A month later, the Civil Human Rights Front, which organises the July 1 protest march to mark the return of Hong Kong to China, put the attendance number at 110,000. Police counted 19,300 protesters at the peak of the rally.

Crowd size was at the centre of the Trump administration’s first stormy White House media briefing last Saturday. It was debated on social media, amid an array of side-by-side images and accusations of doctored photographs.

Why is estimating crowd size so difficult – and so contentious?

History tells us disputes over crowd estimates are nothing new. The politicisation of crowd sizes in the United States has its roots partly in the Million Man March on the Washington Mall by African Americans in 1995. At the time, the National Park Service, which oversees the Mall, gave official crowd size estimates for all major events that took place there.

After that march, the agency released figures that put the crowd at roughly 400,000. That estimate rankled organisers, who threatened to file a lawsuit for defamation. In 1996, Congress banned the agency from using its funds to count crowds, and the parks service has since refrained from doing so.

Part of the reason crowd-size estimates can be so contentious is that it is one of the few metrics available to gauge an event’s popularity, real or perceived, says Steve Doig, a data journalism professor at Arizona State University.

“I think it’s always meant ... the same thing: the size of my crowd is a measure of how wonderful I am or how wonderful my movement is,” Doig says. “Every political rally of any kind ... one of the first things that anybody who is involved in it wants to do is claim a large and enthusiastic crowd. The counter-argument is often, ‘No, it wasn’t really that big’.”

Doig has researched and taught crowd estimation for several years, but first became interested in the subject while working as a reporter in the mid-1980s, when he was assigned to cover an annual street festival.

“Every year the organisers were claiming it got bigger and bigger and bigger until it got ridiculous,” Doig says. “Half of Miami would have had to be there.”

Realising he could not rely solely on crowd estimates released by organisers, Doig began researching ways for journalists to independently and accurately do so.

He found one of the most commonly used approaches was the “Jacobs Method”, devised in the 1960s by Herbert Jacobs, a journalism professor whose office was in a tower that overlooked a plaza where student protests took place. Conveniently, the plaza was divided by markings on the ground so that it appeared as a grid from above.

“It was like standing on graph paper,” Doig says of the method, not unlike that used by Lee for the Post – but without the use of computers.

From his vantage point, it was easy for Jacobs to estimate how many students were in each grid, calculate an average and then multiply that average by the number of grids to arrive at a rough estimate.

“Measure the area that a crowd is in and then divide by whatever your reasonable density estimate is,” Doig says. “And that produces a reality-based estimate of how many people are likely to be there.”

For the most part, the boundaries of the crowd are stable; the density of the crowd is where there is some room for interpretation. In a loose crowd, where people are about an arm’s length away from each other, Doig estimates about 10 sq ft per person. In a very tight crowd, where people are essentially “back to chest”, Doig allows about five sq ft per person. Anything less is not really believable, he says.

“There are bad estimates where I’ve seen one square foot per person,” Doig says. Even the stampedes in Mecca, where people were crushed in 2015, didn’t reach that kind of density, he adds.

Washington comes with its own challenges when it comes to crowd estimation. In 2015, a newspaper report showed that perspective mattered greatly when estimating crowds on the Mall. When viewing a crowd from a low angle – from the inaugural dais, for example – the spaces between people can easily visually disappear, causing the crowd to look less spaced out.

Because of this, the most accurate way to capture a crowd is to take an image from straight overhead. However, much of the city is under a no-fly zone, preventing aircraft and most aerial photographers from doing so.

About 15 years ago, Curt Westergard and a team developed a tethered, bagel-shaped balloon about four metres in diameter that could record aerial images of large gatherings, ideally from about 27 metres above the ground.

Westergard, the owner of Digital Design and Imaging Service, has since provided crowd estimates for a variety of major events, including last week’s Women’s March on Washington.

When analysing aerial images afterwards, instead of using grids to estimate attendance, Westergard uses computer programs to draw polygons – irregular shapes – around clusters of people with similar density. While he acknowledges “grids work”, Westergard says an irregular polygon can often be more accurate, because crowd density is not always consistent within a grid.

Trump has been known to care, deeply, about the size of his crowds. Throughout his campaign, the mogul frequently inflated the attendance at his campaign rallies: after a stop in Phoenix last July, for instance, Trump claimed he had addressed anywhere from 15,000 to 20,000 people there – even when the local fire department asserted capacity at the event space was capped at 4,200 people.

Before his inauguration, Trump had been vamping up expectations for record crowds at his swearing-in ceremony.

“Inauguration Day is turning out to be even bigger than expected,” he tweeted on January 14.

“Hopefully we’re going to get a million people,” he said later in a video ad. “We’re going to really make a big statement.”

Regarding the Trump inauguration, Metro released figures that said 570,557 people took trips in the underground train system between its 4am opening on Friday it closed at midnight. That compared with 1.1 million trips for Obama’s 2009 inaugural and 782,000 in 2013, according to Metro.

The day following Trump’s inauguration, hundreds of thousands of women descended upon the streets of the US capital in a mass rally for women’s rights that some described also as a mass rebuke of Trump.

The Washington organisers had sought a permit for 200,000 for the gathering. On Saturday, they said as many as a half million participated, dwarfing Friday’s inaugural crowd.

It was only a matter of time before a crowd-related response from Trump and his team was issued. On Saturday afternoon, Trump claimed that, from his view on the dais, attendance at his swearing-in “looked like a million, a million and a half people”.

Using his methods, Doig estimates that the crowd size for Trump’s inauguration was actually about a third of the size of Obama’s 2009 inauguration.

Crowd counting scientists came to the same conclusion in an analysis of photos and videos of the Mall from both events.