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Can You Corral Your Hamster?

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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

From Josh Silverman comes a puzzle that’s out of this world:

When you started your doctorate several years ago, your astrophysics lab noticed some unusual signals coming in from deep space on a particular frequency — hydrogen times tau. After analyzing a trove of data measured at many regular intervals, you compute that you heard zero signals in 45 percent of the intervals, one signal in 38 percent of the intervals and two signals in the remaining 17 percent of the intervals.

Your research adviser suggests that it may just be random fluctuations from two sources. Each source had some fixed probability of emitting a signal that you picked up, and together those sources generated the pattern in your data.

What do you think? Was it possible for your data to have come from two random fluctuations, as your adviser suggests?

Submit your answer

Riddler Classic

From Scott Ogawa comes a riddle about rodents of usual size:

Quarantined in your apartment, you decide to entertain yourself by building a large pen for your pet hamster. To create the pen, you have several vertical posts, around which you will wrap a sheet of fabric. The sheet is 1 meter long — meaning the perimeter of your pen can be at most 1 meter — and weighs 1 kilogram, while each post weighs k kilograms.

Over the course of a typical day, your hamster gets bored and likes to change rooms in your apartment. That means you want your pen to be lightweight and easy to move between rooms. The total weight of the posts and the fabric you use should not exceed 1 kilogram.

For example, if k = 0.2, then you could make an equilateral triangle with a perimeter of 0.4 meters (since 0.4 meters of the sheet would weigh 0.4 kilograms), or you could make a square with perimeter of 0.2 meters. However, you couldn’t make a pentagon, since the weight of five posts would already hit the maximum and leave no room for the sheet.

You want to figure out the best shape in order to enclose the largest area possible. What’s the greatest value of k for which you should use four posts rather than three?

Extra credit: For which values of k should you use five posts, six posts, seven posts, and so on?

Submit your answer

Solution to last week’s Riddler Express

Congratulations to 👏 Alban 👏 of Jacksonville Beach, Florida, winner of last week’s Riddler Express.

Last week, you had a large pile of squares that each had a side length of 1 inch. One square was blue, while all the other squares were white. You wanted to arrange several white squares so they covered part of the blue square but didn’t overlap with each other. (The entire blue square did not have to be covered, while the blue area that each white square covered had to be nonzero.)

What was the greatest number of white squares you could have placed?

First, a quick acknowledgment that this was very similar to a problem posed by Martin Gardner some years ago. Special thanks to reader Brian Kell for pointing this out!

Nevertheless, this still proved to be a very challenging express, with a lot of disagreement: Among the hundreds of submitted responses, 2 percent said the answer was four, 3 percent said the answer was five, 14 percent said the answer was six, 56 percent said the answer was seven, 7 percent said the answer was eight, and 4 percent said the answer was nine. (There was a smattering of other answers as well, including readers who creatively wanted to stack paper-thin squares in the third dimension.)

Nine was not the answer. Remember, all the squares were the same size. So if you placed a 3-by-3 grid of white squares over the blue one and then rotated the grid about its center, at most five of the white squares would have an overlapping area with the blue square.

The majority of readers thought the answer was seven — there must be something to that. Solver Michael Branicky’s five-year-old (!) daughter Lydia may have been the youngest to attempt this puzzle. She got seven white squares to overlap with the blue square by arranging them in a rotated honeycomb pattern:

At this point, we’ve seen that seven squares were possible, while nine squares were not possible. So what about eight squares?

Solver Daniel Thompson illustrated different examples, from one white square all the way up to eight:

arrangement of eight white squares so they all overlap a blue square but not each other

It wasn’t as symmetric as the seven-square solution. But by tilting the various squares just right, it was possible to make room for that eighth white square in the middle.

While eight may not be a perfectly square number, this week it was a perfectly hip number.

Solution to last week’s Riddler Classic

Congratulations to 👏 Richard Guidry Jr. 👏 of Baton Rouge, Louisiana, winner of last week’s Riddler Classic.

Last week, the Riddler Manufacturing Company had an issue with their production of foot-long rulers. Each ruler was accidentally sliced at three random points along the ruler, resulting in four pieces. Looking on the bright side, that meant there were then four times as many rulers — they just happened to have different lengths.

On average, how long were the pieces that contained the 6-inch mark?

The problem as stated was slightly ambiguous. I have it on good authority that, for each ruler, the Riddler Manufacturing Company chose the three random points before doing any slicing. (If you assumed a slice was made after each point was selected, it was possible to get different and equally interesting results.)

Solver Quoc Tran declared that “math is for nerds,” before running 50,000 simulations of broken rulers like a total geek. He found that the length of the piece containing the 6-inch mark followed a rather curious distribution (shown below), with an average of approximately 5.621 inches.

Julian Gerez found a similar distribution, further noting that the unusual shape was due to the mashing together of two distinct cases, depending on whether the 6-inch mark was on one of the two middle pieces or on one of the two end pieces — but more on that in a moment!

So that was the computer geeks; now back to the math nerds. The most rigorous way to solve this is with calculus, integrating the product of each possible length multiplied by its relative probability — essentially calculating the mean value of the theoretical curve that Quoc Tran’s histogram approximates. Emma Knight fearlessly worked her way through those messy integrals, arriving at an answer of 5.625 inches.

An alternative approach was to break the problem down into two cases, depending on which of the four pieces contained the 6-inch mark. If the 6-inch mark was on one of the two end pieces of the broken ruler, that meant the three random breaks were all between the 0- and 6-inch marks (which occurred with a one-in-eight probability) or all between 6- and 12-inch marks (which also occurred with a one-in-eight probability). In other words, 25 percent of the time the three random breaks were all on one side of the 6-inch mark, while the other 75 percent of the time there was a single break on one side of the 6-inch mark and two breaks on the other side.

When all three breaks were on one side, the length of the piece with the 6-inch mark was 6 inches plus the average distance between the 6-inch mark and nearest of the three breaks. Regular solvers of Riddler Classics may know a thing or two about order statistics, particularly the result that when you choose N random values from a uniform distribution between 0 and 1, the expected value of the smallest number is 1/(N+1), the expected value of the second smallest is 2/(N+1), and so on up to the largest number, which has an expected value of N/(N+1). So when the three random breaks all occurred (uniformly) over a 6-inch range, the average distance between the 6-inch mark and the nearest break was one-fourth (since four is one more than three) of the total range, or 1.5 inches. Putting the two pieces together, that meant 25 percent of the time the average length was 6+1.5, or 7.5 inches.

But what about the other 75 percent of the time? As we said, there were two random breaks on one side of the 6-inch mark and one break on the other side. The average distance between the 6-inch mark and nearest among two breaks was one-third (since three is one more than two) of the length, or 2 inches. And the average distance between the 6-inch mark and the single break on the other side was one-half (since two is one more than one) of the length, or three inches. Putting these two pieces together, that meant 75 percent of the time the average length was 2+3, or 5 inches.

No integrals required, and we almost have our answer! Using the linearity of expectation, as solver Steve Gabriel did, the average length was 0.25(7.5) + 0.75(5), or 5.625 inches — the same answer we got from calculus!

But the fun didn’t stop there. Several readers took this puzzle to new heights, looking at how the average length of the piece with the 6-inch mark changed with the number of break points, as well as the average length of pieces containing other points on the ruler like the 1- or 2-inch mark. Laurent Lessard combined these into a single generalization, finding the expected length of the piece a fraction a along a ruler of length L broken at N random points. By computing the distribution for the number of breaks on either side of the selected point and applying order statistics, he found this length to be (2−aN+1−(1−a)N+1)L/(N+1).

One more thing: As N got very large — meaning there were many random breaks in the ruler — the average length Laurent found approached 2L/(N+1) for any value of a. In other words, as soon as you pick a point on the ruler and ask for the average length of the piece containing that point, the answer will be twice the length of the average piece. Again, that’s for any point you pick!

It’s a bizarre paradox, to be sure. My sense is this happens because you are looking for the next break in each of the two directions along the ruler. If anyone happens to make further headway on this, be sure to let Laurent (and me!) know.

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

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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.”

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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

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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

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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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