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Are You Hip Enough To Be Square?



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 Dean Ballard comes a sneaky sorting of squares:

You have a large pile of squares that each have a side length of 1 inch. One square is blue, while all the other squares are white. You want to arrange several white squares so they cover part of the blue square but don’t overlap with each other.

For example, here’s how you could arrange four white squares so they each cover part of the blue square.

four white squares partially overlapping with a blue square but not each other

What is the greatest number of white squares you can place so that each covers part of the blue square without overlapping one another? (The entire blue square does not have to be covered, while the blue area that each white square covers must be nonzero.)

Submit your answer

Riddler Classic

From Angela Zhou comes one riddle to rule them all:

The Riddler Manufacturing Company makes all sorts of mathematical tools: compasses, protractors, slide rules — you name it!

Recently, there was an issue with the production of foot-long rulers. It seems that each ruler was accidentally sliced at three random points along the ruler, resulting in four pieces. Looking on the bright side, that means there are now four times as many rulers — they just happen to have different lengths.

On average, how long are the pieces that contain the 6-inch mark?

Submit your answer

Solution to last week’s Riddler Express

Congratulations to 👏 Richard Dickerman 👏 of Dallas, Texas, winner of last week’s Riddler Express.

Last week, I was hiking on Euclid Island, which was perfectly rectangular and measured 3 miles long by 2 miles wide. I was especially interested in locating the point on the shore that was nearest to my position.

From where I had been standing, there were in fact two distinct points on the shore that were both the closest such points. It turned out the trail I was hiking along connected all such locations on the island — those with multiple nearest points on the shore.

What was the total length of this trail on Euclid Island?

The dead center of the island was a logical place for many solvers to start. It was just 1 mile along the width — in either direction. But before we looked for other such points, what did it mean, mathematically, for there to be multiple nearest points on the shore?

One way to think about it was to grow a circle (kind of like blowing up a balloon) centered at your current position. As the circle got larger and larger, it eventually touched (i.e., was tangent to) the shore. The moment the circle touched the shore, you had to look at how many places it touched the shore. If there was only one such point of tangency, then you weren’t on the trail. But if there were two or more such points, then you were indeed on the trail.

So what did Euclid Island’s trail actually look like? Rebekah Murphy noticed the trail included a segment of length 1 that ran east-west down the middle of the island. Every point on this segment was equidistant from the north and south shores. (Interestingly, the endpoints of this segment were equidistant from three distinct points along the shore.)

But the fun didn’t end there. Ishaan Bhatia noted that the trail also had four additional segments, which ran between the endpoints of the middle segment and the four corners of the island. (While the corners themselves were not part of the trail — after all, they were on the beach — the trail came infinitesimally close, which meant the exclusion of these points didn’t affect your length calculations.)

The animation below puts all five segments together, revealing the entire green hiking trail on Euclid Island. The circle that grows and shrinks demonstrates the points of tangency (i.e., the nearest spots on the beach) for each location on the trail.

At this point, we’re ready to answer the original question, which asked you to determine the total length of the trail. As we already said, the middle segment had a length of 1. Using the Pythagorean theorem (or, rather, the Babylonian Formula), the remaining four segments had a length of √2. That meant the total length of the trail was 1+4√2 miles, or about 6.657 miles.

For extra credit, you looked at Al-Battani Island, which was elliptical rather than rectangular. Al-Battani Island’s major axis was 3 miles long, while its minor axis was 2 miles long. Like Euclid Island, Al-Battani Island had a hiking trail that connected all locations with multiple nearest points on the shore. What was the total length of this trail on Al-Battani Island?

As with Euclid Island, a good place to start was the dead center, which was 1 mile from the two endpoints of the minor axis. And once again, there was a central horizontal segment that ran along the major axis. The challenge here was figuring out just where the trail ended.

It didn’t cover the entire major axis, because when it got too close to the endpoints, there was only a single point of tangency. In other words, you had to find when the circle — centered on an ellipse’s major axis and internally tangent to the ellipse — transitioned between one and two points of tangency, as illustrated below:

hiking trail on Al-Battani Island

And that’s where the math got hairy. If you assumed the ellipse was centered at (0, 0), then it was described by the equation x2/1.52 + y2 = 1. Meanwhile, the equation for a circle with radius R centered at h on the x-axis was (xh)2 + y2 = R2. From there, you had to find coordinates (x, y) that satisfied both equations, but you also needed to use calculus to ensure the circle and the ellipse had matching slopes at these points.

In the end, as long as the center of the circle was between (−5/6, 0) and (5/6, 0), the circle was tangent to the ellipse in two locations. That meant the distance between these two points — about 1.667 miles — was the length of the trail and the answer to the extra credit.

Quite the challenging hike, if you ask me. And if you’d like to explore this problem further, check out an interactive version of this graph on Desmos, courtesy of solver Hypergeometricx.

Solution to last week’s Riddler Classic

Congratulations to 👏 Bill Neagle 👏 of Springfield, Missouri, winner of last week’s Riddler Classic.

Last week, inspired by Kareem Carr, you looked at an alternative definition for addition and followed where it led. To compute the sum of x and y, you combined groups of x and y nematodes and left them for 24 hours. When you came back, you counted up how many you had — and that was the sum!

It turned out that, over the course of 24 hours, the nematodes paired up, and each pair had one offspring 50 percent of the time. (If you had an odd number of nematodes, they still paired up, but one was left out.) So if you wanted to compute 1+1, half the time you’d get 2, and half the time you’d get 3. If you computed 2+2, 25 percent of the time you’d get 4, 50 percent of the time you’d get 5, and 25 percent of the time you’d get 6.

We also redefined exponentiation: Raising a sum to a power meant leaving that sum of nematodes for the number of days specified by the exponent.

With this number system, what was the expected value of (1+1)4?

A good strategy here was to work your way up, one power at a time. First, what was the expected value of (1+1)1? You initially had two nematodes that paired up, and after 24 hours there was a 50 percent chance they’d have one offspring. So the expected value of (1+1)1 was 2.5.

Next, what was the expected value of (1+1)2? In other words, what was the expected number of worms after 48 hours? As we said, after 24 hours, there was a 50 percent chance there were two worms and a 50 percent chance there were three worms. After another 24 hours, the two-worm case had an expected value of 2.5. Meanwhile, the three-worm case resulted in two worms that paired up along with a third wheel. The pair of worms resulted in an expected value of 2.5, and adding the third worm gave an expected value of 3.5. Half the time there would be 2.5 worms, and the other half the time there would be 3.5 worms, which meant the expected value was an even 3.

Let’s do one more: What was the expected value of (1+1)3? Digging through the previous paragraph, we just saw that after 48 hours there was a 25 percent chance of having two worms, a 50 percent chance of having three worms and a 25 percent chance of having four worms. In the two-worm case, the expected number of worms after 72 hours was again 2.5. In the three-worm case, the expected number of worms after 72 hours was 3.5. The four-worm case was more complicated — 25 percent of the time there would still be four worms after 72 hours, 50 percent of the time there would be five worms, and 25 percent of the time there would be six worms. Putting all these cases (and sub-cases) together, the expected value of (1+1)3 was 0.25·2.5 + 0.5·3.5 + 0.25·(0.25·4 + 0.5·5 + 0.25·6), or 3.625.

I’ll spare you all the casework for (1+1)4. After 96 hours, there could be anywhere from two to nine worms. The expected value was 4.40625 worms. Andrew Heairet neatly visualized the breakdown of probabilities over the course of 96 hours:

Mark Girard went beyond 96 hours and found the expected number of nematodes more than two weeks later. This value appeared to increase exponentially over time, which is the perfect segue into the extra credit, which asked you how the expected value of (1+1)N behaved as N got larger and larger. As many solvers correctly observed, the expression goes off to infinity. But the real question was how.

The challenge here related to the “odd worms out,” which sat out reproductive cycles. Were it not for them, the math would have been more straightforward. For each pair of worms, their number either stayed the same or increased by 50 percent over each 24-hour period. Averaging those two possibilities meant the expected number increased by 25 percent every 24 hours. But again, this reasoning didn’t apply, thanks to the occasional one worm who didn’t pair up.

To figure this out, some solvers fit exponential models directly to the data. For example, Hypergeometricx found the growth was intriguingly close to 1.56·(1.23456789)N. Meanwhile, Josh Silverman showed that 2·(1.25)N−1 was in fact a pretty decent approximation.

But Rajeev Pakalapati took the cake, showing — analytically, mind you — that 2·(1.25)N−1 + 0.5 was the limiting behavior of nematode exponentiation. His work is shown below, and you can also follow along via Steve Gabriel’s helpful transcription.

Rajeev Pakalapati's solution to the Riddler Classic

Indeed, this was a fun and challenging foray into an alternate set of operations. Only one submitter felt threatened enough to compare this riddle to an Orwellian dystopia. Needless to say, they didn’t get very far with the math.

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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