WASHINGTON (AP) — Donald Trump says his inauguration will have “an unbelievable, perhaps record-setting turnout.” Organizers of a protest the next day call it the biggest demonstration in history to welcome a new president.

Exactly how many people actually show up for both events will likely never be known.

Counting the number of people at major public events is as much art as science, and there will be no official tally of how many people attend either Friday’s inaugural festivities or the Women’s March on Washington on Saturday.

Here’s a look at why crowd size is so mysterious.

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IT STARTS WITH THE MILLION MAN MARCH

For decades, the National Park Service provided official crowd estimates for gatherings on the National Mall. That changed after the Million Man March in 1995, a gathering of black men meant to show renewed commitment to family and solidarity. The park service estimated 400,000 people attended the march, making it one of the largest demonstrations in history in Washington.

But organizers believed they reached their goal of 1 million participants. Nation of Islam leader Louis Farrakhan, one of the march organizers, threatened to sue the park service, arguing its count was motivated by “racism, white supremacy and hatred for Louis Farrakhan.” Researchers at Boston University did an independent analysis and said the crowd was much bigger, pegging it at more than 800,000.

No lawsuit was filed, but the dispute was enough to get the park service out of the head-counting business. The following year, Congress passed an appropriations bill that barred the agency from spending money to count crowds. While that language didn’t appear in subsequent budgets, refusing to count crowds became park service policy.

The agency still estimates crowd size for its own planning purposes, but does not publicly reveal the figures.

“No matter what we said or did, no one ever felt we gave a fair estimate,” U.S. Park Police Maj. J.J. McLaughlin, who had been in charge of coordinating crowd estimates, said in 1996 when the agency confirmed it would no longer count heads.

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HOW CROWDS ARE COUNTED

Before it stopped, the park service came up with its crowd figures by studying aerial photographs. Using a grid system, the park service would divide the Mall into sections of equal square footage. Then it counted the number of people in each section of the grid by looking at how tightly packed the crowd was and assigning a number of people per square foot.

Not much has changed since about the way crowds are estimated, said Steve Doig, a journalism professor at Arizona State University who specializes in measuring crowd sizes.

The figure that corresponds to a “loosely packed crowd,” Doig said, is 10 square feet per person, “which sounds like a lot but it’s really not.” That means people could reach out and touch those closest to them on all sides.

“A tighter crowd would get down to maybe 6 or 7 square feet per person,” he said. “A scary mosh pit crowd would be maybe 3 to 4 at most.”

Crowd estimates for events in Washington are difficult in general because the city’s lack of tall buildings and security restrictions on aircraft make it tough to get comprehensive aerial images, Doig said.

Christopher Geldart, the District of Columbia’s homeland security director, has said consistently that he expects between 800,000 and 900,000 people to attend Trump’s inauguration. He said he would use the tried-and-true grid method to provide his own crowd estimate on the Monday after the swearing-in.

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WOMEN’S MARCH

The largest-ever demonstration in Washington, according to park service figures, was an anti-Vietnam War protest in 1969 that drew 600,000 people. The Mall can easily accommodate crowds in the hundreds of thousands; between 300,000 and 400,000 regularly attend the July Fourth fireworks.

Organizers of Saturday’s Women’s March on Washington have received a permit from D.C. police for a gathering of 200,000 people. Geldart said he expects the number of participants to exceed that figure, based on data including bus, hotel and train bookings. He cautioned, however, that organizers have not yet told authorities to expect more than the permitted total.

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PAST INAUGURATIONS

The park service did not dispute a widely reported estimate that 1.8 million people came to Washington for President Barack Obama’s first inauguration in 2009, believed to be the largest inaugural crowd in history. The biggest crowd the park service counted at an inauguration was 1.2 million for Lyndon Johnson’s 1965 swearing-in.

Doig isn’t sure about either of those figures. His analysis of aerial photographs led him to conclude that just 800,000 were on the Mall for Obama’s first inauguration, though he did not include the parade route, where thousands more people waited all day to catch a glimpse of America’s first black president.

The estimate was difficult because the density of the crowd varied widely, he said, despite aerial photographs that appeared to show the Mall full of people between the Capitol and the Washington Monument. Further away from the Capitol, people huddled around television monitors, leaving big tracts of mostly empty space.

Any time a crowd reaches into the hundreds of thousands, it’s a noteworthy gathering, Doig said, but boosters will always try to politicize the turnout for their chosen cause.

“One of the things that aggravates me … is the need for it to be huge,” he said. “It sort of drives the numbers into areas where it just clearly is fiction and it undercuts the reality of what really is an amazing crowd.”

Trump clearly cares about posting a big number.

In a video ad posted on social media Wednesday, he said: “Hopefully we’re going to get a million people. We’re going to really make a big statement.”

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Follow Ben Nuckols on Twitter at https://twitter.com/APBenNuckols

TAMPA, Fla. (AP) — The driver of famed monster truck Grave Digger is recovering following a crash at a Florida rally over the weekend.

Fan video shows the truck landing upside down while attempting to do a flip at the “Monster Jam” event at Raymond James Stadium in Tampa Saturday night. The show was stopped while medical personal responded to driver Dennis Anderson.

Monster Jam promoter Feld Entertainment says in a statement that “medical care professionals tended to Anderson on site and then transported him to a local hospital for further evaluation.”

Anderson’s son, Adam Anderson, said on Instagram Tuesday that his father is “getting better,” but didn’t offer details on the injury.

TOKYO (AP) — Caroline Kennedy stepped down Wednesday after three years as U.S. ambassador to Japan, where she was welcomed like a celebrity and worked to deepen the U.S.-Japan relationship despite regular flare-ups over American military bases on the southern island of Okinawa.

Appointed by President Barack Obama in 2013, she had been expected to leave with the coming change in U.S. leadership. President-elect Donald Trump’s transition team has also said that all envoys who were political appointees must step down by Inauguration Day on Friday. Trump has not named a new ambassador yet.

Kennedy ruffled some feathers early in her tenure by tweeting her opposition to Japan’s dolphin hunt, shortly after her embassy issued a statement expressing “disappointment” that Japan’s Prime Minister Shinzo Abe had visited a shrine that memorializes World War II war criminals, among others.

During her time, though, the conservative Abe and liberal Obama found common ground despite coming from opposite ends of the political spectrum.

“She has great skills and authority as a convener, a much needed function in U.S.-Japan relations,” said Kent Calder, the director of the Reischauer Center for East Asian Studies at Johns Hopkins School of Advanced International Studies in Washington, D.C. “She has been more of a network builder than a concrete policy initiator, but that is almost an inevitable role for ambassadors these days.”

He said her legacy includes facilitating Obama’s historic visit to Hiroshima last May, one of two Japanese cities devastated by U.S. atomic bombs in 1945. Kennedy was in Pearl Harbor at the end of last year when Abe reciprocated with a visit to the site of Japan’s 1941 surprise attack that drew America into World War II.

Some will also remember the efforts of the first female U.S. ambassador to Japan to promote literacy and women’s and LGBT rights, and for her visits to the northeast region slowly recovering from a deadly and destructive tsunami in 2011.

“She was true to the Obama administration goals, and she maintained the Kennedy mystique without making it the focal point of her tenure,” said Nancy Snow, a professor of public diplomacy at Kyoto University of Foreign Studies. “I will remember her as a champion of person-to-person exchange and engagement.”

Winning understanding in Okinawa for a reduced but still large U.S. military presence proved an impossible task, and was hampered by a series of incidents from crimes by U.S. base personnel to crashes of U.S. military aircraft.

“In every ambassadorship, there are both enduring issues and unpredictable events,” she told Japan’s largest newspaper, the Yomiuri, in a farewell interview. “In my case, both were linked to Okinawa.”

The daughter of former U.S. President John F. Kennedy arrived in November 2013 to more fanfare than the typical envoy. Thousands of onlookers lined streets to snap pictures and wave as she traveled by horse-drawn carriage to present her credentials to Japan’s emperor. The procession was broadcast live on Japan’s public broadcaster NHK.

Her popularity strained embassy resources, a 2015 U.S. government report found, because of the demands for her participation in events across the country. It noted that the embassy “has now caught up on the backlog of gifts sent to the ambassador in her first six months in Japan.”

Now 59, Kennedy is returning to her Manhattan home with husband Edwin Schlossberg, who split his time between Tokyo and New York. She hasn’t indicated publicly what her future plans are.

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Online: Kennedy’s farewell video: https://www.youtube.com/watch?v=vPcI7txv6V0&feature=youtu.be

PITTSBURGH (AP) — The father in Mike Tomlin regrets the language he used to describe the New England Patriots during the postgame speech Pittsburgh Steelers wide receiver Antonio Brown opted to livestream on social media after a taut playoff victory over Kansas City.

The coach in Tomlin has just as big an issue with one of his team’s biggest stars forcing the Steelers to talk about something other than trying to find a way to finally beat Tom Brady when it counts.

A characteristically blunt Tomlin called Brown’s decision to broadcast to the world more than 17 minutes of Pittsburgh’s giddy locker room celebration — a move that caught Tomlin using a handful of profanities — “foolish,” ”selfish” and “inconsiderate.”

“Not only is it a violation of our policy, it’s a violation of league policy, both of which he knows,” Tomlin said Tuesday.

“So there are consequences to be dealt with from his perspective. We will punish him. We won’t punish us.”

Tomlin took responsibility for his choice of words, though he was unaware of being filmed as he spoke.

During Tomlin’s post-game remarks, he attached an expletive to the Patriots, who earned a full day’s head start on preparing for the Steelers by virtue of beating Houston on Saturday night, 24 hours before Pittsburgh outlasted Kansas City 18-16.

“The responsibility associated with being in this thing, just from a role model standpoint, it’s something that I personally embrace,” Tomlin said.

“It’s something that we as a team and organization embrace. So that’s why the language, specifically, in terms of the content, is regrettable.”

So, too, is the action of the talented if sporadically diva-like Brown. The former sixth-round pick has evolved from a raw project into one of the NFL’s best wide receivers.

This season he became only the second player in league history to post four consecutive 100-catch seasons.

His work ethic is universally lauded even as his Q-rating has skyrocketed. Brown can be found on TV pitching everything from soda to credit cards to video games.

The player who — for reasons he hasn’t yet disclosed — occasionally refers to himself as “Ronald ” also has a devoted social media presence, with more than 1.5 million following on Instagram and an additional 650,000 on Facebook, many of whom got an eyeful and an earful as the Steelers celebrated their first trip to the AFC title game in six years.

While Tomlin has “very little concern” about the content of the video, he has plenty of concern over Brown’s lack of judgment.

“You wear on your teammates when they routinely have to answer questions about things that aren’t preparation or football-related,” Tomlin said.

“It’s our desire for him and for everyone to be great teammates, as well as great players. He is a great player. He is a hardworking player. He is respected, largely, in the locker room for those things. But incidences such as this don’t help him in that regard.”

The Steelers have grown accustomed to Brown’s flights of fancy, whether it’s posing for the Mannequin challenge while meeting with reporters, wearing eye-opening (and fine threatening) cleats or doing over-the-top touchdown celebrations.

The fallout this time around has been mixed. Long snapper Greg Warren said Monday that “AB is AB, he can do what he wants to do.”

Ben Roethlisberger expressed disappointment in Brown during the quarterback’s weekly appearance on 93.7 The Fan.

Either way, the fact that both were forced to respond to questions about Brown before the biggest game of the season only reinforces Tomlin’s point, though Tomlin stressed that Brown is hardly the only athlete capable of becoming a distraction.

“Those things don’t apply exclusively to Antonio,” Tomlin said. “It’s a global thing in regards to professional sports. I think that’s why often times you see great players move around from team to team.

“And I definitely don’t want that to be his story. I am sure he doesn’t want that to be his story. So, he has to address these things that put him and us in positions from time to time, in settings such as this, where it needs to be addressed.”

Tomlin didn’t outline the internal discipline Brown faces other than to say it will not affect his availability this weekend.

New England (15-2) pulled away from the Steelers (13-5) in the second half of a 27-16 victory in Pittsburgh on Oct. 23, a game Roethlisberger missed while recovering from surgery on his left knee.

Roethlisberger will be around this time. And so will Brown, who will almost certainly have his phone’s camera turned off late Sunday evening regardless of the outcome.

“He has to grow from this,” Tomlin said. “He has to.”

NOTES: LB James Harrison is dealing with shoulder and triceps injuries and could be limited early in the week. … TE Ladarius Green remains in the concussion protocol more than three weeks after taking a helmet-to-helmet hit against Cincinnati.

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For more NFL coverage: http://www.pro32.ap.org and http://www.twitter.com/AP—NFL

PARIS (AP) — Former French Prime Minister Manuel Valls has been seen getting lightly slapped on the chin by a young man during a presidential campaign stop in the western region of Brittany.

A video posted on the website of local newspaper Le Telegramme Tuesday showed Valls emerging from a building to applause from a small group of people in the city of Lamballe.

Valls walks toward the crowd shaking hands. At one point, a young man wearing a purple sweatshirt is seen shaking the Socialist candidate’s hand and then with the same hand, giving Valls a small slap on the lower face.

A bodyguard at the candidate’s side immediately clamped the man to the ground.

Valls resigned as prime minister last month to run in this month’s Socialist presidential primary.

DAYTON, Ohio (AP) — A federal lawsuit alleges that officers at an Ohio jail beat a homeless veteran so severely that he ended up in a coma for two months, now needs a wheelchair and is cognitively disabled.

Joseph Guglielmo, 59, filed the civil rights suit against Montgomery County’s commissioners, its sheriff and six employees of the sheriff’s office, the Dayton Daily News (http://bit.ly/2j4mYrs ) reported.

Sheriff Phil Plummer declined to comment on the allegations in the lawsuit.

Guglielmo, who served in the Air Force as an operating room technician and suffered post-traumatic stress disorder, was living at a Dayton homeless shelter when he was arrested in January 2015 after a disagreement with shelter employees, according to the lawsuit.

Guglielmo was booked into Montgomery County Jail and banged on his cell when a sheriff’s office employee was trying to call roll. Guglielmo claims that four corrections officers beat him in his cell and that two others who were outside did nothing to stop an employee from throwing Guglielmo against a concrete wall.

The complaint alleges that some officers stood in the way of a surveillance camera so that the beating wouldn’t be captured on video.

Two of the defendants later found Guglielmo unresponsive and he was taken to the hospital, the suit said.

“When people get booked into the Montgomery County Jail, the jailer is just supposed to keep them safe, no matter how much they’re yelling, or how disoriented they may be because of mental illness, the officers should not react with violence,” said Jennifer Branch, one of Guglielmo’s attorneys. “Instead, they should be providing the medical care.”

Documents allege Guglielmo is now unable to take care of himself and currently lives at a nursing and rehabilitation facility.

Records show he pleaded guilty to aggravated menacing. He was sentenced to 180 days in jail, with credit for two days served and the remainder of that sentence suspended.

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Information from: Dayton Daily News, http://www.daytondailynews.com