## Women’s Tennis Is Finally Getting Young Again

In the French Open final, 19-year-old Markéta Vondroušová, who dazzles with drop shots, will face off with Ashleigh Barty, 23, who is crafty and quick with a ton of energy. Amanda Anisimova, 17, reached the semis by hitting smooth strokes and bullets all day long. And world No. 1 Naomi Osaka lost early in Paris but claimed two Grand Slam titles by the age of 21.

Women’s tennis is, at long last, getting young.

The average age of players who have won at least one WTA title this year is 23.6 years old. That’s the lowest age since 2008 — and it will drop even further after Saturday. At the French Open this year, the final eight women in the event were no older than 28.

This represents a sharp downturn from most of the previous decade. The average age of WTA winners floated between 25.8 and 26.1 between 2013 and 2016. But it also serves as a return to normalcy for women’s tennis, which had long been dominated by youth: The average age of WTA tour players in 1990 was 20.9, as 21-year-old Steffi Graf and 17-year-old Monica Seles finished the season ranked first and second. That average age would not rise above 23.5 for nearly two decades.

The amount of young talent is enormous again. And impatient. Since Serena Williams won her last major at the Australian Open in 2017, only one other Slam winner has been at least 30 years old: Angelique Kerber, who won Wimbledon at age 30 last year. The winners of the other seven majors: Osaka, who won the 2018 U.S. Open at age 20 and the 2019 Australian at 21; Jelena Ostapenko, who won the 2017 French at 20; Garbiñe Muguruza, age 23 when she won the 2017 Wimbledon;5 Sloane Stephens, who won the U.S. Open at 24; Simona Halep, who won the French Open last year at age 26; and Caroline Wozniacki, the 2018 Australia Open winner at age 27.

Part of this may be cyclical. As clusters of young, talented players emerge, a generation of older players who exceeded the normal life expectancy of a tennis career is nearing its end — and we aren’t just talking about the Williams sisters, who are now 37 and 38 years old. Francesca Schiavone, 38, won the French Open in 2010 but retired last September. Roberta Vinci, 36, shocked the world at the 2015 U.S. Open when she beat Serena Williams, who was two wins away from a golden Slam, but Vinci retired in May 2018. Agnieszka Radwanska, 30, who lost the Wimbledon final in 2012, called it quits in November.

Other players might not last that long, either. Samantha Stosur, 35, is in great shape and is an expert in doubles, but she struggles in singles. After next year’s Olympics, 30-year-old Dominika Cibulkova could decide to leave the game, too. Maria Sharapova, 32, has missed most of this season with injury. Victoria Azarenka, 29, is back after struggling since she had a child, but she might never be as powerful as she was in 2013.

After she lost a doubles match this year in Paris, Lucie Safarova announced she had retired from tennis. Now age 32, Safarova had played in Slams since 2005. She never won a major singles title, but she came close at the 2015 French Open, when she lost to Serena Williams in three entertaining sets. Safarova also won five doubles majors with Bethanie Mattek-Sands, including two in Roland-Garros.

“I think that it’s great that the competition is so strong,” Safarova said. “And we have, I think, at least 100 players now are amazing competitors. And you just have to be strong and play your 100 percent to be able to be here.”

On Saturday, Barty and Vondroušová will likely be nervous, as is usually the case for players in a Grand Slam final. But their youth — and their passion — is what this age of tennis is all about.

## You’ve Been Marooned By Kidnappers. Can You Escape At Dawn?

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 Marcus Farbstein and Mark Baird, a logical mystery story, with your very freedom hanging in the balance:

You’re super rich, and you often joke with your cadre of intelligent friends and family about getting kidnapped. You all agree that if you were ever kidnapped, the evildoers would knock you out so stealthily that you’d never feel the blow. Then, one snowy night, you step out of a restaurant and, just as predicted, never feel the blow.

When you stir back into consciousness, it’s night, but it’s not snowy. You find yourself sitting on a beach. The sky is clear, with no moon. In front of you stands a shadowy figure whose face you can’t make out. He throws a blocky rectangular object at your feet. “That’s a satellite phone,” the figure growls. “It’s got one minute of battery left in it. Use that to call your people to let ’em know you’re not dead — but not until daylight.” He tosses a paper bag next to the sat phone. “That’s some sandwiches and water, enough for a few days. That’s salt,” he explains, waving toward the surf. “If your people pay our ransom,” he continues, “We’ll come get you. Otherwise, there won’t be any more paper bags. Remember, wait until daylight to make that call.”

He then turns and climbs into a dinghy in the light surf, starts its outboard motor and zooms away. All this time you’ve been too groggy to do anything but listen. Now you watch as the dinghy disappears into the gloom, its wake a faint wash of phosphorescence that quickly fades. Later, there’s a bare wink of lights at the horizon, presumably the mothership getting underway and leaving.

Even though it’s a moonless night, there’s sufficient starlight to assess your surroundings. Your grogginess is gone and you walk about. You’re on a tiny island, which you estimate is a bit more than a mile by half a mile. There are no trees; it’s all flat sand. You taste the water rolled up by the surf, and it is indeed salt. The air is cool, but not cold. Your wallet, expensive chronometer, keys, cell phone, jewelry and small change are all gone; all you have are the clothes on your back — even your shoes and socks have been taken. The bag contains four sandwiches, all liverwurst with peanut butter on cheap rye bread, and four one-pint bottles of water. No napkins. Your knowledge of astronomy is too weak to try to estimate your location by the stars, but you’re not stupid. Before daybreak, you’ve worked out exactly how you’ll use that minute of time on the satellite phone so that your people, who are also not stupid, will be able to dispatch rescue.

What will you say?

## Riddler Classic

From Theodore James, some further international intrigue:

Mathematician Dr. Lana Gurtin has a problem to solve. She was hired by British intelligence for a top-secret assignment, but things have not gone according to plan. The year is 1942 and the Germans are rolling out a new and powerful tank, the Uberpanzer. Prominently displayed on the back of each Uberpanzer is its serial number, which is simply the number of tanks that had been built when it rolled off the line. So the first tank built has the serial number 1, the second one built has a 2, and so on.

Recently, a number of these new tanks were spotted by British scouts who recorded the serial numbers that they saw. They immediately sent this information to British intelligence, hoping that the serial number data could be used to estimate the total number of Uberpanzers the Germans have built. This is when Dr. Gurtin was brought on to head the project.

But then the unexpected happened. A German spy intercepted the dossier with the data before it could reach MI6. By the time British agents caught up with the spy, most of the data had been destroyed. However, two pieces of information were recovered from the debris. One: The lowest serial number recorded was 22. Two: The highest serial number recorded was 114.

Luckily, Dr. Gurtin knows exactly what to do. Assuming that the original data set was a random sample of serial numbers, what is Dr. Gurtin’s best estimate of the total number of Uberpanzers the Germans have built?

## Solution to last week’s Riddler Express

Congratulations to Tyler Silber of New York City, winner of last week’s Riddler Express!

Last week, you and your friend faced off in a friendly game of Lotería, a traditional Mexican game of chance akin to bingo. You each had a card with a four-by-four grid of images, drawn randomly from a deck of 54 possible images. Each image could appear at most once on a card. A caller randomly drew cards from a deck containing all 54 possible images, and you marked that image off on your card if it appeared there. The game ended when one of the players filled their entire card. What was the probability that either of you ended the game with an empty grid — that is, the odds that none of your images was called?

The chances were minuscule — about $$3.508 \times 10^{-12}$$, or on the order of one in a trillion.

Taylor Firman, this puzzle’s submitter, writes: “The inspiration for this problem actually comes from reality: While on vacation, a friend of mine ended a game with zero matching images and I seemed to be the only one flabbergasted by the odds.” And he discusses how to arrive at these flabbergastingly minuscule odds on his blog.

The solution flows from two simple facts. First, the players’ two grids can’t have any overlap — that is, they can’t share any images. If they did, when one player won, the other player’s grid couldn’t be empty. Second, the deck of images must be ordered such that all of one player’s images come up before any of the other player’s images. The mathematical trick is to combine the probabilities of these two events into our final answer.

For example, we can begin by calculating that, given the first player’s grid, there are 38!/(16!22!) — or about 22 billion — non-overlapping grids that the second player could have, out of a raw total of 54!/(16!38!) — or about 21 trillion — possible grids. These are the formulas for choosing 16 cards from sets of 38 (the 54-image deck minus the 16 images on the first player’s grid) and 54 cards, respectively. We can calculate the number of orderings of the images similarly, using factorials and the tools of combinatorics. I refer you to Taylor’s solution for the gritty details.

You were also asked how the probability of having an empty card at the end of the game changed as the rules of the game changed — for example, if there were more or fewer unique images in the deck. Taylor provided that data, too, in interactive chart form. If the size of the grid was fixed at four-by-four, here’s how the zero-match probability increased as the number of unique images increased:

The version of the chart on Taylor’s site also lets you see how the odds change for different grid sizes.

One in a trillion, you say? Never tell me the odds. Unless, of course, I ask you to in a math puzzle column, in which case, please do tell me the odds.

## Solution to last week’s Riddler Classic

Congratulations to Eric Mann-Hielscher of Brooklyn, New York, winner of last week’s Riddler Classic!

On the excellent British game show “Countdown,” there is a segment called the Numbers Game. You ask for six numbered cards in total — up to four “large” cards, with “small” cards making up the balance. Large cards are drawn at random from a deck containing the numbers 25, 50, 75 and 100. Small cards are drawn at random from a deck containing two each of the numbers 1 through 10. Then a random number generator spits out a three-digit target number, and you have 30 seconds to use addition, subtraction, multiplication and division to combine your six numbers into a total as close to the three-digit number as you can. (You can only use a number as many times as it comes up in the six-number set. You can only use the mathematical operations given. At no point in your calculations can you end on something that isn’t a counting number. And you don’t have to use all of the numbers in your set.) If you can nail the target exactly, we’ll call that game solved.

This riddle was twofold. First, what number of large cards is most likely to produce a solvable game and what number of large cards is least likely to be solvable? Second, what three-digit numbers are most or least likely to be solvable?

Two large cards is best — you can solve about 98 percent of all target numbers. Zero large cards is the worst — you can solve only about 84 percent of the targets. In general, large target numbers tend to be the hardest to solve. Assuming you’ve chosen the optimal two large numbers, 967 is the most difficult target number — you can only solve it about 89 percent of the time. There are a few smaller target numbers that you can always solve.

Given the huge number of combinations of cards and target numbers, this riddle is a problem for a computer to solve. Well, for you and your computer. Our winner Eric, along with solvers Benjamin Phillabaum, Ryan Vilim and Ben Weiss, were kind enough to share their code.

Solver Jason Ash plotted the results of his programmatic solution for each number of large cards. You can see that the chances of solving tend to go down as the target number gets bigger, and that two large cards is the best choice — its chances of solving hover closest to 100 percent.

And solver Austin Buscher provided another look at the problem, charting the frequency of combinations capable of yielding each three-digit number. “The histogram shows what we would intuitively expect,” he writes. “Smaller numbers can be computed more often. There are also peaks at multiples of 25, 50 and especially 100, another intuitive byproduct.”

Now if only I could do all of this on TV in less than 30 seconds …

## 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]

## Most Americans Agree That WWII Was Justified. Recent Conflicts Are More Divisive.

Welcome to Pollapalooza, our weekly polling roundup.

## Poll of the week

On Thursday, Americans both at home and abroad commemorated the 75th anniversary of D-Day, the seaborne invasion of France that helped usher in an Allied victory in World War II. The widespread observance of the occasion was hardly surprising; most Americans see World War II as a proud moment in U.S. history. A full 66 percent of Americans said they thought the U.S. role in World War II was “completely” or “somewhat” justified, according to a YouGov poll out this week, while just 14 percent said it was “not very” or “not at all” justified.

That makes World War II the most popular U.S. military engagement of the eight that the poll asked about, as measured by the net share of people who said it was justified (+52). The American Revolution came in second (+47); 61 percent said it was justified and 14 percent said it was not justified. (Yes, apparently after nearly 250 years, there are still some Loyalists out there!) Americans’ perceptions of more recent wars are more complicated. For some of those conflicts, a greater share of people said they see them as unjustified than see them as justified. One of those is the Vietnam War, which 55 percent said was not justified and 22 percent said was justified.

The Civil War is the third-most-supported war in the poll, with 54 percent of Americans saying it was justified and 22 percent saying it wasn’t. Perhaps surprisingly, about the same share of Southerners said the war was unjustified as residents of other regions. Respondents from the South said the Civil War was justified by a 52-percent-to-23-percent margin. As anyone who has seen a Confederate flag in Maine can tell you, Confederate sympathies can be found in all corners of the country.

Similarly, there weren’t big partisan differences on the U.S.’s involvement in the Civil War. About the same share of Republicans (61 percent) as Democrats (57 percent) said they thought it was justified.

But there’s a catch. These numbers may have less to do with the historical reasons behind the war and more to do with Americans’ philosophies about going to war in general. In answer to a different question, 59 percent of Republicans told YouGov that there is “often” or “always” a justification for war, while 29 percent said there is “rarely” or “never” a justification. Among Democrats, those numbers were 21 percent and 66 percent, respectively. 10 Unsurprisingly, then, Republicans are more supportive on net than Democrats of every conflict YouGov asked about — but the gap between the parties is smallest on the Civil War.

##### More Republicans than Democrats think wars are justified

Share of each party that said a given conflict is “completely” or “somewhat” justified vs. “not very” or “not at all” justified, according to a May 21-22 poll

Democrats Republicans
Conflict Justified Not justified Net Justified Not justified Net Diff.
Afghanistan 26% 55% -29 56% 27% +29 R+58
First Gulf War 28 47 -19 58 20 +38 R+57
Korean War 30 40 -10 56 17 +39 R+49
Vietnam War 14 68 -54 37 44 -7 R+47
World War I 51 27 +24 68 14 +54 R+30
American Revolution 60 18 +42 74 7 +67 R+25
World War II 67 17 +50 77 9 +68 R+18
Civil War 57 24 +33 61 18 +43 R+10

Source: YouGov

Overall, Democrats tended to view older wars (World War II and earlier) as justified, bringing them in closer agreement with Republicans. After the Civil War, the parties are closest together on World War II and the American Revolution. But there is more partisan polarization over more recent wars. For example, Republicans said by a +29 net margin that the current military engagement in Afghanistan is justified, while Democrats said it was not justified by the same margin (-29). The Vietnam War is the only conflict in the poll that both parties said was not justified. But a large majority of Democrats felt very strongly about it while Republicans were divided, so there remains a significant gap between the parties.

## Other polling bites

• In a press conference on May 29, then-special counsel Robert Mueller said, “If we had had confidence that the president clearly did not commit a crime, we would have said so.” In a survey released this Monday, HuffPost and YouGov asked Americans who were aware of that statement whether they thought Mueller’s report had or had not cleared President Trump of any wrongdoing. Seventy-four percent of Trump voters said they thought it had. They were more likely to say that than Trump voters who were not aware of Mueller’s statement.
• So far in 2019, several states have passed “fetal heartbeat” laws that ban abortions after fetal cardiac activity can be detected, which can be as early as six weeks into a pregnancy. A new USA Today/Ipsos poll found that Americans oppose these laws 55 percent to 45 percent. In addition, as Missouri may soon become the first state without an abortion clinic, Americans said 73 percent to 27 percent that they don’t want every abortion clinic in their state to close.
• Morning Consult’s continuous poll of the Democratic presidential primary found that the share of Democrats who say “women’s issues” are their top priority rose from 6 percent to 14 percent in the month of May.
• In a new poll from Monmouth University, 27 percent of Americans said someone in their household did not receive needed medical care in the past two years because of the cost. And 20 percent reported that the need to hold onto their health insurance plan had prevented them from trying to change jobs at least once in the past 10 years.
• An Opinium poll in the United Kingdom has put the nascent Brexit Party in first place for U.K. parliamentary elections for the first time. When asked which party they would support, 26 percent said the Brexit Party, 22 percent said Labour, 17 said the Conservatives and 16 percent said the Liberal Democrats. Multiple other polls in the past couple of weeks have shown a similar four-way pileup at the top, a remarkable shift from the traditional two-party dominance of Labour and the Conservatives. It reflects how much Brexit has come to dominate U.K. politics — the Liberal Democrats and Brexit are explicitly pro- and anti-European Union, respectively, while Labour and Conservative opinion is muddled.

## Trump approval

According to FiveThirtyEight’s presidential approval tracker, 41.5 percent of Americans approve of the job Trump is doing as president, while 53.6 percent disapprove (a net approval rating of -12.1 points). At this time last week, 41.2 percent approved and 54.2 percent disapproved (for a net approval rating of -13.0 points). One month ago, Trump had an approval rating of 42.7 percent and a disapproval rating of 52.4 percent (for a net approval rating of -9.7 points).

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

## Get To Know The 2019 U.S. Women’s National Team

This year’s U.S. women’s national team is a mix of familiar faces and young talents. A dozen players from the team’s winning run in the 2015 Women’s World Cup have returned, including eight of the 11 players who started the final match against Japan. But this is the first World Cup for some of the team’s key players, like midfielder Rose Lavelle, forward Mallory Pugh and defender Abby Dahlkemper.

But who are they, both on and off the field? We’ve been diving through their stats, interviews and social media musings to put together our guide to the team’s likeliest impact players. The U.S.’s path to a record fourth World Cup starts on June 11. Don’t get caught flat-footed.

## Tobin Heath

As a youngster, Heath was drawn to the Brazilian style of soccer, with its flair and panache. Her favorite player growing up was Ronaldinho, the Brazilian legend who played as if the ball were glued to his foot. Heath took her ball with her everywhere, sending it through the legs of chairs in her house — and also her mom. That informed the style that would become her hallmark with the Portland Thorns and the national team.

One of the most technically proficient players on the team, the 31-year-old Heath tends to create space out of nowhere thanks to those crafty dribbling skills. Fans call her the Nutmeg Queen because of how often she humiliates defenders by passing the ball through their legs to bypass them. Nutmegs are sadly not a stat tracked by any data-keepers, but Heath’s dribbling is among the world’s best, standing sixth in international play in completed dribbles per 90 minutes since 2017. Some of them surely went through the five-hole.

## Alex Morgan

Morgan is the most marketable and famous member of the team, with an off-the-field empire — a movie, a streaming series and a line of books — that continues to grow regardless of whether she’s scoring goals. Luckily for the USWNT, Morgan enters this World Cup as a healthy starter for the first time after off-and-on injury woes, which means she could finally have the breakout performance at a World Cup that fans have been waiting for.

The 29-year-old Orlando Pride striker is at her best when she races behind backlines and takes on goalkeepers, but she also tends to check back and hold up the ball for other talented attackers. Sometimes that means she doesn’t score as many goals as fans or critics would want, even if she’s setting up the players around her. But with 28 goals in her last 36 games for the USWNT, it’s a good bet that Morgan will add to her tally in France.

## Megan Rapinoe

As notable as “Pinoe” is on the pitch, she’s just as well known for her outspokenness off it. She joined Colin Kaepernick’s protest of kneeling during the national anthem in 2016 until U.S. Soccer took steps to ban it, which turned the affable and gregarious Rapinoe into a divisive figure. She’s also been outspoken against President Trump but says there’s no conflict in wearing the United States crest over her heart; the openly gay native of conservative Redding, California, says she is “a walking protest.”

On the field, both for the USWNT and Reign FC, she plays with a similar confidence. She’s unpredictable, she can score in different ways, and she sometimes takes over games with her individual brilliance, with a right-footed delivery that stands up against any in the game. At 153 caps, the 33-year-old Rapinoe is one of the veterans on this squad, having competed in the two previous World Cups, and she memorably assisted Abby Wambach in 2011 on the latest goal in World Cup history, a 122nd-minute equalizer to advance past Brazil in the quarterfinal.

## Julie Ertz

Fans may remember Ertz as Julie Johnston from the 2015 World Cup, but in 2017, the Chicago Red Stars defender married Philadelphia Eagles tight end Zach Ertz. (Her teammates and coaches still call her J.J.) In the last World Cup, she was a center back along the backline, but she is a defensive midfielder now. In her new role, Ertz shields the backline while pushing up into the attack, where she has proven to be the team’s best header of the ball. Since 2017, the only Americans with more headed goals than Ertz in international play are Carli Lloyd and Lindsey Horan.

Ertz’s role for the USWNT is more crucial this time around because there isn’t another like-for-like player on the roster who provides both the mobility and physicality that Ertz does. The 27-year-old puts her body on the line so often that finishing games with gauze to stop any bleeding has become something of a joke among USWNT fans.

## Lindsey Horan

Lindsey Horan took an unusual path to the national team. She was the first American woman to go pro straight out of high school, forgoing a scholarship from the powerhouse program at UNC to sign with Paris Saint-Germain at 18 years old. Back then she played as a striker, but after coming back to the U.S. to play for the Portland Thorns, she has transformed into an all-around midfield threat.

The 25-year-old can score goals, and she influences the game by controlling the midfield and disrupting opponents. In the NWSL last year, Horan was involved in 555 duels across 24 matches — 190 more duels than any other player in the league — and she won the ball 321 times, which is 116 more than anyone else. Horan is among the league’s best at progressing the ball at her feet, a plus ball-winner in central midfield, and by far the most effective long passer in the league.

## Rose Lavelle

Lavelle has a bulldog dog named Wilma that she tweets about a lot, and her social media presence is filled with videos of her performing choreographed hip-hop dances with her teammates. The Washington Spirit midfielder always seems to be having fun, and she plays soccer the same way.

Of the World Cup debutants, Lavelle could play the most crucial role. The 24-year-old is a classic “No. 10” player, which means she’s a playmaker who can unlock defenses with slick passes and creativity. She is one-of-a-kind on the USWNT’s roster, and coach Jill Ellis is counting on her to create goals. Lavelle is among the most creative passers on the team, passing less often than her midfield teammates but aggressively when she does. This is her first major tournament, and how she responds to the big stage could make or break the U.S. team.

## Abby Dahlkemper

Dahlkemper’s national team career nearly ended right after it started. A few weeks after her first USWNT call-up in 2016, she contracted a serious sepsis infection and worried that she might lose her leg. She was bedridden for six weeks until surgery and intravenous antibiotics cleared the infection. By mid-2017, she was back on the national team radar, where she has been a fixture as a center back ever since.

The 26-year-old has been one of the best defenders in the NWSL lately, and for the national team, she plays a specific role of pinging direct balls up the field to striker Alex Morgan. For her play with the North Carolina Courage, Dahlkemper has stood out among NWSL center backs for both the volume and precision of her passing. During the past two NWSL seasons, she has completed the third-most long balls per 90 minutes among center backs, and her 55 percent completion rate ranks second.

## Crystal Dunn

What position does Dunn play? Good question. She’ll start for the USWNT as a left back, but it’s not unusual to see Dunn flipped over to the right side, shifted to the central midfield or up the field as a winger — sometimes all in the same game. Even as a left back, she’ll often be high up the field when the U.S. has possession, giving the team an asymmetrical look of having an extra attacker on the left and just three defenders in the back.

It makes sense — Dunn, 26, plays as an attacker for the North Carolina Courage, and she has traditionally been one of the most productive attacking players in the NWSL. Among league players who aren’t pure strikers,1 only Tobin Heath can match her 0.81 nonpenalty goals and assists per 90 minutes in the last two seasons.

## Kelley O’Hara

O’Hara’s journey to right back was a long one. She broke into the national team in 2010 as a forward. For the 2012 Olympics, she was converted into a left back. By the time the U.S. won the World Cup in 2015, she was in the midfield. For her club teams — currently the Utah Royals; before that, Sky Blue FC — she’s been shuffled around nearly as much. Now, the 30-year-old has returned to the back line, but on the right side.

Given her experience elsewhere on the pitch, O’Hara has as strong an attacking mindset as any defender could. Look for her to bomb up the field and combine with Rose Lavelle and Tobin Heath in front of her. O’Hara is a key creative outlet and playmaker for the U.S., and there’s a good case that no fullback in the world does more to support her midfield in possession: She has tallied 4.9 progressive passes played and received per 90 minutes since 2017, behind only France’s Amel Majri and O’Hara’s American teammate Crystal Dunn for most in international competition.

## Becky Sauerbrunn

A lover of cats and science fiction books, Sauerbrunn — captain of the Utah Royals FC — is the cerebral foundation of the U.S. back line and the most experienced veteran on the defense.

She was a late bloomer on the national team — her big break came at 25 years old when she replaced an injured player. But she has been a fixture with the USWNT since, and she was the NWSL defender of the year in 2013, 2014 and 2015. She just turned 34, and this may be her last World Cup, but she’ll be needed as the leader of a relatively young back line. At 158 caps, Sauerbrunn is the third-most capped player on the team, and she’ll be the second-most capped starter behind Alex Morgan.

## Alyssa Naeher

For two decades, the U.S. has put two strong personalities in goal: Hope Solo and Briana Scurry. Naeher, 31, takes a different, quieter approach. Perhaps the most controversial thing the Chicago Red Stars keeper has ever said is that she prefers to do The Wall Street Journal crossword puzzle with her morning coffee – sorry to The New York Times.

Naeher isn’t the shot-stopper Solo was during the USWNT’s previous World Cup cycle — Naeher’s save percentage for the U.S. since Feb. 13, 2015, has been 71 percent compared with Solo’s 79 percent, according to Opta Sports.2 But Naeher plays a bigger role in the team’s offense, completing about 14.3 passes per 90 minutes for the national team over the past two years, compared with Solo’s 8.7 per 90 in 2015-16. Naeher’s greater skill with the ball at her feet is apparent in her long passing: She completes 4.2 passes over 25 yards long to Solo’s 2.6, and she has a higher completion rate on those long passes, 48 percent to 42 percent.

## Carli Lloyd

Anyone who only tunes into the Women’s World Cup every four years may be surprised to see Lloyd — the hero of 2015 — listed as a substitute. But the Sky Blue FC midfielder will turn 37 soon after the tournament ends, and coach Jill Ellis has determined that the younger attackers on the roster are a better stylistic fit. Lloyd, a no-nonsense New Jerseyite who is known for training even on vacation and making her own ice baths wherever she goes, isn’t happy about being a substitute, and she’s not afraid to show it.

Lloyd’s age certainly hasn’t hurt her form. Even as her role and minutes have been reduced, she’s been incredibly effective in a U.S. kit. Her 10 nonpenalty goals since 2017 is third-best for the US despite more limited playing time in the last three years.

## Samantha Mewis

When U.S. Soccer conducted a written test to gauge the soccer IQ of players of both genders, Mewis scored in the top 1 percent. Credit all those books she reads — the North Carolina Courage midfielder says it’s her favorite hobby, and Harry Potter is her favorite series. At 5-foot-11, she is the tallest field player the USWNT has ever had.

Mewis, 26, may not be the best at any single thing, but she is very good at everything she’s asked to do, and she can slot into any central midfield role that might open up. She also isn’t afraid to fire shots from distance, which forces backlines into a tough decision of how to defend her. Mewis can win a tackle and play a pass like her fellow midfielders, but what sets her apart most notably are the one to two shots from outside the penalty area that she unleashes per match — of the USWNT players since 2017, only Rapinoe (of course) has tried her luck from range more often.

## Christen Press

Press was a standout in college as a striker, setting Stanford’s all-time scoring record at 71 goals. But she only broke into the U.S. national team when she gave up on trying to make it — she went to play in Sweden, where she figured U.S. scouts wouldn’t see her. In the process, she rediscovered her love for soccer. Although the 30-year-old now plays stateside in the NWSL for the Utah Royals, she has said that she still likes to enjoy the Swedish coffee break known as fika, and she’s a proponent of meditation.

It hasn’t been easy for Press to find a place on the team. She couldn’t supplant Alex Morgan as the team’s go-to striker, and in 2015 she was thrust into a wide midfielder role where she never looked comfortable. Now, she has embraced being a winger and has been a consistent threat off the bench. She has averaged 0.78 nonpenalty goals and assists per 90 for the U.S. since 2017 while providing nearly one-third of her international minutes as a substitute.

## Mallory Pugh

Pugh broke into the USWNT at 17 years old, and she’s been a regular ever since, so it’s easy to forget that she is the second-youngest player on the team, having just turned 21 in April. She left UCLA to turn pro with the Washington Spirit before ever playing in an official game, following in the footsteps of fellow Coloradan Lindsey Horan.

While expectations have been heaped on Pugh as the future of the team, she figures to be a “super-sub” off the bench for the Americans this summer. Pugh, like Press, is one of the weapons that sets the U.S. apart from the rest of the world, and she has been among the 10 most efficient scorers in international football in the last two years.

Annette Choi contributed visuals to this story.

Check out our latest Women’s World Cup predictions.

## Rafael Nadal Is Playing Mind Games With You

When you think of Rafael Nadal, you might think of a player who hits balls with hellacious topspin and grinds out points on clay. His RPMs and his sweat grab the glory. But the 11-time French Open champion uses a few insidious tricks that go beyond the obvious strokes and traditional tactics.

All of Rafa’s ways and means traveled to Roland Garros in 2019 — the energy, the rituals, the patterns of play — it’s all been put to use in another run to the semifinals, this time at age 33. He’ll need every tactic at his disposal, the conspicuous and the cunning, as he takes on Roger Federer and potentially Novak Djokovic after that.

Here are three examples of the subtle mental maneuvers that Nadal makes against his opponents.

## He makes them wait

Strictly from a length-of-match standpoint, Nadal is one of the slowest tennis players of this era. And there are a host of things Nadal does to extend matches — and possibly distract and annoy his opponents in the process.

The ultimate creature of habit, Nadal starts managing time with his first step on court. When the chair umpire prepares to toss the coin and the presence of both players is required in the middle of the court before the start of the match, Nadal is typically the second to arrive — after a delay of several seconds while he goes through his routines with water bottles.

Once the match is underway, Nadal’s pre-point rituals have been fodder for everything from complaints to comedy routines. Even the typically chilled-out Roger Federer has been critical of the time that Nadal takes between points. By rule, players are limited to 25 seconds between points. Beginning this year at most events, the sport put in formal, visible shot clocks in an attempt to keep servers from abusing the rule.

According to an analysis by Melbourne, Australia-based Data Driven Sports Analytics of more than 140 matches each for Nadal, Federer and Djokovic from 2008 through this year, Rafa averaged 26.1 seconds between points when serving — the longest of the so-called “Big 3.”3 Nadal’s average time between points is over the limit — and that’s just an average, which means that he regularly serves beyond the 25-second rule. Chair umpires can use their discretion in starting the clock, so, clearly, Nadal is getting some wiggle room.

Nadal finds a way to play on Rafa time when he’s returning as well, going through a catalog of rituals and often turning his back on the server or lifting his racquet until he’s ready to receive.

The overall effect is that Nadal asserts his own pace of play, which can be legitimately discomfiting for opponents.

## He conditions them like Pavlov

One of the hardest things to do in professional tennis is return serves. Speeds regularly top 125 mph, and then there’s the spin. Professional tennis players also excel at “spot serving” — landing serves in precise locations. They most often hit close to the lines of the service box, placing the serve at angles to inflict the most damage. Those most-visited, go-to spots are either up the T, which is the middle of the court, or out wide, which is on the outer edge of the service box. Servers work hard to place their serves effectively on both the deuce side (serving from the right side to the servers’ left and returners’ right), when the game score is usually tied, and on the ad side, when one player is always ahead.

Nadal has a curious modus operandi when serving on the deuce side. The effect is Pavlovian: It conditions his opponents for one thing and then kills them with another.

Because Nadal is a left-handed server, the natural play for him in this situation is to serve up the T. The ATP Tour has collected serve placement data from 2011 to 2019 for Masters 1000 events, which are just beneath Grand Slams and the Tour Finals in terms of stature, ranking points and prize money. According to that data set, on clay, Nadal’s first serve has been up the T 56.4 percent of the time on the deuce side. His success rate for this location — meaning how often he wins the point — is a healthy 68.5 percent. Much less often — 27 percent of the time — Nadal takes his first serve out wide from the deuce court on clay. And in that spot, his success rate is eye-popping, 74.8 percent. In 2019 alone, it’s up to 79.2 percent.

Why would Nadal use a serve that is statistically so successful for him so infrequently? It’s possible that the tactic is about mentally conditioning the returner, greasing the tracks as it were, and then flipping his pattern when he really needs it.

Indeed, data from that same set of ATP Masters tournaments reveals this morsel about Nadal on clay: He habitually hits his first serve from the deuce court up the T on nearly all scores. The most notable exception: when he’s down 15-40. When Nadal faces two break points against him, his primary service pattern switches to his secondary, “money” spot — the out wide. At 15-40, he goes T only 39.7 percent of the time, and his primary pattern becomes out-wide, at 44.9 percent.

How does the King of Clay perform with that deuce court, out-wide serve down two break points? Put simply, he crushes: 82.9 percent of the time he wins the point.

It’s a smart and subtle way to serve in the situation, and his success rate suggests that his opponent, potentially predisposed to Nadal’s T serve, does not see it coming.

Granted, it’s also a bit of a “tell” for anyone lucky enough to find himself with a pair of break points against Nadal — those guys should look for the out wide serve. But more than that, it reveals a mental game-within-the-game orchestrated by Nadal.

## He turns balls hit to his backhand side into forehand winners

Nadal’s forehand is his biggest weapon. Opponents try to dodge it at all costs, which means avoiding hitting the ball to the ad court as much as possible against the southpaw. A good way to understand the baseline is to divide it into four vertical zones — two in the deuce court and two in the ad court. In tennis, these zones are sometimes labeled A, B, C and D, with A being the out wide in the deuce court, all the way to the D zone, which is the out wide in the ad court.

If you’re Nadal’s opponent and you’re trying to avoid his forehand, you would hit to zones A and B (the ad court). And that’s where he gets you.

The King of Clay is also the King of Running Around His Backhand to Hit Forehand. He’s an expert at it. Indeed, he loves to run around his backhand to hit forehand so much that in some matches, he has hit about the same number of winners from what would be his natural backhand side of the court — zones A and B — than from his normal, left-handed forehand side of the court — zones C and D. Through five rounds at this year’s French Open, 54 percent of Nadal’s forehand winners (46/85) have been hit as run-around forehands from zones A and B, according to officially recorded statistics from Roland Garros.

By comparison, consider the right-handed Djokovic, the number one player in the world. Through the first four rounds at Roland Garros this year, 42 percent of Djokovic’s forehand winners have been run-around forehands (14/33).

Nadal is like a spider looking to snare a rally ball, and players would be ill-advised to hit toward Nadal’s backhand unless they can be sure he’ll only be able to use his backhand. At the same time, better not hit too far out wide or the errors will flow.

Just when you think you know Nadal, think again. He will bend your mind more than he bends the ball.

## How Trevor Bauer Remade His Slider — And Changed Baseball

Travis Sawchik is a FiveThirtyEight staff writer. His new book “The MVP Machine: How Baseball’s New Nonconformists Are Using Data to Build Better Players,” co-authored with The Ringer’s Ben Lindbergh, is available this week. In it, they examine how outsiders (and a few forward-thinking insiders) are employing unconventional ideas along with new data from new technology to lead a bottom-up revolution in improving skill levels. We’re publishing an excerpt of the book on how Cleveland Indians pitcher Trevor Bauer, a trailblazer in player development, used new technology like the high-speed Edgertronic camera, which he introduced to baseball — along with some stealthy reconnaissance — to fuel his 2018 breakout. It was Bauer who ushered a new, game-altering field into the sport: pitch design.

As April went on, Bauer became increasingly frustrated with his slider. Each time he sat in front of his locker after games and pulled up the pitch’s horizontal movement on his smartphone, he noticed it declining. His off-season project was failing. Although he’d had a fine first month (2.45 ERA, 46 strikeouts and 16 walks in 40 1/3 innings), he was frustrated with the pitch’s lack of improvement and doubted he could sustain success without it.

On April 1, his slider averaged 6.7 inches of horizontal movement. In his second start on April 7 versus the Royals, the horizontal movement fell to 6.2 inches, and then to 4.5 inches on April 12 at Detroit. He wanted 10 inches of horizontal movement. Although he was pitching relatively well, he was also facing some of the weakest teams in baseball. In terms of performance against extra-divisional opponents, the 2018 AL Central was the second-weakest division of all time, and Bauer didn’t have to face its best team. On April 20 at Baltimore, the horizontal movement averaged 5.1 inches. He also wanted the slider to have zero inches of vertical depth to keep it on the same plane as his fastball for as long as possible, creating deception. Instead it was breaking 3 to 4 inches. The slider was behaving like a less effective version of his curveball.

The Indians did not travel with their Edgertronic camera, Bauer says, and its use at home was sporadic. There were occasions when the camera wasn’t used because a Fox Sports TV camera occupied the camera well behind the plate. “The TV guys were down there,” Bauer recalls the club explaining. “I don’t give half a fuck,” he said. “Tell the TV guys to get out of there.”

The Indians’ video personnel also lacked familiarity with the device, and as a result, some bullpen sessions were not recorded properly. It was all maddening to Bauer. There was another potential use of the camera that the Indians weren’t fully exploiting, although the Astros—and Bauer before them—had done so previously: intelligence gathering. Bauer didn’t just want to study himself; he also wanted to study the other best pitchers in baseball. The ironic thing about criticism from past teammates like Miguel Montero was that Bauer didn’t believe he had all the answers. He was always searching for better information and practices.

On his own team, Corey Kluber and Clevinger had two of the best sliders in baseball. As Bauer designed his slider in the winter, he had studied their grips, which he had filmed the previous season. And on April 13, there was another pitcher he wanted to study: Toronto’s Marcus Stroman.

Stroman and Bauer had long admired each other from a distance. Prior to being selected in the first round of the 2012 draft, Stroman called Bauer a “pioneer to the pitching world” on Twitter. Stroman said he used to watch tape of Bauer’s UCLA starts in his Duke dorm room. Bauer cited Stroman as his favorite non-Cleveland player to watch. Now Bauer wanted his elite slider.

Leading up to that start on a frigid April evening, Bauer hounded the video staff and its coordinator, the white-haired and studious looking Bob Chester, to make sure they got Edgertronic footage of Stroman. But in the first inning, as Clevinger pitched for the Indians, neither Chester nor the camera was in the camera well behind home plate. If this start wasn’t filmed, Bauer would be livid. Between innings, Chester and the Edgertronic appeared in the camera well. As Stroman started pitching, Chester attached the camera to a tripod directly behind home plate. He then left the area. But the camera view was obstructed. This was Bauer’s one chance to get a look at Stroman and his slider on the Edgertronic. The first inning was over, and there was no usable footage. But Chester, realizing the error, returned in the second inning and repositioned the camera more to the left of the plate, allowing for an unobstructed view of Stroman. It was perhaps the most important repositioning of a high-speed camera in the brief history of pitch design.

After the game, Bauer immediately dove into the video: eleven minutes and fifty-one seconds’ worth of Stroman throwing pitches, a global shutter capturing every detail in thousands of frames per second. In May, he shared the video with Travis and motioned to Stroman’s right hand. “You see his thumb?” Bauer said. “It slips really early.”

That was the key. As Stroman’s upper body rotated and his right hand came through to throw, his thumb lost contact with the ball first, with his index and middle fingers still in contact before release. The middle finger was on the far side of the ball, the index finger behind the ball. Bauer paused the video.

“His middle finger never gets to the front of the ball. It just kind of brushes the side of it,” Bauer said. “Then you can only see his pointer finger appear once there is separation between the ball and his hand. … The pointer finger pushes the ball [to Stroman’s left], which puts more of a sidespin component on the ball. When I saw this video I was like, I have to find a way to get my thumb to slip earlier while my hand is still behind the ball.”

Two days later, Bauer threw his bullpen indoors, in a concrete bunker in the depths of Progressive Field. He commandeered the Edgertronic and filmed the session. One of his first throws resulted in a nearly perfect gyroball; not what he wanted. His index and middle fingers wrapped too much around the ball. His thumb was still in the way. To get his thumb to leave the ball earlier, he tried tucking it under the ball. This would push the axis up, he hoped, creating more sidespin and resulting in more lateral movement. One of his first attempts resulted in a pitch that got away from him; had a right-handed batter been in the box, it would have hit him in the helmet. He adjusted, applying more pressure with his index and middle fingers on the far side of the ball, running along the wide part of the figure-eight seam. Progress. The axis tilted up slightly, but not enough.

Bauer took his experiment to the mound over the following four starts at Baltimore on April 20 (5.1 inches of horizontal movement), against the Cubs at Wrigley Field on April 25 (4.8), at home against the Rangers (5.4), and at Yankee Stadium (4.4). It still wasn’t working. A day before his May 11 start at home against Kansas City, Bauer discovered something while lobbing balls back toward the infield during batting practice. Instead of gripping the “horseshoe” part of the seam with his middle finger as he had since the winter—and as most pitchers do when throwing a slider—he tried throwing one with a two-seam fastball grip, only tucking his thumb like Stroman. He spread his index and middle fingers over a narrower stretch of parallel seams and tucked his thumb and locked his wrist as he always did.

“I was like, holy shit,” Bauer recalls. “I could definitely see that the axis was different. I’m throwing the ball toward the collection bucket. I just flipped a couple. I saw a left-handed turn. … I said, I’m going to try that tomorrow.”

The grip allowed his thumb to get out of the way, to create a pitch with a more vertical axis. His index finger made the last contact with the ball, just brushing it, to create an element of sidespin. He had the gyro spin and sidespin mix he was seeking. He knew he couldn’t get a perfect north–south axis. He was hoping to create an axis pointed toward him at about 60 degrees. Most pitchers don’t experiment in games. They save experimentation for bullpens. But Bauer doesn’t mind failing before thousands. That’s one benefit of having a low threshold. You don’t care what anyone thinks if you are doing what you believe is right. If Bauer listened to his pitching coach and was always agreeable, he argues, he would not have a career. “And people wonder why I have the reputation of not listening to coaches,” he says.

On May 11, Bauer used the new grip for two innings, the second and third. His slider began to dart horizontally. As he examined the data after the game, he saw that one slider had moved eight inches. He was thrilled, at least until he learned that only one inning had been captured on the Edgertronic. While the pitch moved like he desired, he couldn’t control it.

“I switched back [during the game] because I had no idea where it was going,” he says. “I was in self-preservation mode. I switched back to my old grip, which was more comfortable, but it didn’t have the profile.”

The start was a disaster: he allowed five runs and eleven hits in 4 2/3 innings. But it may have been his most important start of the season. The grip had given him the movement he wanted. The Indians next traveled to Detroit, where Bauer threw his bullpen. He felt he was able to replicate the grip and axis. Those watching the bullpen session had their doubts that it was an improvement. “In my head I was throwing a freaking party,” Bauer says. “I went with it exclusively, and the movement profile was drastically different.”

With greater confidence in his new slider grip, Bauer took the mound on an afternoon getaway day at Comerica Park in Detroit. The stands at the home of the rebuilding Tigers were mostly empty. With two outs in the first inning, Bauer faced a 2–2 count against Nick Castellanos. Bauer threw a slider with the new grip. Behind the plate, catcher Roberto Perez held his glove outside and just below the strike zone. The pitch appeared to make a left-handed turn as it neared the plate, darting into Perez’s glove. Castellanos swung and missed. In the second inning, Bauer threw another two-strike slider to the Tigers’ John Hicks, a league-average hitter. Again, the slider appeared to be headed for the center of the plate before breaking to the left. Hicks also swung and missed.

One reason Bauer had longed for a slider was to pair it with his two-seam fastball, the laminar express. Because they both have little vertical movement, the two pitches could share the same path, or tunnel, for much of their journeys toward the plate but break horizontally in opposite directions too late for batters to adjust. To start the fourth inning, Bauer threw a 95 mph, two-strike two-seamer to the right-handed- hitting Pete Kozma. The pitch’s axis allowed for a smooth spot to develop near the nose of the ball. As it neared the plate, the turbulent air on its backside compelled it to break back toward the plate and catch the outside part of the strike zone. Kozma gave up on the pitch, thinking it was outside. Instead, he stared at strike three.

With two outs in the seventh, Bauer again faced Hicks. On a 2–2 count, Bauer threw an excellent slider. It held its plane, masquerading as a fastball bound for the outside corner before darting away. Hicks whiffed to end the inning. Bauer looked businesslike as he walked off the mound, as if he had done this before. But he hadn’t, and inside his head he was celebrating. It was arguably the finest start of Bauer’s career: eight innings, four hits allowed, no runs, no walks, and ten strikeouts. All ten of the Ks had come via whiffs on sliders or batters looking at his comeback two-seamer. That day, Bauer threw his new slider sixteen times and induced eight swings-and-misses: an outstanding 50 percent whiff rate, the best of all his pitches in the outing.

The pitch had an average horizontal movement of 8.6 inches, nearly what he wanted and roughly double where it had been in the previous six starts. The slider also averaged 0.3 inches of vertical movement relative to gravity. It was nearly perfect.

Through his first nine starts, Bauer owned a 2.59 ERA and a 2.82 FIP. He had struck out sixty-seven batters in fifty-nine innings and allowed just forty-five hits and only in his last outing had his pitches felt fully operational. He was as close as he’d ever been to becoming what he thought he could be.

Excerpted from “The MVP Machine: How Baseball’s New Nonconformists Are Using Data to Build Better Players” by Ben Lindbergh and Travis Sawchik. Copyright © 2019. Available from Basic Books, an imprint of Hachette Book Group, Inc.

## James Holzhauer Broke ‘Jeopardy!,’ But Is Broken So Bad?

Note: This article discusses the results of the June 3, 2019, episode of “Jeopardy!”

James Holzhauer claims not to remember many particulars of how he lost on “Jeopardy!” for the first time, other than that he blanked on a clue about the city of Albany and his opponent quickly took control of the board, landing a game-changing Daily Double. Before long, it was all over. Monday’s episode marked the end of Holzhauer’s two-month reign as one of the winningest, and certainly the most radical, champions in the decades-long history of the trivia game show “Jeopardy!”

Holzhauer finished Monday’s game in second place with $24,799 behind Emma Boettcher’s$46,801. But during his 32-win run, he averaged about $77,000 per game — an average nearly identical to the record for the single richest game ever played before he took the podium in early April. In the process, he laid claim to the entirety of the top 10 highest-scoring games of all time, including one single half-hour haul of$131,127. It was a historic run driven by immaculate trivia knowledge, disciplined strategy and calculated aggression.

But other records will forever remain just out of reach. Holzhauer’s streak ended with total winnings of $2,462,216 — less than$60,000 shy of Ken Jennings’s record $2,520,700 which was amassed over a nearly incomprehensible 74 straight wins in 2004. Holzhauer will sit second on the all-time money list until the arrival of some other great champion. (Or he might sit there forever, which seems more likely.)2 “I played every day exactly according to my game plan, so I have no regrets,” Holzhauer told FiveThirtyEight a few days before the fateful episode aired. Holzhauer rewrote swaths of the show’s record books. But his biggest contribution may be to “Jeopardy!” strategy. Holzhauer exploited the game’s Daily Doubles to their fullest, hunting them down and betting big on them. Over his 32 wins (and one loss), Holzhauer — a professional sports bettor from Vegas — not only got significantly richer but likely changed how the venerable game show will be played. Holzhauer was such an effective and alien force that opponents began to mimic his style out of desperation, like growling at a hungry lion in hopes of scaring it away. They hopped wildly around the game’s board whenever they could, picking big-dollar clues early, searching madly for the Daily Doubles and betting big when they found them — just the sort of unalloyed aggression that had quickly become Holzhauer’s trademark and the fuel for his success. “Many of my opponents played like I do, but I’m not sure they would have done so without provocation,” Holzhauer said. “You don’t want to inadvertently make your opponents play a better strategy. In a sense, I may have helped bring about my own downfall.” Life as a longtime “Jeopardy!” champion is a strange one, chronologically speaking. Holzhauer has been watching the world wonder when his streak will end, all the while knowing exactly when it would happen. On its taping dates, the show records five episodes back-to-back, with just a change of clothes in between. The episodes don’t air until much later. If he could alter time, maybe buy a time machine with that$2,462,216, would he have approached the game any differently?

“The only things I would do differently from the start of my run: never wear a sport coat, which interfered a little with my buzzer form, and use gel insoles in my dress shoes,” Holzhauer said.
“Both were fixed by the second taping day.”

The insoles seem to have worked. Holzhauer has earned a spot in the pantheon of the “Jeopardy!” greats, and he gives himself prominent placement there. “I think there is a nebulous top three of Ken Jennings, Brad Rutter and me,” he said. “Ken’s 74-game streak remains the most impressive achievement in the show’s history.”

Rutter is no slouch, either — he has won more money than any other “Jeopardy!” contestant, and he’s a man who has never lost to a human. Rutter’s initial winning streak was ended by the show’s rules at the time, which limited a defending champion to five appearances. But between those appearances and the show’s Tournament of Champions, Ultimate Tournament of Champions and Million Dollar Masters, Rutter won $4,688,436. (Here’s a free idea for the “Jeopardy!” producers: Holzhauer vs. Jennings vs. Rutter in the Ultimate Tournament of Ultimate Champions.) A couple of days after we first emailed, however, Holzhauer followed up to amend his initial assessment. “Amidst all the people comparing me to Ken and Brad, I totally forgot about the two greatest Jeopardy champions of all time: Cindy Stowell, who won six games while dying of cancer, and Eddie Timanus, who’s … blind and was an undefeated five-time champ in his initial run. It’s impossible for me to compare myself to them, so perhaps they should be in their own category.” Holzhauer’s plan for now is a return to normalcy. “The 19-year-old version of James would be thrilled by the opportunities” that the winning streak has brought his way, “but married parent James is hoping to keep his home life settled.” The game that made him famous, however, has been left unsettled. Lots of esteemed competitive pursuits have been “broken” lately. Baseball. Basketball. Even the spelling bee just last week. An innovative strategy or an outlier talent can deeply alter the games we’ve played for decades. In the process, the cadence or tenor of the game might be rendered unrecognizable for someone who hadn’t seen it in a few years. These innovative strategies are often driven by mathematical analysis, data and statistical rigor — things that a sports bettor from Vegas must embrace in order to eat. I asked Holzhauer if “Jeopardy!” now belonged in this category of sabermetrically altered pursuits. “I can see the parallels, for sure,” he said. “At its heart, all these shifts are just attempts to increase your chances of winning. Why would anyone not want to maximize their chances?” Plenty of outlets have written that, thanks to Holzhauer, “Jeopardy!” is now broken. But there’s art in that. While the game may look a bit different than it did before, it may also be closer to perfection — to an ideal expression of trivia game-show strategy. Broken is beautiful. From ABC News: ## How Tottenham Could Shock Us One More Time This could finally be the year when a true underdog wins the Champions League again. The last time a club not among the world’s very elite hoisted the trophy was in 2012, when Chelsea won it all.1 In the six years since, the ruling class of Bayern Munich, Barcelona and Real Madrid captured Europe’s most prestigious club tournament each year. In this context, if Liverpool triumphs on Saturday, despite being clearly among the best teams in the world according to the FiveThirtyEight Soccer Power Index, the Reds would still be breaking ground. But the real underdog here is Tottenham. Spurs rate as the 12th-best club in the world by SPI. And in many ways, the story of Tottenham is the incredible journey it took just to get close to the world top 10. Spurs languished in the middle of the English Premier League table through most of the 2000s, and unlike Manchester City, which also made the leap to the EPL elite in this millennium, Tottenham didn’t luck into an owner willing to spend at a massive deficit to take the club up the ladder. Instead, Spurs had to ratchet up their spending slowly and play the market carefully. Of the Premier League’s current “big six” clubs,2 only Tottenham has come close to breaking even on its transfer dealings over the past decade — receiving about as much money in transfer fees as it has paid out. In this same time, Tottenham has managed to increase its wage spending from about$85 million in 2009-10 to \$187 million reported last season. The money saved on transfers has, to a significant degree, been reinvested into wages. By carefully managing the club’s incoming and outgoing funds on transfers, chairman Daniel Levy has built up the quality of the squad and paid to keep better and better players.

And it’s worked. Tottenham has finished at least sixth in every season this decade, including now four consecutive berths in the Champions League.3 In the previous decade, Spurs finished in the top six only twice (back-to-back fifth-place finishes in 2005-06 and 2006-07). Tottenham has become a regular Champions League competitor through a process of careful business management backing up good player development.

This same “do more with less” mentality is reflected in the team’s performances this season under the management of Mauricio Pochettino. Faced with injuries up and down the squad, Pochettino has had to improvise. His team has given at least 1,500 minutes to 14 different outfield players,4 most among the big six sides in the Premier League. Tottenham’s 10 most-used outfield players have covered only 66 percent of the team’s total minutes, the smallest share among the big six.

The team’s injuries have hit the midfield particularly hard: Eric Dier missed most of the second half of the season after an appendectomy, Harry Winks has missed time with ankle and groin injuries, Victor Wanyama was unavailable for most of the season with knee problems, and Mousa Dembele was injured in November and has not played for Tottenham since.5 Dier, Dembele and Wanyama were three of Spurs’ four leading midfielders in tackles and interceptions won per 90 minutes this season; without them, the club had no true ball-winners in the center of the pitch. The loss of these more defensively sound midfielders changed Tottenham’s approach, forcing Pochettino to dial back his preferred high-pressing, high-possession style. The team’s pressing rate has dropped to its lowest level under Pochettino at 47 percent.6 Instead, Pochettino has had to develop a more counterattacking approach.

Tottenham has remained one of the best attacking teams in the league, with about 50 open-play expected goals created — good for fifth overall. But unlike the other clubs that rank high in this metric, Spurs complete surprisingly few passes into or within the penalty area.

Arsenal completed 459 open-play passes into or within the penalty area to Tottenham’s 310. And yet Spurs have created only about three fewer expected goals from open play.

This speaks to an unusually direct attacking style. Tottenham no longer pins its opponents back with a strangling high press but instead builds its attacks quickly to free individual players to run at the defense, in an attacking style that has been compared to college football’s “air raid.” To make passes inside the penalty area, a team must have multiple players already in the vicinity, and if one of these high-risk passes is left incomplete, fewer players are in position to stop a potential counterattack. Without the ball-winners in midfield to recover possession, such a deliberate approach would put Tottenham under too much pressure in transition. So instead, the team works more directly, bypassing midfield and quickly feeding a forward to face a defender one-on-one.

This is the attacking style that Liverpool will have to defend against on Saturday in Madrid. Liverpool has consistently been able to use its midfield press to control matches and hold possession. But against Spurs, Jurgen Klopp’s side will have to be particularly wary of the direct counterattacks that have become the team’s go-to play. If Liverpool takes risks to overload Spurs defensively in possession, looking for some of these killer penalty area passes, it will be the Reds who are at risk of being countered.

The table may be set for a more slow-paced and tactical Champions League final than either Klopp’s or Pochettino’s reputations might suggest. In the most recent meeting of the two sides on March 31, there were just 2.6 expected goals created from 25 shots. This is almost exactly average for a Premier League match7 but is far from the high-scoring barn-burner one might hope for from the matchup. If Liverpool won’t risk sending extra runners to support its forwards, while Tottenham invites some pressure and looks to counterattack, the game may slow down as it did at Anfield in March.

The final, then, could come down to one team or the other relying on its superstar forward for a moment of magic. For Liverpool, this is not such a complicated question. Mohamed Salah had a great season, has returned to fitness after a head injury and is expected to be ready to go at full strength. Harry Kane is a bigger question for Tottenham. The English striker suffered his fifth ankle injury in the past three seasons against Manchester City in April and has not played since. He has rejoined the team and traveled to Spain, but can Spurs hope to get a vintage Kane performance?

Last season when Kane returned from an ankle injury, his performance fell off badly. He had been running hot, averaging 0.98 expected goals and assists per 90 minutes in the 10 matches before he got hurt, and he dropped to about 0.73 in the 10 matches after his return. The same thing happened again this season after Kane suffered an ankle injury against Manchester United: His shot statistics declined from 0.88 xG+xA/90 to 0.52. But his ankle injuries in the fall of 2016 and spring of 2017 didn’t result in such a drastic downturn in performance. In 2017, he seemed to come back stronger than he was before the injury, going on a rampage down the stretch with 1.04 xG+xA/90 compared with 0.74 beforehand.

Tottenham’s counterattacking style, and the team’s efficiency in turning passes around the penalty area into scoring chances, run best through an elite center forward. If Kane can return to form within a single game, that would give Spurs the best chance at a Champions League update. Getting an upset win probably requires a little bit of good fortune, and there is no better possible stroke of fortune for Tottenham than a fit Harry Kane.

Check out our latest soccer predictions.

## Does Toronto’s Game 1 Win Spell Doom For The Warriors?

sara.ziegler (Sara Ziegler, assistant sports editor): One game into the NBA Finals, and #WarriorsIn4 is already over. But what a first game! The Toronto Raptors led for most of the contest but weren’t able to put away the Golden State Warriors until the very end.

tchow (Tony Chow, video producer): I’m gonna be honest. I was second screening Game 1 because my eyes were glued to the Scripps National Spelling Bee. I learned some new words that I’m gonna try to sneak in here, so you all better have your dictionaries ready — I’m about to drop some

neil (Neil Paine, senior sportswriter): Omg, Tony

sara.ziegler: Tony

Though, I’m not gonna lie, I turned to that after the game was over.

neil: Fortunately, the NBA can’t declare an eight-way tie for the championship. (Sorry, Celtics.)

sara.ziegler: Chris, you’re in Toronto right now. What was the game like up close?

chris.herring (Chris Herring, senior sportswriter): The atmosphere was incredible, and loud — both during the game and then pretty wild after. The fans here are insane.

I think the game was what we hoped it would be, after years of watching relatively uncompetitive series with a team that couldn’t defend Golden State well enough. The Raptors’ defense is no joke, and it challenged the Warriors all game long. Toronto presents real problems for a club missing someone like Kevin Durant.

neil: Yeah, Chris, this was the Warriors’ 20th-worst shooting game of the season by effective field-goal percentage. They still managed to get to the line, but they had a lot of turnovers, and Toronto held the non-Steph Curry scorers mostly in check. Fred VanVleet even did an admirable job keeping Curry from truly exploding.

chris.herring: The Warriors shot 23 percent on contested shots last night, the worst mark they’ve had in a playoff game in the Steve Kerr era, according to ESPN Stats & Information Group.

neil: And you have to think that Durant — one of the best tough-shot makers in history — would have boosted that some.

chris.herring: Yeah. I’m really curious as to where Curry is going to have problems with VanVleet — we mentioned in our preview that he’d done very little scoring this season — averaging just 10 points per 100 possessions when VanVleet is the man defending him. That continued last night.

sara.ziegler: FiveThirtyEight’s most valuable player (valuable in the most literal sense), Pascal Siakam had an amazing NBA Finals debut, scoring 32 points on 14-of-17 shooting. How surprised were you at how well he played?

tchow: You could say Siakam was shining bright like a pendeloque last night.

neil: LOL

Our model definitely thinks highly of Siakam as a player, but I’m not going to say I saw him scoring 30+ going in. He had 32 points on 82 percent shooting!!! That’s the fourth-best shooting percentage in a 30-plus-point finals game EVER, according to Basketball-Reference.com.

chris.herring: The Warriors got a lot of questions here about Siakam after last night’s performance. Draymond Green said it’s clear that Siakam is “a guy” now — meaning that we might not have treated him as a difference-maker before, but we sure as hell will now.

sara.ziegler: He’s a now.

chris.herring: Golden State basically acknowledged leaving certain guys open to begin the game in hopes of taking away Kawhi Leonard. That process worked, in a way. Leonard wasn’t efficient.

But as a result, everyone else — particularly Siakam and Marc Gasol, who played brilliantly — got going. Danny Green was also himself again. And Golden State was never able to turn off that faucet.

neil: Siakam might be a problem for the Warriors going forward. They didn’t have many good options to stop him. He scored 16 directly on Draymond. He also showcased his dangerous range as a 3-point shooter when rotations broke down or he trailed the play.

chris.herring: I understand why GSW was willing to take that gamble with Siakam. He’s become very good from the corners but is right around 30 percent — if not worse — from above the arc. The real issue was letting him get whatever he wanted in transition. He was 5 of 5 in transition and hit 11 shots in a row at one point — the longest streak in a finals game over the last 20 years. As good as he is, that simply can’t happen in a game like that if you’re the Warriors.

Golden State gave credit to Siakam but also largely chalked the game up to them not having seen this Raptors club before. They hadn’t played since early December, and Toronto has added Gasol, while Kawhi obviously took turns in and out of the lineup to rest.

sara.ziegler: Yeah, no one was expecting this from Siakam, so game-planning it would have been tricky.

chris.herring: I feel like I should get my apology in now.

Although I don’t know if I’m apologizing to a person or an algorithm.

neil: Or are you apologizing directly to CARMELO Anthony? Lol.

chris.herring: Our model narrowly had Toronto winning this series. I ruled that possibility out pretty swiftly last week.

But Thursday’s game was enough for me to think that their defense is good enough to win the series — particularly if Durant doesn’t return, and perhaps even if Durant is back but doesn’t jell right away after the long layoff.

neil: I wanted to go back to what you said about loading up to stop Kawhi. Klay Thompson and Andre Iguodala did a good job limiting his efficiency, although it seems like that played a little into Toronto’s hands. Jackie MacMullan had a great reaction story about just how many other efficient options the Raptors have now if a team tries to focus too much on Kawhi.

Only two of the seven Raptors who played at least 10 minutes averaged fewer than 1.2 points per individual possession, according to Basketball-Reference. (For reference, the Warriors as a team averaged 1.17 points per possession in the game.)

sara.ziegler: And even with Kawhi bottled up, he still scored 23.

neil: And! I worry about Iguodala’s health after he came up limping late. He did the bulk of the job guarding Leonard.

tchow: So far, it looks like he’ll be OK, though.

chris.herring: Yeah. That was the one other concern we mentioned in the preview: While the Warriors clearly could use Durant on offense, their defense becomes really, really thin on the wings without him. Especially if Iguodala is hurt or isn’t himself. This is now the second time he’s been banged up — he didn’t play in Game 4 against the Blazers, either.

Speaking of Durant: The Raptors’ starting front court outscored Golden State’s 75-18.

neil:

sara.ziegler: Wow

How much of a problem is that for the Warriors? If there’s no scoring help for Steph and Klay?

neil: Certainly Draymond wasn’t much of a factor. Yes, he got the rare 10-10-10 triple double, but he also shot 2 of 9 from the floor and was a minus-8.

chris.herring: They’re now 29-2 when he records a triple-double.

neil: And both losses have come this postseason.

chris.herring: I think what we saw yesterday is this: The Warriors, without KD, don’t have anyone who can shoot outside of Curry and Thompson.

chris.herring: I think Quinn Cook is probably the most reliable guy outside of those two.

neil: How weird is it to think about the Warriors not having enough shooting?

chris.herring: That’s where Durant’s ability to get his own shot comes in handy. He forces enough defensive attention to where he can play other guys open. Generally speaking, Steph often commands a second defender’s attention, so that’s enough to get someone else open and get the ball moving. It’s a tougher task when the other team can guard him and everyone else straight up.

sara.ziegler: And Klay doesn’t really create his own shots.

chris.herring: We haven’t talked much about DeMarcus Cousins’s return, but that’s both the blessing and the curse of having him

You hope he can create an occasional double-team. But by the same token, his spot could have been used on a guard — and I think some people were of that opinion when they first got him: that the Warriors might have been better served by having another shooter.

neil: Yeah, I thought the Warriors might go smaller and take somebody like Gasol out of the game, but either Kevon Looney or Jordan Bell played most of the game, and Gasol logged nearly 30 minutes. Meanwhile, Cousins played eight minutes and didn’t really do much of note.

chris.herring: He looked a little rusty, but he made a few really nice passes.

It’s tough to get your first playing time in weeks and weeks at this level, in the finals. Same may be true of Durant, honestly, if and when he comes back.

tchow: It feels like it might be too soon to judge Cousins, but this is the problem of reintroducing someone like him back into the lineup during the finals.

chris.herring: Exactly.

neil: And that might be one of the ways our model was overrating the Warriors. It considered him one of the biggest talents of the series, which is true, but didn’t factor in the injury comeback.

chris.herring: Not to mention the fact that Golden State has been better with Cousins off the court this season.

Albeit with Durant playing more often than not.

tchow: Yeah, Neil, it probably did overrate the Warriors because of his return. He ranked as the fifth most valuable player (behind Curry, Durant, Leonard and Lowry) according to our projections.

neil: And at full health, that might be true in terms of skills. But that was a lot to expect with him easing back into playing.

sara.ziegler: While Cousins did play a bit, the other injured Warrior was spotted high-fiving teammates behind the scenes. What did you make of Durant traveling with the team?

neil: It has to be an encouraging sign for his chances of returning sooner rather than later, right?

sara.ziegler: Is there a chance he plays in Game 2?

chris.herring: No, it sounds like he won’t. Kerr was pretty firm about him needing to practice before having a chance to play.

They’ll have another two practices — today and again on Saturday — before Game 2. But it doesn’t sound like he’ll be ready to practice here in Toronto before they suit up again Sunday.

neil: The good thing about the finals is the sheer gap in days between games.

Game 1 on a Thursday — Game 2 … all the way on Sunday.

sara.ziegler: He has at least resumed “basketball activities,” which is my favorite phrase in all of basketball.

neil: That reminds me, I need to go to the gym and “resume basketball activities” as well.

sara.ziegler:

So what do the Warriors need do to even the series?

neil: Well, it seems obvious that Siakam won’t be down for 30+ again, so they have that going for them.

chris.herring: Be a little less focused on stopping Kawhi to make sure that the other Raptors don’t overtake Jurassic Park again.

And they have to slow Toronto down in transition, where the Raptors can be wildly efficient.

It’s more of a question as to what they do differently on offense. But getting more stops and creating more opportunities to get out and run off those misses will ease some of that concern, I’d think.

neil: Yeah, and that probably played a part in Toronto’s 24-17 disparity on fast-break points as well. Not enough stops turning into chances the other way.

tchow: They have to play with rhathymia. (Am I using that right?) Just be the fun-loving Warriors we know.

sara.ziegler: LOL

tchow: I also agree with Neil in that the Warriors could afford to play smaller and get Gasol out of the game. He’s been solid all playoffs like an imbirussú for the Raptors. Otherwise, the Raptors could embarrass you again. Calembour intended.

(OK, now I’m just forcing it.)

neil: Tony, you’re banned from watching the spelling bee at work ever again.

chris.herring: It’s a lot tougher for the Warriors to dictate the tempo without Durant. Playing smaller alone doesn’t get it done if you don’t have enough shooting to force the Raptors to come out and guard you on the perimeter.

sara.ziegler: It’s interesting to me, too, that Kyle Lowry didn’t add much on offense again. He had as many field goals as charges forced. If he heats up, that’s a different wrinkle for Toronto.

neil: Lowry continued his trend of being associated with strong Raptors play (+11) despite garbage individual stats.

chris.herring: Frankly, if they’re getting what they got from everyone else — Green, Gasol and Siakam — they don’t need Lowry to do anything but bring energy. He had massive moments in that last series, and he’s always going to give you what he has on defense.

It also helps a ton that VanVleet can stay attached to Curry so well in the minutes that Lowry is taking a breather.

tchow: VanVleet was draped over Curry like a ferraiolone and actually guarded Curry for more possessions than Lowry in the end (33 possession vs. 16).

chris.herring: O_______o

neil: I’ve come around on this, Tony, and I applaud your spelling work here.

tchow: Can we all pretend to be a marmennill for a minute? What do you think is going to happen now? Do the Warriors still three-peat? Do the Raptors pull this off?

sara.ziegler: Our model (which accepts Chris’s apology) now has the Raptors at 63 percent to win it all. That feels right to me.

chris.herring: The Raptors are the lone team that the Warriors haven’t beaten this season, and they have now won all three matchups against Golden State. I expect Golden State to respond. But stuff will get SO interesting if Toronto takes Game 2 as well.

neil: 63 percent kinda makes more sense than our pre-series projection, to be honest. Home teams that win Game 1 of the NBA Finals win the series 78 percent of the time, historically. So this suggests that Toronto has far less of a talent edge than the typical home team that takes a 1-0 finals lead. Which is definitely true.

tchow: This is anecdotal, but I was chatting with my cousin who lives in Toronto during last night’s game, and he said: “There’s just one guy outside our building somewhere screaming at the top of his lungs, ‘Let’s go, Raptors, over and over.” I can’t imagine what that guy will scream if the Raptors pull this off. That city is gonna be WILD.

neil: I love seeing how excited Toronto fans are. (Drake aside.) Nav Bhatia was going nuts trying to distract Warrior free-throw shooters.

chris.herring: I decided to walk home last night, about 35 minutes to my hotel. These two people were shouting “Let’s go, Raptors!” for entire blocks. I thought it was a crowd of people, and it was actually just those two guys.

But between that, and all the car horns going off last night, people are on a noisy cloud here right now. Sort of how Milwaukee was to begin the last series. So we’ll see how it plays out.

tchow: The city is gonna be as loud as a large flock of emberizines.

From ABC News:

Check out our latest NBA predictions.

## Can You Win The Lotería?

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,8 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 Taylor Firman, the unluck of the draw:

Lotería is a traditional Mexican game of chance, akin to bingo. Each player receives a four-by-four grid of images. Instead of a comically large rotating bin of numbered balls, the caller randomly draws a card from a deck containing all 54 possible images. If a player has that image on their grid, they mark it off. The exact rules can vary, but in this version, the game ends when one of the players fills their entire card (and screams “¡Lotería!”). Each of the 54 possible images can only show up once on each card, but other than that restriction, assume that image selection and placement on each player’s grid is random.

One beautiful day, you and your friend Christina decide to face off in a friendly game of Lotería. What is the probability that either of you ends the game with an empty grid, i.e. none of your images was called? How does this probability change if there were more or fewer unique images? Larger or smaller player grids?

## Riddler Classic

From Ben Wiles, a mathematical trip across the pond:

My favorite game show is “Countdown” on Channel 4 in the UK. I particularly enjoy its Numbers Game. Here is the premise: There are 20 “small” cards, two of each numbered 1 through 10. There are also four “large” cards numbered 25, 50, 75 and 100. The player asks for six cards in total: zero, one, two, three or four “large” numbers, and the rest in “small” numbers. The hostess selects that chosen number of “large” and “small” at random from the deck. A random-number generator then selects a three-digit number, and the players have 30 seconds to use addition, subtraction, multiplication and division to combine the six numbers on their cards into a total as close to the selected three-digit number as they can.

There are four basic rules: You can only use a number as many times as it comes up in the six-number set. You can only use the mathematical operations given. At no point in your calculations can you end on something that isn’t a counting number. And you don’t have to use all of the numbers.

For example, say you ask for one large and five smalls, and you get 2, 3, 7, 8, 9 and 75. Your target is 657. One way to solve this would be to say 7×8×9 = 504, 75×2 = 150, 504+150 = 654 and 654+3 = 657. You could also say 75+7 = 82, 82×8 = 656, 3-2 = 1 and 656+1 = 657.

This riddle is twofold. One: What number of “large” cards is most likely to produce a solvable game and what number of “large” cards is least likely to be solvable? Two: What three-digit numbers are most or least likely to be solvable?

## Solution to last week’s Riddler Express

Congratulations to Adam Martin-Schwarze of Sequim, Washington, winner of last week’s Riddler Express!

Last week we met a soccer coach who was trying to assemble a team of 11 players in a very specific way. He had an infinite pool of players to choose from, each of whom wore a unique number on their jersey such that there was one player for every number. That number also happened to be the number of games it took on average for that player to score a goal. The coach wanted his team to average precisely two goals per game, and he also wanted his weakest player to be as good as possible. What number does the ideal weakest player wear? What are the numbers of the other 10 players the coach should select?

The weakest player selected for the team wears the number 24. The other 10 players wear the numbers 1, 5, 6, 8, 9, 10, 12, 15, 18 and 20.

Let’s quickly check that everything adds up correctly. If a player’s number is 5, say, then that player scores an average of 1/5 goals per game. If their number is 6, they average 1/6 goals per game, and so on. So our team as a whole averages 1/1 + 1/5 + 1/6 + 1/8 + 1/9 + 1/10 + 1/12 + 1/15 + 1/18 + 1/20 + 1/24 = exactly 2 goals per game, just like the coach wanted!

I’m not aware of a more elegant method of solving this coaching conundrum than basic guess-and-check. There are many possibilities to consider, but one thing we do know is that the fractions we’re adding up to try to get to a sum of 2 are of a specific type: 1 divided by a whole number, which are also known as Egyptian fractions. These made a prominent appearance in the so-called Rhind papyrus, which also happens to be the oldest known collection of math puzzles.

We also know that we want the worst player on the team to be as good as possible — that is, to have as big a fraction as possible — and that there are 11 players on the team. So first we might check 1/1 + 1/2 + 1/3 + … + 1/11, i.e., the best possible team, but that sum equals about 3, too big for our coach. We then might check the sums of the possible sets of fractions between 1/1 and 1/12, and then the possible sets between 1/1 and 1/13, and so on, rejigging the sums until we find something that gets us to exactly 2. It turns out that none of these will add up to exactly 2 until we get to testing out those fractions between 1/1 and 1/24, and specifically those fractions listed above.

Solver David DeSmet shared a handy computer program he wrote to churn through all these possibilities, and Martin Piotte shared his thorough accounting of the possible teams.

## Solution to last week’s Riddler Classic

Congratulations to Curtis Bennett of Long Beach, California, winner of last week’s Riddler Classic!

Last week, three astronauts were at the edge of their Mars lander, staring down at the surface of the red planet. Each wanted to be the first human to step foot on the planet, but they wanted to pick who it would be using a fair and efficient method. They could, for example, use a fair coin, assign each astronaut an outcome — heads-heads, heads-tails and tails-heads — and flip the coin twice. If the result was tails-tails, they could simply restart the process. However, that method could take a long time and there was exploration to be done.

Another approach, however, would be to use an “unfair coin” — one for which the probabilities of heads and tails are not equal. Is it possible to make a fair choice among three astronauts with a fixed number of flips of an unfair coin? You were able to set the coin’s probability of heads to any number you like between 0 and 1. You could flip the coin as many times as you like, as long as that was some known, fixed number. And, you could assign any combination of possible outcomes to each of the three astronauts.

Indeed it was possible, though who knew coin flips could get so complicated. This puzzle’s submitter, Dean Ballard, walks us through his solution:

What makes this problem interesting is that at first glance it appears to require searching an overwhelmingly large space of possibilities. The trick to solving this more easily lies in a simplifying assumption. Instead of dealing with three different probability functions for the three different astronauts, we can assign two of them sets of head-tail combinations that will give them the equal chances of winning, independent of the weighting of our coin. This way, we can only worry about two things at once rather than three. We will need at least four coin flips to make this work.

Let $$p(H)$$ be the probability that our specially designed coin lands heads. With four flips we have 16 possible outcomes: HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH, THHT, THTH, TTHH, HTTT, THTT, TTHT, TTTH and TTTT. Note that subsets of these 16, such as {HHHT, HHTH, HTHH, THHH}, all have the same probability — in this specific case, $$p(H)^3 (1 – p(H))$$. So let’s call these “3H1T” — three heads, one tail.

Expressing the 16 outcomes this way gives us: one 4H, four 3H1T, six 2H2T, four 1H3T and one 4T, or in another piece of shorthand, [1, 4, 6, 4, 1]. Let’s say we assign Astronaut A just the HHHH and TTTT outcomes, and evenly divide the other 14 between Astronauts B and C. This gives us A = [1, 0, 0, 0, 1], B = [0, 2, 3, 2, 0], and C = [0, 2, 3, 2, 0]. Each of these defines a probability function for each astronaut. (Note that the sum A + B + C = [1, 4, 6, 4, 1], so all outcomes have been assigned.) The probability for Astronaut A equals 1 when $$p(H) = 0$$ or $$p(H) = 1$$, so it is concave up. The function for B and C equals 0 when $$p(H) = 0$$ or $$p(H) = 1$$, so it is concave down. Since both functions are continuous, as long as A’s value is less than B’s and C’s value when $$p(H) = 0.5$$ (which is true in this case), there must be a solution between 0 and 0.5, and another between 0.5 and 1.

Here is what those solutions look like graphically. On the x-axis is $$p(H)$$ and on the y-axis is the probability that an astronaut wins the contest.

Let’s quickly check the solution at point A, where the $$p(H)$$, the probability of our coin landing heads, equals about 0.24213. As we mentioned above, Astronaut A gets to take the first step if the coin lands with four heads or four tails. This happens with probability $$0.24213^4 + (1-0.24213)^4$$ = 0.3333, or a third, which is exactly what we want. Since the other two astronauts have been assigned probabilistically equivalent outcomes, we know they must have equal chances, which must also be a third. So we’ve successfully devised a fair method that will give us a result in a known and fixed number of unfair coin flips!

For extra credit, you faced the same question but with five astronauts. I’ll spare you all the gory details, but suffice it to say that solver Zach Wissner-Gross devised one method that used eight flips of a coin that came up heads about 81.7 percent of the time. The colors correspond to the lucky astronaut who will get to make history based on the flips shown on the axes of the diagram. So, for example, if the coin came up eight straight heads — relatively likely given the weighting of the coin — then Astronaut Yellow gets to take those first steps.

## 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]