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Can You Feed The Hot Hand?



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 Drew Mathieson comes an exploration of basketball’s historied hot hand:

This season, on the way to winning her fourth WNBA championship in her 17-year career, Sue Bird made approximately 50 percent of her field goal attempts. Suppose she and Seattle Storm teammate Breanna Stewart are interested in testing whether Bird has a “hot hand” — that is, if her chances of making a basket depend on whether or not her previous shot went in. Bird happens to know that her chances of making any given shot is always 50 percent, independent of her shooting history, but she agrees to perform an experiment.

In each trial of the experiment, Bird will take three shots, while Stewart will record which shots Bird made or missed. Stewart will then look at all the trials that had at least one shot that was preceded by a made shot. She will randomly pick one of these trials and then randomly pick a shot that was preceded by a made shot. (If there was only one such shot to pick from, she will choose that shot.)

What is the probability that Bird made the shot that Stewart picked?

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Riddler Classic

Now that LeBron James and Anthony Davis have restored the Los Angeles Lakers to glory with their recent victory in the NBA Finals, suppose they decide to play a game of sudden-death, one-on-one basketball. They’ll flip a coin to see which of them has first possession, and whoever makes the first basket wins the game.

Both players have a 50 percent chance of making any shot they take. However, Davis is the superior rebounder and will always rebound any shot that either of them misses. Every time Davis rebounds the ball, he dribbles back to the three-point line before attempting another shot.

Before each of Davis’s shot attempts, James has a probability p of stealing the ball and regaining possession before Davis can get the shot off. What value of p makes this an evenly matched game of one-on-one, so that both players have an equal chance of winning before the coin is flipped?

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Solution to last week’s Riddler Express

Congratulations to 👏 Frank Probst 👏 of Houston, Texas, winner of last week’s Riddler Express.

Last week, the residents of Riddler City were electing a mayor from among three candidates. The winner was the candidate who received an outright majority (i.e., more than 50 percent of the vote). But if no one achieved this outright majority, there would be a runoff election among the top two candidates.

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

The “uniformly distributed” wording in the problem was ambiguous and was interpreted several ways by readers. How can you randomly choose three numbers between 0 and 100 that add up to 100? Here, I will write about three popular interpretations.

First, imagine randomly picking three values between 0 and 100, and call them x, y and z. Each choice of (x, y, z) corresponded to a point in a cube that measured 100 by 100 by 100. But only some points in this cube had coordinates that summed to 100 — those that also lay on the plane x+y+z = 100. This intersection between a cube and a plane might have been hard to visualize — it was an equilateral triangle (shown below). If you divided this triangle into quarters, then three of those quarters had one value (x, y or z) that exceeded 50. That meant the probability of a runoff, with no voting shares that exceeded 50, was 1/4.

That was one interpretation. Another way to “uniformly” pick three values was to draw a number line between 0 and 100 and break it into three segments by randomly picking two points, a and b. Assuming b was greater than a, the three lengths that summed to 100 were a, ba and 100−b. The challenge was to find when each of these three values exceeded 50 inside the triangle defined by 0 ≤ a b ≤ 100. Each of the three inequalities (a ≥ 50, ba ≥ 50 and 100−b ≥ 50) carved out a quarter of the larger triangle. And so, once again, that meant the probability of a runoff was 1/4.

Yet another way to “uniformly” pick three values was to go ahead and pick three numbers between 0 and 100 (again, let’s call them x, y and z) and then “normalize” them — that is, divide each number by x+y+z and multiply by 100, so that they were guaranteed to add up to 100. Like before, each choice of (x, y, z) corresponded to a point in a cube. But this time, to avoid a runoff, you needed one of the values to exceed the sum of the other two, meaning it would exceed 50 percent of the sum of all three numbers. There were three regions in the cube where a runoff would not occur: x > y+z, y > x+z and z > x+y. Each region made up one-sixth of the cube, so all together they represented half the cube, as shown below. In the other half, a runoff was necessary. So according to this interpretation of the problem, the answer was 1/2. (Some solvers, like Benjamin Dickman, noted that this approach was identical to finding the probability that three lengths from a random, uniform distribution satisfy the triangle inequality.)

For extra credit, you wanted the probability of a runoff when there were N candidates instead of three. Once again, the answer depended on your interpretation of the problem. Based on the first interpretation, this general solution was 1−N/2N−1 (nicely explained by Josh Silverman), since each of the N candidates had a 1/2N−1 chance of winning an outright majority. Similarly, based on the second interpretation, the answer was again 1−N/2N−1. But based on the third interpretation, the answer was 1−1/(N−1)!.

That’ll do it for Riddler City’s mayoral election. Don’t forget to vote in any other elections that may be happening!

Solution to last week’s Riddler Classic

Congratulations to 👏 Asher S. 👏 of Chicago, Illinois, winner of last week’s Riddler Classic.

Last week, you were playing a modified version of “The Price is Right.” In this version’s bidding round, you and two (not three) other contestants had to guess the price of an item, one at a time.

The true price of this item was a randomly selected real number between 0 and 100. Among the three contestants, the winner was whoever guessed the closest price without going over. For example, if the true price was 29 and you guessed 30, while another contestant guessed 20, then they would be the winner even though your guess was technically closer.

In the event all three guesses exceeded the actual price, the contestant who had made the lowest (and therefore closest) guess was declared the winner.

If you were the first to guess, and all contestants played optimally (taking full advantage of the guesses of those who had gone before them), what were your chances of winning?

At first, this three-player game might have seemed unsolvable. As the first to guess, you would want to know what the second and third players’ strategies would be. But their strategies depended on yours and on each other’s. Was there any way out of this mess?

Indeed there was. One approach was to work backwards. Suppose you (the first player) guessed a price A and the second player guessed a price B. What should the third player do? For now, let’s assume A was less than B. The third player would then choose from among three options:

  • Guess a value of zero, in which case they’d win if the true price was between 0 and A — a range of A.
  • Guess a value infinitesimally greater than A, in which case they’d win if the true price was between A and B — a range of BA.
  • Guess a value infinitesimally greater than B, in which case they’d win if the true price was between B and 100 — a range of 100−B.

But which of the three values should the third player guess? Whichever corresponded to the greatest range. (If A had been greater than B, there would again be three options, but with A and B reversed.)

So for any combination of A and B, the third player’s strategy was known. Next, it was time to look more closely at the second player.

For each value of A, the second player could figure out their chances of winning for any B they picked, since they now knew what the third player would do given A and B. For each A, the second player would then pick a B that maximized their own chances of winning.

At last, we’re back to you, the first player. By now, you knew exactly what the second and third players would do in response to any guess A. As with the second player, that meant you had to pick the value that maximized your own chances of winning.

Amidst all this strategizing, I neglected to mention just what these optimal guesses were. In the end, your best guess was two-thirds of 100 (~66.7). Then the second player’s best move was to guess one-third of 100 (~33.3), and the third player’s best move was to guess anything less than that (e.g., zero). All players had a one-third chance of winning. If you deviated from these optimal values, the second and third players would both have the advantage over you, each with a greater than one-third chance of winning.

Some solvers said they would have guessed a price that was one-third of 100. But as Emma Knight observed, Player 2 could then have guessed anything less than that and Player 3 slightly more. That would have meant Player 2 still had a one-third chance of winning, while Player 3’s chances went up to two-thirds, leaving you with nothing. Player 2 might not have chosen to sabotage your hopes of winning, but why leave it to chance?

Finally, Keith Wynroe of Edinburgh, Scotland offered a neat extension to this problem, asking how the three players’ strategies might change if the goal was not simply maximizing one’s chances of winning, but rather maximizing the expected value of the prize won. According to Keith, while this shift would incentivize all three players to bet higher values, no one’s chances of winning actually changed.

After all was said and done, it turned out to be a fair game. How sweet.

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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