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Is The Price Right?



Welcome to The Riddler. Every week, I offer up problems related to the things we hold dear around here: math, logic and probability. Two puzzles are presented each week: the Riddler Express for those of you who want something bite-size and the Riddler Classic for those of you in the slow-puzzle movement. Submit a correct answer for either,1 and you may get a shoutout in next week’s column. Please wait until Monday to publicly share your answers! If you need a hint or have a favorite puzzle collecting dust in your attic, find me on Twitter.

Riddler Express

As you may have seen in FiveThirtyEight’s reporting, there’s an election coming up. Inspired, Vikrant Kulkarni has an electoral enigma for you:

On Nov. 3, the residents of Riddler City will elect a mayor from among three candidates. The winner will be the candidate who receives an outright majority (i.e., more than 50 percent of the vote). But if no one achieves this outright majority, there will be a runoff election among the top two candidates.

If the voting shares of each candidate are uniformly distributed between 0 percent and 100 percent (subject to the constraint that they add up to 100 percent, of course), then what is the probability of a runoff?

Extra credit: Suppose there are N candidates instead of three. What is the probability of a runoff?

Submit your answer

Riddler Classic

This week, we return to the brilliant and ageless game show, “The Price is Right.” In a modified version of the bidding round, you and two (not three) other contestants must guess the price of an item, one at a time.

Assume the true price of this item is a randomly selected value between 0 and 100. (Note: The value is a real number and does not have to be an integer.) Among the three contestants, the winner is whoever guesses the closest price without going over. For example, if the true price is 29 and I guess 30, while another contestant guesses 20, then they would be the winner even though my guess was technically closer.

In the event all three guesses exceed the actual price, the contestant who made the lowest (and therefore closest) guess is declared the winner. I mean, someone has to win, right?

If you are the first to guess, and all contestants play optimally (taking full advantage of the guesses of those who went before them), what are your chances of winning?

Submit your answer

Solution to last week’s Riddler Express

Congratulations to 👏 Nick Russell 👏 of Vancouver, Canada, winner of last week’s Riddler Express.

Last week, you were helping me park across the street from a restaurant for some contactless curbside pickup. There were six parking spots, all lined up in a row.

While I could parallel park, it definitely wasn’t my preference. No parallel parking was required when the rearmost of the six spots was available or when there were two consecutive open spots. If there was a random arrangement of cars occupying four of the six spots, what was the probability that I had to parallel park?

With six spots and four cars, there were 6 choose 4, or 15 cases to consider. Some solvers, like Lisa Fondren of Montrose, Michigan, listed them all out and counted how many required parallel parking.

But there were other ways to find the solution that didn’t require working through every case. Rather than considering where the four cars were, solver Libby Aiello equivalently looked at where the two empty spots were. Among the total 15 combinations, there were five in which the two empty spots were adjacent: the first and second spots, the second and third, the third and fourth, the fourth and fifth, and the fifth and sixth.

There were also five combinations in which the last spot was open, since there were five spots from which to choose the other open spot. Combining these two cases (having two consecutive open spots and having the sixth spot open), there appeared to be 10 combinations that didn’t require parallel parking.

But that wasn’t quite right. As Libby noted, one combination — when the fifth and sixth spots were open — was counted twice, since two consecutive spots were open and the sixth spot was open. Subtracting one to account for this double counting meant there were nine combinations that didn’t require parallel parking, and six combinations that did. Therefore, the probability I had to parallel park was 6/15, or 40 percent.

I was pleased to see how many readers solved this combinatorics challenge. Now if only that many drivers could successfully parallel park…

Solution to last week’s Riddler Classic

Congratulations to 👏 Douglas Thackrey 👏 of Loulé, Portugal, winner of last week’s Riddler Classic.

Parking cars was one thing — parking trucks was another thing entirely. Last week, you looked at a very long truck (with length L) with two front wheels and two rear wheels. (The truck was so long compared to its width that you could consider the two front wheels as being a single wheel, and the two rear wheels as being a single wheel.)

You were asked to determine the truck’s turning radius, given the angles by which you could turn the front or rear wheels. First, you considered what would happen if the front wheels could be turned up to 30 degrees in either direction (right or left), but the rear wheels did not turn.

There was no doubt among readers that this was a geometry puzzle, but the challenge lay in translating the constraints on the wheels (i.e., only turning a certain amount or not at all) into the resulting motion of the truck.

As suggested by the term “turning radius,” the key was to think about the circular motion of the truck. When the driver rotated the front wheels a full 30 degrees in one direction and drove forward, both the front and the rear of the truck would move in circles. The front of the truck always made a 30 degree angle with the tangent line to the circle it was moving around.

Meanwhile, the rear wheels of the truck couldn’t turn. That meant the truck’s rear was always tangent to the circle it was moving around.

If that wasn’t clear, here’s an animation to illustrate how the truck was moving:

Truck driving around in a circle. The front wheels can turn 30 degrees and the back wheels cannot turn. The radius of the front (outer) circle is twice the length of the truck.

The green line segment represents the truck, and the circles represent the paths of the truck’s front and rear. Sure enough, the angle between the truck and the tangent line (represented by the white segment) is always 30 degrees. This means that the front of the truck is moving around a wider circle than the rear — and, consequently, that the front of the truck moves faster than the rear!

At this point, calculating the turning radius was a matter of geometry and trigonometry. If the green segment was doubled in length so that it formed a chord within the larger circle, the 30 degree angles meant that this chord was one side of an inscribed regular hexagon, whose sides all equal the circle’s radius. And so if the truck had length L, the turning radius — that is, the radius of the circle around which the front of the truck moved — was 2L. (Solvers who gave the turning radius for the truck’s midpoint or rear and explained their reasoning were also given full credit.)

That was the case when you could only turn the front wheels. You were also asked for the turning radius when both the front and rear wheels could be independently turned up 30 degrees in either direction. To make the tightest possible circle, the front and rear wheels were both rotated the full 30 degrees, but in opposite directions, allowing the front and rear to move along the same circle. Again, here’s an animation:

A truck driving around a circular track. Both the front and rear wheels turn 30 degrees. The radius of the circle equals the length of the truck.

This time, the truck made up a complete chord that was a side of an inscribed regular hexagon. That meant the turning radius was equal to the truck’s length, L. (Again, solutions for different locations on the truck were also accepted.)

A few solvers, including Laurent Lessard and Josh Silverman, tackled the general version of this problem, in which the front wheels could be turned an angle θ1 and the rear wheels could be turned an angle θ2. The turning radius at the front of the truck was L·cos(θ2)/sin(θ1+θ2), while the turning radius at the rear was L·cos(θ1)/sin(θ1+θ2).

These formulas checked out for both questions in the riddle. And when neither wheel could turn (i.e., θ1 and θ2 were both zero), the turning radius went to infinity, which also made sense. In that case, you’d just have to keep on truckin’.

Want more riddles?

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

Want to submit a riddle?

Email Zach Wissner-Gross at riddlercolumn@gmail.com


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Toronto FC hoping to make MLS Cup run having spent much of 2020 far from home



On a recent Thursday in Hartford, Conn., Toronto FC goalkeeper Quentin Westberg pondered the dichotomy of wanting to reach MLS Cup on Dec. 12, but also desiring to see his family again. Meanwhile, Jim Liston, the team’s director of sports science, was planning a trip to Lowe’s to buy 15 garbage cans so players could have an ice bath after training. As for manager Greg Vanney, he was fretting about his team’s health and the lack of practice time their schedule was affording.

Such is the life of a team as it attempts to not only navigate its way through the COVID-19 pandemic, but has been forced to do it away from home.

Due to travel restrictions between the U.S. and Canada, TFC — like the league’s other two Canadian teams, Montreal Impact and Vancouver Whitecaps — set up a “home” base in the U.S. for the remainder of the season; Toronto were stationed in Hartford. (Vancouver Whitecaps took roost in Portland, ground-sharing with Timbers, while Montreal Impact split use of New York Red Bulls’ facilities in Harrison, N.J.) This was on top of nearly every team spending nearly a month inside a bubble back in July at the MLS is Back Tournament outside Orlando, Florida.

The Reds spent about seven weeks back in Toronto as they played a series of matches against Canadian teams. In mid-September, the remainder of the regular season — and the temporary move to Hartford — beckoned. The vagabond nature of the campaign is what led Liston to joke that he was willing to discuss “whatever five seasons” the team has been through so far. But for Vanney and the players, the campaign has required a special kind of focus.

“A lot of what we’ve done here, and what we try to preach here is just control the controllables, and don’t get too drawn into the things you can’t,” Vanney told ESPN. “Roll with it, and make the best out of whatever the situation is.”

Stream FC Daily on ESPN+
– 2020 MLS Playoffs: Who’s in, schedule and more
– MLS on ESPN+: Stream LIVE games and replays (U.S. only)

Toronto has largely succeeded in spite of its odyssey. While there was disappointment at missing out on the Supporters’ Shield to the Philadelphia Union, TFC went 7-3-2 during its Hartford sojourn and finished with the second-best record in the league. But the challenges have still been immense. Simply being out of one’s home environment is difficult enough, but the time spent away from family and loved ones weighs heavy on the psyche, even as Vanney has given players the occasional trip back to Toronto — under quarantine — to reconnect with loved ones.

“It’s just very different, very challenging and emotionally exhausting,” Westberg said of his experience while based in Hartford.

Westberg has arguably had it tougher than most. The TFC goalkeeper is married with four children, including a baby girl who was born in June. For that reason, Westberg and his wife, Ania, made the decision at the end of September that it would be better for her and their kids to head back to his native France so they could be surrounded by family. Westberg called it “the least bad decision,” but there are difficulties nonetheless.

“I’m a very even person, and this year has challenged me a lot,” he said. “I’m still pretty even, but I keep a lot to myself and for sure there’s some difficult days, seeing your family [struggle] from your absence.”

The inability to be home has affected the players and staff in other ways. In Toronto, there are ways of disengaging from the game. Being with friends, loved ones or even in familiar surroundings can be the best medicine in terms of forgetting a bad game or training session. But in Hartford, at the team’s hotel, that escape is nearly impossible even as players try to distract themselves by reading or taking online classes.

“You don’t really unplug,” Westberg said. “You FaceTime family, or this or that, but it’s too short. You’re 100 percent focused on your soccer, and your whole day basically relies on being ready for whatever soccer activity that you have next, whether it’s practice or game. It’s good for your physique, it’s optimal for the way you eat and the way you [train]. But mentally, you’re not as fresh as your body.”

That isn’t to say there are only negatives to the separation. There is also an us-against-the-world mentality that Toronto has adopted, given that their players and personnel are experiencing the season in a way that is vastly different than most other teams. The team staff has done what it can to make their surroundings a home away from home, whether it’s personalizing the locker rooms at Rentschler Field or having hotel staff brand the surroundings in TFC colors. The hotel went so far as to bring in a barista who could consistently give the players their coffee fix. Supporters groups have even sent down banners in a bid to convey the fact that the players are remembered.

The care that TFC takes for players has extended to families back home, with the club supplying meals to loved ones three times a week.

On the logistical side, Liston made sure that one of the gyms used at MLS is Back was brought to TFC’s hotel in Hartford, and he remarked that the food at the hotel is “arguably the best we’ve ever had on the road.”

There have also been efforts to create new routines. Assistant coach Jason Bent, aka DJ Soops, has been in charge of the pregame music selection for the past 18 months — no easy feat for a squad that has a considerable international presence. In Hartford, Bent has set aside Thursday nights to spin music in one area of the hotel. He’ll even go live on Instagram or Twitch for those who prefer to relax in their rooms.

“[We] opened it to players and staff and basically anyone that’s part of our bubble to come relax, listen to music and just enjoy each other’s company,” Bent said. “I enjoy making people happy so if it’s helping everyone even in the slightest, I have no problem arranging the set and spinning.”

For Vanney, the pandemic and operating outside of the team’s home market has meant any number of challenges. He said the team has used three different training facilities in Hartford, with varying field conditions. He recognizes that the trips home are vital for the mental health of his players and staff, but any breaks also mean less time spent on the practice field. The compressed schedule, which at times involved games every three or four days, has had an impact as well. Even the best-laid plans in terms of squad rotation were impacted as minor injuries began popping up.

“We end up with a lot of guys in different positions because they need special kinds of treatment or care to help them get fit and back to health,” Vanney said. “So it ends up being a lot of different things kind of going on all at once, and that’s been the challenge of it.”

Recovery from matches has been complicated by the fact that TFC doesn’t have access to the same level of facilities that it does at home — hence Liston’s emergency trip to Lowe’s to fashion impromptu ice baths for the players. Then there are the different ways the players occupy themselves on the road as compared to home, especially amid the pandemic.

“There’s really no life outside of the hotel,” Liston said. “[At home], you may go walk the dog in the afternoon or go for a walk with your wife or friend or girlfriend or family and you’re out and about. The recommendation [here] is to kind of stay put. So you’ve got a really active population and pro athletes, who we’re asking them to be sedentary the rest of the time, kind of stay in the hotel from a COVID and safety standpoint. That’s not optimal for recovery either.”

There are also the creature comforts of home that are no longer available on the road, which can impact sleep.

“Sleep is the number one tool for recovery, and that’s definitely been a challenge,” Liston said. “We do well-being questionnaires and the scores on quality of sleep, and hours of sleep, just drop.”



Tom Barlow and Brian White seal Toronto’s fate in a 2-1 win for New York Red Bulls. Watch MLS on ESPN+.

Another change has been same-day travel, which has drawn mixed reactions from the TFC players and staff. Vanney and Westberg are generally in favor, saying it reminds them of when they each played in France. Flying back the same night also means a training day isn’t lost. Liston has a different perspective in that he prefers arriving the day before, and then leaving the same day.

“I think [same-day travel] makes for a really long day,” he said. “And there’s definitely a negative impact on performance, taking three bus rides and a plane ride before your game. You’re getting home — it can be 12:30, but it could also be 1:30 in the morning, and that’s where you know our well-being scores and sleep hours and quality just disappear. When you have so many games in succession, you can’t make up the sleep.”

With the playoffs set to begin for TFC on Nov. 24, the end is in sight, even as it makes for a complex — and even conflicting — set of emotions.

“This is the tricky part. I miss them a lot,” Westberg said of his family. “But in a way I want to see them as [late] as possible in December, because obviously, there’s this idea that we want to do well in the playoffs and we want to keep going. TFC has a history of setting high standards and high expectations. It’s a heavy load to carry but also an exciting one.”

Win or lose, it’s a season they’ll never forget.


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Bettman: NHL is mulling temporary realignment



The NHL is considering a temporary realignment of its teams for the 2020-21 season due to COVID-19 travel restrictions, according to commissioner Gary Bettman.

Bettman said Tuesday that restrictions on travel across the Canadian border, as well as “limitations in terms of quarantining when you go from certain states to other states” within the United States, could mean the NHL creates a more regionalized alignment for its upcoming season.

“As it relates to the travel issue, which is obviously the great unknown, we may have to temporarily realign to deal with geography, because having some of our teams travel from Florida to California may not make sense. It may be that we’re better off — particularly if we’re playing a reduced schedule, which we’re contemplating — keeping it geographically centric and more divisional-based; and realigning, again on a temporary basis, to deal with the travel issues,” Bettman said during a 2020 Paley International Council Summit panel with fellow commissioners Adam Silver of the NBA and Rob Manfred of MLB.

The NHL board of governors has a meeting scheduled for Thursday which will provide a progress report and possible recommendations for a season format, based on talks between the league and the NHL Players’ Association. The target date for starting next season remains Jan. 1.

Bettman said the league is considering a few scheduling options for the 2020-21 season. Something that’s off the table: playing the entire season in the kind of bubbles the NHL had in Toronto and Edmonton, Alberta, to complete last season. But Bettman said teams opening in their own arenas is a possibility, along with a modified bubble.

“We are exploring the possibility of playing in our own buildings without fans [or] fans where you can, which is going to be an arena-by-arena issue. But we’re also exploring the possibility of a hub. You’ll come in. You’ll play for 10 to 12 days. You’ll play a bunch of games without traveling. You’ll go back, go home for a week, be with your family. We’ll have our testing protocols and all the other things you need,” he said.

Bettman also indicated that the NHL is exploring “a hybrid, where some teams are in a bubble, some teams play at home and you move in and out.”

The NBA’s board of governors unanimously approved a deal with the players’ union that sets the stage for a season that will open on Dec. 22 and with a reduced schedule of 72 games. Silver said that the commissioners are in communication on COVID-19-related issues, especially the NBA and the NHL, since the two leagues’ teams share arenas and, in some cases, team owners.

Silver said he senses that the NBA will have fans in many of its buildings this season.

“We’re probably going to start one way, where we’re maybe a little bit more conservative than many of the jurisdictions allow,” he said. “What we’ve said to our teams is that we’ll continue to work with public health authorities. Arena issues are different than outdoor stadium issues. There will be certain standards for air filtration and air circulation. There may be a different standard for a suite than there will be for fans spaced in seats.”

Silver said there will be standardized protocols that are consistent from arena to arena, such as proximity between players and fans: “In certain cases, for seats near the floor, we’re going to be putting in testing programs, where fans will certify that they’ve been tested — some within 48 hours, some within day of game.” While Silver supported a continued expansion of the NBA postseason through its play-in tournament, Bettman said that he’s not in favor of expanded playoffs or “playing with the fundamentals of the game.” The NHL had 24 teams in its postseason last summer.


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The Battleground States Where We’ve Seen Some Movement In The Polls



With apologies to The Raconteurs, the presidential race continues to be “steady as she goes,” with little sign of tightening despite a plethora of new polls. FiveThirtyEight’s presidential forecast gives Joe Biden an 89 in 100 shot at winning the election, while President Trump has just an 11 in 100 chance. This makes Biden the favorite, but still leaves open a narrow path to victory for Trump, for whom a reelection win would be surprising — but not utterly shocking.

At the same time, we also have fewer polls from live-caller surveys, which have historically been more accurate and have shown slightly better numbers for Biden, than polls that use other methodologies, such as polls conducted primarily online or through automated telephone calls. Nevertheless, while the overall picture has shifted only a little in recent days, a few battleground states have seen at least some movement in their polls, which has slightly altered the odds Biden or Trump wins in each of those places.

What election stories need to get more coverage | FiveThirtyEight Politics Podcast


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