Riley McCarron Laments ‘Stupid Play’ As Muffed Punt Mars Patriots Debut

FOXBORO, Mass. — Riley McCarron’s New England Patriots career got off to an inauspicious start Sunday afternoon at Gillette Stadium.
McCarron, who was promoted from the Patriots’ practice squad this week primarily to handle punt return duties during Julian Edelman’s four-game suspension, muffed a punt late in New England’s season opener against the Houston Texans.
Houston recovered the fumble, retained possession at New England’s 16-yard line and scored six plays later, with quarterback Deshaun Watson connecting with wide receiver Bruce Ellington on a 5-yard touchdown to cut the Patriots’ lead to seven points.
The Patriots held that lead to win 27-20, but McCarron still was frustrated with himself after the game.
“I made a stupid play,” the 25-year-old wide receiver told reporters. “I should have handled the ball. It’s as simple as that. I did not make the decision on a fair catch, obviously, but I know the one I should have made now.”

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He added: “I just dropped the ball. It’s not a good feeling. I would like that not to happen anymore this season.”
The miscue was the highlight of an otherwise nondescript afternoon for McCarron, who was making his NFL debut after spending most of last season on the Patriots practice squad. He fair-caught three other punts, and Tom Brady’s only pass to him on offense fell incomplete.
“It obviously is not a good feeling, but credit to the defense,” McCarron said. “They did a great job. I have great support from (my) teammates. They told me to just keep my head up.”
Patriots running back and co-captain James White said he offered McCarron some words of support after his blunder.
“I went up and tapped him on the head,” White said. “I told him, ‘The play is over. We’ll move on from it. The defense is going to try and get a stop, and we might need you that next drive, so you can’t be down in the dumps.’ But Riley’s a good football player. We’ve all made mistakes in some key moments so you’ve definitely got to try and lift him up and keep his spirits because we still had a football game to play.”

Some Patriots Weren’t Thrilled With Fourth-Quarter Officiating Vs. Texans

FOXBORO, Mass. — The New England Patriots beat Houston handily Week 1, but some fourth-quarter officiating nearly let the Texans back into the game.
Cornerback Stephon Gilmore was flagged twice for holding while covering wide receiver DeAndre Hopkins late in the fourth quarter when the Texans were inside the red zone.
The first holding penalty came on second-and-11 and gave the Texans a first down from New England’s 5-yard line. The second holding penalty came on third-and-2 and gave the Texans another new set of downs from the 1-yard line. They scored two plays later.
“I don’t know,” Gilmore said. “The first one, maybe. The second one, I don’t know. You can’t touch them nowadays. Just gotta find a way to win. (Hopkins) pushed off a lot, but they’re not gonna call that. You just gotta play it strong through and make plays when they come your way.”
Gilmore had an interception and did a nice job limiting Hopkins’ production while shadowing the star receiver.
Safety Duron Harmon was flagged for unnecessary roughness after an incomplete pass for a helmet-to-helmet hit on Hopkins with 38 seconds left in the game on second-and-11. The penalty pushed the Texans from their 1-yard line to the 16-yard line. They made it to the 43-yard line before attempting a failed hail mary.

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“I felt like it was a clean tackle,” Harmon said. “All honestly, I felt like it was shoulder. My shoulder hit his back. You know what I’m saying? It’s a bang-bang play. We’ll see, but I feel like it was a clean tackle.”
Officials were particularly liberal with flags pertaining to player safety during the preseason.
“Oh yeah definitely gray area, but they just want to make the game safe,” Harmon said. “We’ll look at it as a team tomorrow and see what coach (Bill) Belichick says. If I need to switch something up, I’ll switch something up. If not, I’m just going to keep trying to play aggressive and keep playing the way I know how to play football.”
The Patriots wound up winning 27-20 despite the calls. But the penalties also made the game closer than it needed to be.
League officials in New York also didn’t contact officials in Foxboro quickly enough on a completion to Patriots tight end Rob Gronkowski they wanted to challenge. So, the Patriots also benefitted from officiating.

Patriots Notes: James Develin Contributes In Unexpected Way For Undermanned Pats

FOXBORO, Mass. — Some assorted notes and nuggets from the New England Patriots’ 27-20 victory over the Houston Texans on Sunday at Gillette Stadium:
— One of the game’s unlikely offensive standouts was Patriots fullback James Develin, who caught four passes on four targets for 22 yards in the win.
Develin, whose primary role is as a lead blocker for Patriots running backs Rex Burkhead, Jeremy Hill and James White, never had caught more than two passes in any game in his NFL career. He caught just 10 all last season.
Yet on Sunday, the bruising 2017 Pro Bowler finished with more receptions than Burkhead, Chris Hogan and Cordarrelle Patterson combined. Football is weird sometimes.
“I’m just trying to go out there, do what the coaches ask me to do, get as open as I could and catch every ball that came to me,” Develin said after the game. “Thankfully, I was able to do that (Sunday), but we have a ton more that we can work on offensively, and I’m excited to get back to work.”
Develin’s teammates certainly appreciated his contributions, which came while tight end Jacob Hollister and running back Sony Michel both were sidelined with injuries. Wide receiver Julian Edelman also was unavailable due to his four-game suspension.
“It’s huge,” tight end Rob Gronkowski said. “I mean, we really don’t have that many guys right now in the tight end room, wide receiver room. We’re low on running backs, so we need to use everyone, and he’s been always preparing and been always ready.
“Every time since I (can) remember since he’s been here, he’s always been ready to catch the ball, to get the first down, how many yards we need. He works hard, he plays hard, he blocks hard. It’s great to have him in the room.”
White added: “He’s out there on the field more and more each week. He’s a very versatile guy, and he made some key big plays for us (Sunday). I think a lot of guys just tried to step in there and make plays when the ball came their way.”
— Unfortunately for Develin, he also was involved in an unfortunate play early in the third quarter.

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After Tyrann Mathieu recovered a Rob Gronkowski fumble, Develin and Hill both tried to tackle simultaneously. Develin’s shoulder made contact with the side of Hill’s right knee, knocking the running back out of the game.
Hill’s brief statement on social media and Develin’s somber postgame comments both suggested the injury was a significant one.
“That’s incredibly unfortunate,” Develin said, “but we’ve just got to keep pressing on and just try to work as hard as we can and play hard in his honor.”

Gods Plan
— Jeremy Hill (@JeremyHill33) September 9, 2018

— White, a former high school teammate of Phillip Dorsett’s at St. Thomas Aquinas in Florida, was thrilled to see the Patriots wideout enjoy a breakout performance Sunday. Dorsett caught a career-high seven passes on seven targets for 66 yards and a touchdown.
“I’ve known that guy since high school,” White said. “He works extremely hard. He (joined the Patriots) last year after training camp, so a lot was kind of thrown at him fast. I think he got better as the year went on. I think he took the challenge this offseason to learn the offense as best he can and just be in the right spots and win in man (coverage). I think he went out there and showed it (Sunday) that he can be a good player for us.”
— In a surprise move, Patriots defensive end Derek Rivers was a healthy scratch in what would have been his first regular-season NFL game.
Rivers, a 2017 third-round draft pick who missed his rookie season with a torn ACL, evidently is behind fellow second-year pro Keionta Davis on the depth chart. Davis made his NFL debut Sunday and made an impact as a pass rusher, finishing with one QB hit.

Rob Gronkowski Addresses Rumor Of Preventing Trade From Patriots

FOXBORO, Mass. — It would be tough for Rob Gronkowski to make it any clearer: he’s happy to be a member of the New England Patriots.
Gronkowski’s future with the Patriots — and in the NFL in general — was called into question immediately following New England’s Super Bowl LII loss when the star tight end hinted at retirement.
The uncertainty only grew larger when the Pats reportedly shopped the five-time Pro Bowl selection leading up to the 2018 season, but reports surfaced Sunday that Gronkowski shut down all trade talks involving him, allegedly letting the organization know he’d rather walk away from the game than play for another team.
While the latest report on Gronk was the talk of the football world leading up to the Patriots’ matchup with the Houston Texans, the narrative quickly shifted to No. 87’s play once the game unfolded. Gronkowski ended up making quite an impact in New England’s season opener, as his seven catches for 123 yards and a touchdown helped lift the Pats to a 27-20 win.

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After the game, Gronkowski was a bit befuddled by the most recent surge in chatter surrounding his future. The 29-year-old only has his mind centered on being an impact player for the Patriots, which must come as music to the team’s ears.
“I was wondering why they asked me that on the field right after the game,” Gronkowski said. “I was like, ‘Where is this coming from again?’ No, I’m just glad to be here, glad to be part of this team, glad to be part of this organization. And all that talk, all that stuff, I mean I would love to put it in the past. Whatever it is, whatever reports keep coming out because I’m here and I’m here to stay and I’m here to keep playing ball with the Patriots and just enjoying my time here.”
Unfortunately for Gronkowski, it’s hard to imagine his wishes will be granted, as he’ll almost certainly still field questions about his future as the season progresses. But if he continues to dominate on the field like he did Sunday, he might just be able to limit questions to the only topic he wants to talk about: football.

Patriots Slot Cornerback Jonathan Jones Was Secret Weapon In Week 1 Win

FOXBORO, Mass. — After the offseason additions of Jason McCourty, Duke Dawson, Keion Crossen and JC Jackson, New England Patriots cornerback Jonathan Jones became a forgotten man this summer.
Jones began training camp on the physically unable to perform list following a postseason foot injury and appeared in just one preseason game, playing 10 defensive snaps. But he beat out several cornerbacks, including McCourty, Dawson, Crossen, Jackson, Cyrus Jones, Ryan Lewis and Jomal Wiltz, for the Patriots’ slot cornerback role in Week 1.
He certainly didn’t look rusty Sunday afternoon, when he blanked the receivers he was covering, and broke up two passes, in the Patriots’ 27-20 Week 1 win over the Houston Texans.
“It’s great to have Jon back,” Patriots head coach Bill Belichick said Sunday. “Jon’s a really good football player. He has a big role for us in our substituted defenses against multiple receivers. He does a great job in the kicking game as well, as he always has. That’s where he kind of made his mark, got started, and that’s led to a lot more opportunities for defensive playtime.
“He’s taken advantage of those. He’s done a really good job for us. It’s good to have him back. I thought our secondary competed pretty well against a good group of receivers. (Bruce) Ellington, obviously (DeAndre) Hopkins, the tight ends, the backs. They split (Tyler) Ervin out, had a decent amount of empty there. They challenged us pretty good with the passing game, and obviously the quarterback with extended plays. I thought our secondary competed well and that’s what it took in this game.”

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Jones, a third-year player, is still relatively new to playing the slot. He was an outside cornerback at Auburn, and he barely played defense during his rookie season. Jones had a solid stretch of games last season before Eric Rowe returned from injury and passed him on the depth chart. Now Jones is in the driver’s seat for the role all season. He mostly covered Texans slot receiver Bruce Ellington in his 37 snaps Sunday.
“Experience,” Jones said. “That’s one thing you can’t buy is experience, so game after game, I’m just going to try to keep getting better and not make the same mistakes and improve and make plays.”
Jones also made a game-changing play on special teams, when he downed Ryan Allen’s punt at the 1-yard line with less than a minute to play in the fourth quarter. The Texans were only able to get to their own 43-yard line before attempting a failed hail mary.
“It made a big difference in this game because they were out of timeouts and that added what would’ve been touchback yardage, so another 19 yards to the drive,” Belichick said. “It was a huge play. It ran about eight or nine seconds off of the clock. In the end, I think what they were hoping for, what I would be hoping for offensively is that you can get close enough offensively to throw the ball into the end zone and hope to make a play.
“The ball was far enough away from the end zone that Watson couldn’t really get it there. Had that ball gone into the end zone and they had another 19 yards, I don’t know, maybe it could’ve gotten to the end zone. It was a big play. It was a great kick. It was a great job by Jonathan. Ryan’s done a real good job for us putting the ball down tight for us.”

How The Mystics Can Pull Off An Upset In The WNBA Finals

It seemed easy enough to count out the Washington Mystics at different junctures during the WNBA playoffs.
Elena Delle Donne’s knee injury looked flat-out brutal — and potentially season-ending — when it happened in Game 2 of the semifinals last week. Things appeared bleak when the Mystics were pushed to the brink of elimination after a Game 3 loss in which Delle Donne sat out to rest the bone bruise. And with the Atlanta Dream hosting the decisive Game 5 on Tuesday, and Delle Donne hobbled but playing, Washington still didn’t seem to be in the driver’s seat.
Nonetheless, here the Mystics are, in their first WNBA Finals, a stage on which they’ll again be underdogs — this time against a skilled Seattle club that had the league’s best record. Yet despite what figures to be an uphill battle for Washington, the Mystics have something of a blueprint to work off of in hopes of winning the title when the finals begin Friday night. Above all else, they’ll need to grind the games to a halt as best they can to maximize their chances.
Yes, some of that slowdown would be aimed at helping out former league MVP Delle Donne, who worked tirelessly to make it back from the knee injury last round. But the Mystics’ offense thrives while playing at one of the slowest paces in the WNBA — a contrast from Seattle, which likes to push the tempo when possible.
Washington takes more possessions into the last four seconds of the shot clock than any team, and it’s found success that way. In fact, according to data from Synergy Sports Technology, the Mystics lead the WNBA in efficiency when the clock is down to its last four seconds — not only in terms of points per possession but also in score-frequency rate and free-throw rate.2 That could come in handy given that the team’s already-slow attack has been even more tortoise-like since Delle Donne returned. (While her presence is huge for this offense, her scoring is down in the two games since her injury — 14.5 points per game, off from the 21 she averaged in the regular season. Delle Donne also averaged 30 points per game against Seattle during the regular season.)
The Mystics and Storm have faced off three times this year. Seattle took the first two, while Washington won the August game, which featured the slowest pace of the three, at just 73 possessions per 40 minutes. For context, Seattle’s offense — arguably the best in the NBA — usually is far faster than that (it averaged 83 possessions per 40 minutes during the regular season). Also noteworthy: The Mystics grabbed an eye-popping 35 percent of their own misses that night — up from a season average of 26 percent — giving them ample second chances and allowing them to win the time-of-possession battle with the Storm, who would prefer to get up and down. Given all this, taking care of the ball and limiting the number of possessions against the perceived on-paper favorite — which will host the first two games of the series at KeyArena — would be ideal for the Mystics.
Still, there are a couple of unusual factors that could make the matchup tougher to call one way or the other. For starters, there wasn’t a whole lot of playoff experience on either of these teams entering this postseason — let alone WNBA Finals experience. So there could be some initial jitters. The Mystics, who played their entire regular season home slate at Capital One Arena in D.C., had to host their semifinal round home games at George Washington University because of renovations at their normal home, and now scheduling conflicts will push them to George Mason University — which is in Virginia — for the WNBA Finals.
If there’s a safe bet to be made in these finals, though, it’s that the 3-point shooting will be plentiful. Nine of the league’s top 25 3-point shooters this season, in terms of accuracy, are represented by these two teams alone.3
Another: To have a chance of winning the series, Washington must find a way to at least contain 2018 MVP Breanna Stewart. She scored 25 points in each of the Storm’s first two games against the Mystics this season, but she was held to just 10 — tied for her second-lowest total of the season — during Seattle’s loss in D.C. last month. The 24-year-old, who ranked in the 99th percentile of offensive efficiency this season, according to Synergy, is almost unstoppable. But the Mystics found some success in double-teaming Stewart in the post, a scenario in which the Storm scored only a third of the time — and committed a turnover 17 percent of the time.

With Delle Donne at less than 100 percent, Kristi Toliver (who scored 38 points combined in the first two games against Seattle) and Ariel Atkins (who averaged 16 points against Seattle this season while shooting 47 percent) will likely need to shoulder more of the offensive burden against the Storm. Meanwhile, Sue Bird — who’s won two WNBA titles already with Seattle, the most recent in 2010 — will almost certainly stretch Washington’s defense after her late, ice-in-veins showing in Tuesday’s Game 5, marking Diana Taurasi’s first loss in a winner-take-all game.
With all the spacing on the court in these finals, both teams could find it difficult to truly settle in on the defensive end. And against an uptempo team like Seattle, that wants to push the other way in transition, Washington taking every possession to the end of the clock could throw the Storm even more out of rhythm. It could be Washington’s best bet here, as the Mystics and Delle Donne vie to overcome their last, and most daunting, hurdle.

Politics Podcast: How Useless Are Supreme Court Confirmation Hearings?

By Perry Bacon Jr., Oliver Roeder and Galen Druke, Perry Bacon Jr., Oliver Roeder and Galen Druke and Perry Bacon Jr., Oliver Roeder and Galen Druke


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After watching Supreme Court nominee Brett Kavanaugh’s confirmation hearings for the better part of a week, FiveThirtyEight’s Perry Bacon Jr. and Oliver Roeder join the FiveThirtyEight Politics podcast to share what they learned. When it comes to Kavanaugh’s judicial philosophy or even general worldview, it wasn’t much. Most of the meaningful speaking was done by the senators.
You can listen to the episode by clicking the “play” button above or by downloading it in iTunes, the ESPN App or your favorite podcast platform. If you are new to podcasts, learn how to listen.
The FiveThirtyEight Politics podcast publishes Monday evenings, with occasional special episodes throughout the week. Help new listeners discover the show by leaving us a rating and review on iTunes. Have a comment, question or suggestion for “good polling vs. bad polling”? Get in touch by email, on Twitter or in the comments.

The Serve-And-Volley Works, But Women’s Tennis Players Aren’t Using It

Women at this year’s U.S. Open have displayed incredible athleticism and shrewd strategy. We’ve seen blistering serves and masterful groundstrokes in the nearly two weeks of play. But we’ve been missing a powerful, ancient weapon: the serve-and-volley. The tactic is all but extinct from women’s tennis, despite still being incredibly effective.
Through the quarterfinals in Flushing Meadows, on points in which women followed their serve by rushing to the net for a volley, they won a remarkable 76 percent of the time, according to U.S. Open data from IBM. But confoundingly, the strategy was deployed only 84 times — less than 1 percent of all points played.
Serena Williams, for her part, flashed the potential of the serve-and-volley in her semifinal win Thursday over Anastasija Sevastova. Williams came to the net five times immediately after serving, taking four of those points.
After the match, Williams said she usually approaches the net “only to shake hands,” but she wanted to try something different against Sevastova. It was a reminder of just how potent the tactic can be.
The serve-and-volley appears to be used more on the men’s side of late. At this year’s U.S. Open, men served and volleyed 488 times through the quarterfinals, for just less than 3 percent of all points played. Though they couldn’t quite equal the gaudy success rate of the women, they still did very well with the tactic, winning 66 percent of the time. No matter who’s doing it, statistically, the serve-and-volley is an effective way to win points.
Why this play is underutilized has long been a topic of discussion in tennis. Some have chalked up the change in tactics to advanced racquets and strings that improve passing shots, while others have blamed slower courts at Wimbledon and this year’s U.S. Open.4 But there’s no doubt that women have been giving serve-and-volley the cold shoulder.
During the late 1970s and ’80s, Martina Navratilova dominated the sport using an attacking serve-and-volley style with monsterous success, propelling her to 18 Grand Slam Championships. Even in the 1990s, serve-and-volley was still in a heyday in the women’s game. To illustrate, look at Wimbledon. It’s a place where serve-and-volley has historically been a favorite tactic because of the ability of the server to dominate on grass, where the ball bounces are low. Returners must put loft on their shots to ensure they clear the net. That’s a feast for a volleyer.
Wimbledon first released serve-and-volley data in 1997, and it included tournament totals for the first time in 2002. Jana Novotna served and volleyed her way to the final that year, losing to Martina Hingis. For the tournament, Novotna employed the serve-and-volley 339 times, winning 213 of those points. In the final against Hingis, she used it 76 times — almost as often as the entire women’s field through five rounds at this year’s U.S. Open. It’s an astounding drop of a tactic and a big shift in the way women play tennis.
Wimbledon is the only Grand Slam that releases historical data, but we can look at its last 17 tournaments to track the decline of serve-and-volley as a strategy in the women’s game:

Assuming the point does not end on an ace or double-fault, serving players essentially have two choices: serve and move forward or serve and stay at the baseline. Since the 2000s, players who prefer to remain mainly at the baseline have taken over tennis, despite the unshakably consistent success rates for serve-and-volley over the years. But the difference between the strategies has been stark at this year’s U.S. Open: Through the quarterfinals, women have won just 48 percent of points at the baseline — 28 percentage points less than the share of points won on serve-and-volley.
All sports go through trends. As the games change, different approaches fall in and out of vogue. The Philadelphia Eagles won the Super Bowl last year on the strength of the RPO — run/pass option — in which backup quarterback Nick Foles had the option of handing off or passing. But the play wasn’t new. The option has been around in college football since the 1960s. But it’s taken a backseat as more teams have gone to pro-style offenses.
This happens all the time in sports: What’s old becomes new again. Tennis is no different. Strategies experience revivals.
Women in tennis have been relegating themselves mostly to the baseline. But perhaps Williams’s use of the serve-and-volley Thursday will spark a new trend.

I’d Like To Use My Riddler Lifeline

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,5 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 Tom Hanrahan, some big-money game show strategizing:
You are a contestant on “Who Wants to Be a Riddler Millionaire.” You have already made it to a late round: You could walk away right now with $250,000. But there are two potential questions still to go that you can try to answer. You could earn $500,000 if you get one right and then walk away, or $1 million if you nail them both. If you attempt any answer and miss, you go home with $10,000.
Luckily, you still have two of your lifelines:

The 50/50: The host reduces the four possible answers to two; one of them is the correct one and the other is randomly chosen from among the other three answers.
Ask the Audience: The studio audience submits their own guesses. You know historically that the correct answer will be chosen by the plurality 50 percent of the time; while 30 percent of the time the right answer finishes second; 15 percent third; and 5 percent last. Additionally, if there are only two answers available to the audience, they pick the correct one more often 65 percent of the time.

The problem: You’re burned out. All the pressure and questions you’ve already answered have made you a babbling mess. You assess that you would have no clue on the last two questions, so you’ll be guessing randomly.
What is your best strategy to play, or stop, or use your lifelines to maximize your expected winnings?
Submit your answer
Riddler Classic
From Jerry Meyers, a conundrum timed to kick off the football season!
My son recently started collecting Riddler League football cards and informed me that he planned on acquiring every card in the set. It made me wonder, naturally, how much of his allowance he would have to spend in order to achieve his goal. His favorite set of cards is Riddler Silver; a set consisting of 100 cards, numbered 1 to 100. The cards are only sold in packs containing 10 random cards, without duplicates, with every card number having an equal chance of being in a pack.
Each pack can be purchased for $1. If his allowance is $10 a week, how long would we expect it to take before he has the entire set?
What if he decides to collect the more expansive Riddler Gold set, which has 300 different cards?
Submit your answer
Solution to last week’s Riddler Express
Congratulations to Bridger Conklin of Washington, D.C., winner of last week’s Riddler Express!
Last week, Riddler Nation had just legalized sports betting, and everyone was excited to bet on the national pastime: competitive coin flipping. Every coin-flipping match is between two teams. Each team selects a two-coin sequence of heads and/or tails to look for, and they simultaneously flip their own coin over and over until one team finds its sequence. (If both teams find their sequences at the same time, they start over and flip until only one team finds it.) First to be the only team to have found its sequence wins.
When you arrived, you saw that the Red Team had chosen the sequence “heads-tails,” and the Blue Team had chosen “heads-heads.” You can get even odds on either team. Which team should you put your money on?
If you like winning, you should bet on the Red Team and its head-tails sequence.
Understanding why the Red Team has an advantage might be easiest if we’re armed with a diagram. Solver Paul Sim wrote in to share his, shown below.

Each branch of the tree depicts a flip of either heads or tails, and a team’s victory along a given path of branches is highlighted with that team’s color. The Red Team has six victories possible in these first four flips, while the Blue Team has only four. “The Red Team’s natural advantage,” Paul wrote, is “in being able to overlap their failures with their next attempt.”
Solver Steven Fellows shared another way to visualize this solution, shown below. Each team starts the game at the top of the flowchart, flips the coin and follows the steps shown. The Blue Team’s disadvantage can be see by the long arrow up the left-hand side of its chart. If the Blue Team fails to get its second head, it must go back to the beginning of the game, in essence. If the Red Team fails to get its tail, it only needs to go back one step.

According to the simulations of many solvers, all of this means that the Red Team wins this game about 59 percent of the time — a good deal for your even-money bet.
Solution to last week’s Riddler Classic
Congratulations to Peter Wiggin of San Francisco, winner of last week’s Riddler Classic!
Last week, Acey and Deucy wanted to play a game of Nim. An example game goes like this: Two players start with three heaps — one with three counters, one with four counters and one with five counters. They take turns choosing a heap and removing counters — at least one counter, but they can also take multiple counters or the whole heap. The player who takes the last counter from the last heap wins.
The problem was that Acey and Deucy were experts at this game, and knew that with perfect play, the player who went first was guaranteed to win. Boring! But they still wanted to play, so they introduced some randomness. They’d roll three dice, and use those three numbers to determine the number of counters placed in the three piles to start the game.
But this randomized Nim game still gives an advantage to one of the players. To make this random-start game actually fair, if Acey goes first, how much should Acey bet against Deucy’s nickel?
Acey should wager 40 cents.
That’s because Deucy is eight times less likely to win than Acey. There are 6×6×6 = 216 different games of Nim these two might play, given the rolls of the three dice. With perfect play, the first player will win 192 of these games. That’s 192 wins to 24 losses for Acey, or a ratio of 8-to-1. Therefore, Acey’s 40 cents wagered against Deucy’s nickel would be a fair bet.
The somewhat trickier bit is to figure out why 192 of the games are wins for the first player. One way to get there would be through an exhaustive computer search of the 216 games, and Ziling Zhou shared an example. Another would be to think about the idea of “nim-sum,” which in this case transforms the dice’s numbers into binary and adds them up without carry. Because the two players take turns, converting things into binary helps us solve the puzzle. For example, the nim-sum of rolls of 5 and 3 is 101 + 011, without carry, which equals 110, or 6. It has been proven that the first player in a game of Nim is guaranteed to win so long as the nim-sum of a given game is not zero. The sum is zero in the following games: (1,2,3), (1,4,5), (2,4,6) and (3,5,6). There are 3×2×1 = 6 ways to roll each of those four games, for a total of 24 wins for Deucy, and 192 wins for Acey.
Good luck, Deucy.
Want to submit a riddle?
Email me at [email protected]

Significant Digits For Friday, Sept. 7, 2018

You’re reading Significant Digits, a daily digest of the numbers tucked inside the news.

$1 fine
Jeffrey Winder, who was convicted of punching Jason Kessler, the organizer of the “Unite the Right” rally in Charlottesville, Virginia, was ordered by a jury to pay a $1 fine. He faced a maximum sentence of a year in jail and a $2,500 fine. [NPR]

$124 fine
In related news, a man who was accused of kicking a seagull that tried to eat his cheeseburger was ordered to pay a $124 fine. Seagulls, the AP reports, are protected by federal law. The man said it was a mistake, and he was trying to shoo the birds to, I presume, protect his delicious meal. [AP]

$100 million of stock
In August, The New Yorker published allegations of sexual misconduct against Les Moonves, the CEO of CBS. Now, according to sources familiar with the talks, Moonves is negotiating his exit from the network. His contract reportedly demands a $180 million severance package, while CBS board members are offering $100 million in stock. One. Hundred. Million. Dollars. [NBC News]

1.24 elasticity score
My colleagues Nathaniel and Nate (we have fun here) are back with their latest Election Update. This version brings us elasticities! It’s a term of art, and one that measures which states and districts swing the most with the national mood. The most elastic district in the country is Michigan’s 5th, which is basically the index finger of the state’s mitten shape. It’s elasticity score is 1.24, meaning that for every 1 percentage point the national political mood shifts toward a party, the district is expected to move 1.24 percentage points in that direction. The least elastic is Pennsylvania’s 3rd, in the southeastern part of that state, with a score of 0.72. [FiveThirtyEight]

1,800 miles from the East Coast
Hurricane Florence, which became the first major hurricane of the season, was, as of midday yesterday, churning in the Atlantic Ocean some 1,800 miles from the East Coast. But it could have an effect on the eastern seaboard of the U.S. in about a week, and the odds of that happening have ticked up, according to the latest weather models. Whether Florence is a “direct hit, a graze or near-miss is unknowable” for now because it’s so far away. But we here at Significant Digits HQ in New York City will keep a close eye on it. [The Washington Post]

2 million low-income Americans
Nearly 2 million low-income Americans would lose their benefits under a farm bill being considered by the House, according to the nonprofit research firm Mathematica. That bill would alter the eligibility criteria for 42 million recipients of the Supplemental Nutrition Assistance Program, or SNAP. That 2 million figure includes nearly 500,000 households with young children. [The New York Times]

If you see a significant digit in the wild, please send it to @ollie.
CORRECTION (Sept. 7, 2018, 10:41 a.m.): An earlier version of this post incorrectly described the location of Pennsylvania’s 3rd Congressional District. It is in the southeast of the state, not the northwest.