plurality — 44 percent — said it has made the country less safe.
  • A HuffPost/YouGov survey asked Americans their views on trade and the new United States-Mexico-Canada Agreement, finding that many Americans are relatively unaware of the trade deal, which the Senate passed on Thursday. Forty-four percent said they favored the free trade agreement with Canada and Mexico, another 44 percent said they weren’t sure what they thought of it and 11 percent opposed it. Two explanations for why so many Americans weren’t sure? Four out of five respondents said they had heard little or nothing at all about the trade deal. And Americans are also generally somewhat uncertain about trade agreements between the U.S. and other countries — 43 percent said such deals are a good thing, while 20 percent said they were a bad thing and 37 percent said they weren’t sure.
  • In a survey of American attitudes toward vaccines, Gallup found that there is still uncertainty about the debunked claim that vaccines can cause autism. Although only 10 percent said that certain vaccinations can cause autism and 45 percent said they do not, a plurality — 46 percent — said they weren’t sure. However, 86 percent said that the diseases vaccines prevent are more dangerous than the vaccines themselves.
  • In a new poll from YouGov measuring attitudes toward the Confederate flag among Americans, a plurality of respondents in two former Confederate states — Arkansas and Louisiana — said that the flag represented heritage more than racism. Of the other nine states that were part of the Confederacy, pluralities in seven said that the flag represented racism more than heritage, while respondents in Alabama and Florida were split evenly.

Trump approval

According to FiveThirtyEight’s presidential approval tracker, 42.4 percent of Americans approve of the job Trump is doing as president, while 52.9 percent disapprove (a net approval rating of -10.5 points). At this time last week, 41.9 percent approved and 53.3 percent disapproved (for a net approval rating of -11.4 points). One month ago, Trump had an approval rating of 43.2 percent and a disapproval rating of 52.5 percent, for a net approval rating of -9.3 points.

Generic ballot

In our average of polls of the generic congressional ballot, Democrats currently lead by 5.9 percentage points (47.0 percent to 41.1 percent). A week ago, Democrats led Republicans by 6.6 points (47.5 percent to 40.9 percent). At this time last month, voters preferred Democrats by 6.6 points (47.2 percent to 40.6 percent).

Can You Track The Delirious Ducks?

Welcome to The Riddler. Every week, I offer up problems related to the things we hold dear around here: math, logic and probability. Two puzzles are presented each week: the Riddler Express for those of you who want something bite-size and the Riddler Classic for those of you in the slow-puzzle movement. Submit a correct answer for either,<a class="espn-footnote-link" data-footnote-id="11" href="#fn-11" data-footnote-content="<p>Important small print: Please wait until Monday to publicly share your answers. In order to 👏 win 👏, I need to receive your correct answer before 11:59 p.m. Eastern time on Monday. Have a great weekend!</p> “>11 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

The Riddler Football League (RFL) playoffs are upon us! As the coach, you’ve devised a new strategy for scoring after a touchdown. Your team will line up 2 yards away from the goal line in such a way that it could attempt either a 1-point conversion or a 2-point conversion. (Unlike other football leagues, the distance is the same for both types of conversion, and you need not announce which you’ll be attempting.) Your opponent can only properly defend against one of those two possibilities, so they’ll have to guess.

If you attempt a 1-point conversion and the other team defends against it properly, you’ll score 90 percent of the time. If they don’t defend it properly, you’ll score 100 percent of the time.

If you instead attempt a 2-point conversion and the other team defends against it properly, you’ll score 40 percent of the time. If they don’t defend it properly, you’ll score 60 percent of the time.

To tell your team which they should attempt, your team’s offensive coordinator will communicate to your team’s captain the probability with which they should attempt each. For example, the coordinator might say: “Go for 1 with a 51 percent chance, and go for 2 with a 49 percent chance.” Using a random number generator, the captain will then ultimately decide to go for 1 point or 2 points. (Naturally, every athlete in the RFL has a random number generator handy.)

However, given all the spying that occurs in the RFL these days, you can assume that the offensive coordinator’s message will also be heard by your opponent — that means the defense knows the exact probability with which you’ll attempt either conversion. Your opponent also knows the probability of you scoring in each of the four scenarios listed above.

With all that said, what strategy will maximize the average number of points you’ll score (i.e., how often should your team go for 1 or 2)? What should your opponent’s defensive strategy be? How many points will you score, on average, after each touchdown?

Submit your answer

Riddler Classic

After a long night of frivolous quackery, two delirious ducks are having a difficult time finding each other in their pond. The pond happens to contain a 3×3 grid of rocks.

Every minute, each duck randomly swims, independently of the other duck, from one rock to a neighboring rock in the 3×3 grid — up, down, left or right, but not diagonally. So if a duck is at the middle rock, it will next swim to one of the four side rocks with probability 1/4. From a side rock, it will swim to one of the two adjacent corner rocks or back to the middle rock, each with probability 1/3. And from a corner rock, it will swim to one of the two adjacent side rocks with probability 1/2.

If the ducks both start at the middle rock, then on average, how long will it take until they’re at the same rock again? (Of course, there’s a 1/4 chance that they’ll swim in the same direction after the first minute, in which case it would only take one minute for them to be at the same rock again. But it could take much longer, if they happen to keep missing each other.)

Extra credit: What if there are three or more ducks? If they all start in the middle rock, on average, how long will it take until they are all at the same rock again?

Submit your answer

Solution to last week’s Riddler Express

Congratulations to 👏 Nathan Holmes-King 👏 of Fremont, California, winner of last week’s recent Riddler Express.

Last week, you were asked to find a fraction (with a whole number numerator and denominator) that was greater than 1/2020, less than 1/2019 and with the smallest possible denominator.

Solver Amy Leblang used an algebraic approach, looking for a fraction a/b (where a and b are whole numbers) such that 1/2020 < a/b < 1/2019. Flipping all the fractions, you can rewrite this inequality as 2019 < b/a < 2020. Finally, multiplying through by a gives us 2019a < b < 2020a.

Again, our goal is to find the denominator b that’s as small as possible, and that will happen when a is also small. If we let a = 1, then there’s no whole number b that sits between 2019 and 2020, so that won’t work. But if a = 2, then we’re looking for a value of b between 4038 and 4040, which means b = 4039. Larger values of a will produce larger values of b, which this riddle wasn’t asking about. In other words, 2/4039 is the correct answer.

Many solvers, like Angela Zhou, observed that this question was straightforward if you knew a thing or two about Farey sequences, which are ordered sets of fractions between 0 and 1. For the Farey sequence in which all the denominators are at most 2020, the fractions 1/2020 and 1/2019 are “Farey neighbors,” meaning they’re next to right next to each other in the sequence. This riddle is effectively asking you to identify the first Farey sequence where 1/2020 and 1/2019 are no longer neighbors — that is, one where there’s another fraction between them. That fraction will be what’s called the mediant of the fractions on either side, generated by adding the numerators and the denominators: (1+1)/(2020+2019) = 2/4039.

Just to be extra sure that’s the right answer, solver Nolan Gannage wrote code to search all the fractions between 1/2020 and 1/2019 whose denominators were also 50,000 or less. Sure enough, the one with the smallest denominator was 2/4039.

That’s one of the Riddler maxims: “When in doubt, code it out.”

Solution to last week’s Riddler Classic

Congratulations to 👏 Austin Calico 👏 of Ashland, Kentucky, winner of the last week’s Riddler Classic.

Last week, you looked at an alphanumeric code inspired by “Gematria,” where words were assigned numerical values based on their letters. Each A was worth 1 point, each B was worth 2 points, and so on. The value of a word was then the sum of the values of its letters. For example, RIDDLER had an alphanumeric value of 70, since R + I + D + D + L + E + R became 18 + 9 + 4 + 4 + 12 + 5 + 18 = 70.

But what about the values of different numbers themselves, spelled out as words? The number 1 (ONE) had an alphanumeric value of 15 + 14 + 5 = 34, and 2 (TWO) had an alphanumeric value of 20 + 23 + 15 = 58. Both of these values were bigger than the original numbers.

Meanwhile, if we looked at larger numbers, 1,417 (ONE THOUSAND FOUR HUNDRED SEVENTEEN) had an alphanumeric value of 379, while 3,140,275 (THREE MILLION ONE HUNDRED FORTY THOUSAND TWO HUNDRED SEVENTY FIVE) had an alphanumeric value of 718. These values were much smaller than the original numbers.

If we considered all the whole numbers that were less than their alphanumeric value, what was the largest of these numbers?

First off, this question was a little ambiguous. The intent was to find the largest number N that was less than its “Gematria score” of N, which we’ll call G(N) — that is, N < G(N). However, you could also have read the question as asking for the largest value of G(N) among numbers N where N < G(N). The majority of readers answered the first question, but we’ll address both here.

Almost all solvers wrote code for this one, with the general strategy of (1) systematically describing how numbers are codified as words in English, and (2) scoring those words. A few solvers, like Cameron Shelton, took the time to work it all out by hand. In the words of Cameron, this “gives me a better feel for the problem and because I enjoy it more.” Bravo, Cameron!

Either way, the answer turned out to be 279, or TWO HUNDRED SEVENTY NINE, which had a Gematria score of 284. To confirm this result, here’s a graph from solver Jason Ash showing the scores for the numbers from 1 to 500.

First 500 Gematria scores

Sure enough, 279 is the last number above the dotted line, meaning it’s the greatest number to exceed its Gematria score. Interestingly, 80 is the smallest number that’s less than its Gematria score: EIGHTY is only worth 74 points. And for those of you who had the alternate interpretation of the original question, the number above the dotted line that’s worth the most points is 277, or TWO HUNDRED SEVENTY SEVEN, which is worth a whopping 307 points.

If you’re curious to see the scores of Gematria scores beyond those of the first 500 numbers, then solver Quoc Tran has an animation for you, showing the scores from ONE to ONE HUNDRED THOUSAND:

Finally, James Chapman took this riddle even further, solving it for multiple languages (not just English). James found that Finnish, French, and Polish each had answers just shy of 400. Dan Miller even went on to suggest using Roman numerals — maybe next time!

Want more riddles?

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

Want to submit a riddle?

Email Zach Wissner-Gross at [email protected].