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Can You Save Some Cold Pizza?



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 Dan Levin comes a hazardous riddle for the ironclad and eagle-eyed:

The U.S. Open concluded last weekend, with physics major Bryson DeChambeau emerging victorious. Seeing his favorite golfer win his first major got Dan thinking about the precision needed to be a professional at the sport.

A typical hole is about 400 yards long, while the cup measures a mere 4.25 inches in diameter. Suppose that, with every swing, you hit the ball X percent closer to the center of the hole. For example, if X were 75 percent, then with every swing the ball would be four times closer to the hole than it was previously.

For a 400-yard hole, assuming there are no hazards (water, sand or otherwise) in the way, what is the minimum value of X so that you’ll shoot par, meaning you’ll hit the ball into the cup in exactly four strokes?

Submit your answer

Riddler Classic

From Dean Ballard comes a pernicious pizza puzzle:

Dean ordered a personal pizza that was precisely 10 inches in diameter. He ate half of it, and he wants to save the remaining semicircle of pizza in his refrigerator. He has circular plates of all different sizes, so to save space in his fridge, he’ll place the pizza on the smallest plate that holds the entire semicircle (i.e., with no pizza hanging off the plate).

Unfortunately, the smallest plate that can hold half of a 10-inch pizza is just a circle with a 10-inch diameter. So much for saving space.

But Dean has a thought: If he cuts the pizza, he can squeeze both of the resulting pieces onto a smaller circular plate — again, with no pizza hanging off the plate and without the pieces lying on top of each other.

If Dean makes a single straight slice, what is the diameter of the smallest circular plate onto which he can fit the two resulting pieces?

Extra credit: Dean wants to save even more space in his fridge. So instead of one straight slice, he will now make two straight slices. First, he will cut the semicircular pizza into two pieces. Then, he will take one of those pieces and make his second slice, giving him a total of three pieces. What is the diameter of the smallest circular plate onto which he can fit all three pieces?

Submit your answer

Solution to last week’s Riddler Express

Congratulations to 👏 Graham E. McGrath 👏 of Boston, Massachusetts, winner of last week’s Riddler Express. (This problem might have hit a little too close to home for Graham, who is a lab technician and uses centrifuges for a living!)

Last week, you were doing your best not to break a microcentrifuge, a piece of equipment that separates components of a liquid by spinning around very rapidly. Liquid samples were pipetted into small tubes, which were then placed in one of the microcentrifuge’s 12 slots evenly spaced in a circle.

Animation of a centrifuge with 12 slots spinning around.

For the microcentrifuge to work properly, each tube had to hold the same amount of liquid. Also, importantly, the center of mass of the samples had to be at the very center of the circle — otherwise, the microcentrifuge would not be balanced and likely break.

You needed to spin exactly seven samples. In which slots (numbered 1 through 12, as in the diagram above) could you have placed them so that the centrifuge was balanced?

Balancing centrifuges is certainly not a new problem, and was previously addressed on the Numberphile YouTube channel, with Holly Krieger explaining the solution.

In the case of 12 slots, it was always possible to balance the centrifuge for any even number of tubes. All you had to do was arrange them into pairs and place each pair on diametrically opposite ends of the circle. The center of mass of each pair was at the center of the circle, which meant the total center of mass was also at the center.

Alas, the problem asked how you could place seven tubes, and seven is not an even number. Several readers suggested adding an eighth tube, which was against the spirit of the puzzle.

As for odd numbers of tubes, balancing a single tube was not possible. However, it was indeed possible to balance three tubes by arranging them evenly around the circle so they were the vertices of an equilateral triangle (e.g., in positions 1, 5 and 9 in the diagram above). It was further possible to balance five tubes by again arranging three so they formed an equilateral triangle (e.g., in positions 1, 5 and 9) and another two on opposite sides (e.g., in positions 2 and 8). Again, because the centers of mass for both the trio and the pair were at the center of the circle, their combined center of mass was also at the center of the circle.

Returning to the original puzzle, it was possible to balance seven tubes by arranging three in an equilateral triangle, another two on opposite sides and the last two on a different pair of opposite sides. (Alternatively, some solvers noted that placing seven tubes was equivalent to choosing the five slots that didn’t have tubes, which was essentially the same problem as placing five tubes.) There were several correct answers here — more on that in a second — one of which was slots 1, 2, 5, 6, 8, 9 and 12, found by solver Alicia Zamudio.

For extra credit, you had to find the total number balanced arrangements for seven tubes. It turned out that there were exactly 12 arrangements, all rotations of each other. Solver Reece Goiffon verified this by finding all the ways seven vectors on a unit circle could add up to zero:

Meanwhile, in what I believe is a first for The Riddler, Chris Sears wrote code in Commodore 64 BASIC, checking all 792 (i.e., 12 choose 7) ways to place seven tubes in 12 slots.

The general version of this problem, with N slots and k tubes, was addressed in the aforementioned Numberphile video and was further proven by T. Y. Lam and K. H. Leung back in 2000. As for how many arrangements there are for given values of N and k, there’s an OEIS sequence for that! (Thanks to solver Eric Thompson-Martin for spotting it.)

When N = 12 and k increased from 0 to 12, the number of balanced arrangements was 1, 0, 6, 4, 15, 12, 24, 12, 15, 4, 6, 0 and 1. If you look closely, you’ll see that the sequence is symmetric, since placing k tubes was exactly the same problem as choosing where not to place k tubes.

I encourage readers to verify these results for themselves. But not the grad students or technicians out there. I don’t need angry lab directors emailing me about broken centrifuges.

Solution to last week’s Riddler Classic

Congratulations to 👏 Lowell Vaughn 👏 of Bellevue, Washington, winner of last week’s Riddler Classic. (Lowell is finally joining his son in the winner’s circle.)

Last week, you were introduced to an online game, Guess My Word, in which you tried to guess a secret word by typing in other words. After each guess, you were told whether the secret word was alphabetically before or after your guess. The game stopped and congratulated you once you had guessed the secret word.

The secret word was randomly chosen from a dictionary with exactly 267,751 entries. If you had this dictionary memorized and played the game as efficiently as possible, how many guesses should you have expected to make to uncover the secret word?

Most solvers recognized that the most efficient strategy was a binary search, where you first guessed the middle word in the dictionary. If the secret word came later, you then guessed the middle word in the second half of the dictionary. But if the secret word came earlier, you guessed the middle word in the first half of the dictionary. With each guess, you narrowed the possibilities by approximately half, and there was also a chance that the guess itself was the secret word.

The sequence of guesses could therefore be structured as a complete binary tree whose top node was your first guess (the exact middle word in the dictionary). If the secret word came earlier in the dictionary, you moved to the right; if the secret word came later in the dictionary, you moved to the left. Eventually, you’d find the secret word.

Now that you had the optimal guessing strategy, all that was left was calculating the average number of guesses. In your binary tree, one word — the middle word in the dictionary — would be guessed on the very first attempt. Two words required two guesses, four words required three guesses, eight words required four guesses, 16 words required five guesses, and so on. For each additional guess, there were twice as many words.

It turned out that the number of words in the dictionary, 267,751, was 1 + 21 + 22 + 23 + … + 216 + 217 + 5608. And each term in the sum required one more guess than the term before it. To find the average number of guesses, you had to divide each of these terms by 267,751 and then multiply by the corresponding number of guesses. In the end, the average was approximately 17.042 guesses.

Several solvers, like Michael Branicky, Josh Silverman and Rajeev Pakalapati, generalized this method, finding a formula for the expected number of guesses for a dictionary with any number of words. Meanwhile, Emma Knight went in a completely different direction — recursion — still arriving at the same solution.

Thanks again to Oliver Roeder for introducing Riddler Nation to this fun word game. (If only it weren’t so darn addictive.)

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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