How many people were actually protesting against far-right extremism across Germany over the last few days? The numbers may range from hundreds of thousands to 1.4 million depending on where you look or who you ask.

Protester numbers also varied greatly in specific cities: In Berlin, organizers spoke of 350,000 attendees, whereas police calculated over 100,000. In Munich, authorities said 100,000 had attended, while organizers said more than twice as many people turned up.

These discrepancies have raised questions about the accuracy of such estimates, especially among social media users and far-right politicians.

And during heightened tensions in Germany, they have become more than just numbers. They are being used as part of a political message to promote or discredit the significance of these protests.

Why have these numbers become so controversial?

Far-right politicians and commentators have used these discrepancies to delegitimize demonstrations, particularly one in the northern German city of Hamburg that was ended prematurely because of overcrowding.

Local police said around 30,000 people were present at the event's start, while the AFP news agency later reported about 50,000. Organizers spoke of up to 80,000. And far-right politicians were quick to criticize these variations and suggest manipulation.

Georg Pazderski, a member of the far-right Alternative for Germany (AfD) party, cited different reports about the protest, with participant estimates that ranged from 30.000 to 130.000. He then said that "figures and images are being falsified" and spoke of a "staging against the right."

Pazderski was not the only far-right politician to comment on the number of protesters, and DW's fact-checking team analyzed and debunked similar claims made this week.

So, why do numbers vary so much?

 Figures police and organizers offer for demonstration often differ — albeit not always to such a degree — and the demonstrations against the far right were no exception.

For instance, Berlin's police spokesperson Martin Halweg told the daily newspaper Der Tagesspiegel that discussions about these divergences "is almost an everyday occurrence" in the German capital.

But in this particular political and social context, the problem has become more evident in part because of the sheer number of protests nationwide. In only a few days, demonstrations were called in over 100 locations across Germany, with full squares and streets in major cities and plenty of people coming and going.

This makes it particularly hard for authorities and organizers to determine how many people turned up accurately. In fact, experts have stressed that these figures are no more than basic estimates or rough calculations based on different methods.

Particular interests might also play a role when determining numbers. While organizers tend to be more optimistic about attendance, police estimates are often more cautious, sometimes even "underestimating" participant numbers. These differences ultimately play a role in how successful a protest appears. 

Studies like this one from the Konrad Adenauer Foundation have shown that "for organizers of political protest events, comparative quantification is always a form of success control." 

How are protester figures calculated?

One method for large demonstrations is to analyze aerial photographs taken from helicopters or drones. 

An average number of people per square meter can be calculated, and the figure is then extrapolated to the total area. Software programs can also be used to check video footage.

Another way is by manually counting people. Fridays for Future, which helped organize the anti-far-right protest in Berlin, positioned helpers in specific locations to calculate the number of participants, according to Der Tagesspiegel. Then, they used maps to come up with an overall figure.

But as the movement's spokesperson Samira Ghandour said, "there was no beginning and no end," with people being diverted to side streets.

This shows the fundamental challenges of estimating protest attendance: when, where and how to make those calculations.

A different result would probably be obtained if the count is made at the beginning of a demonstration or in the middle when more people have gathered. Numbers will also vary if the focus is on the central protest location or if people on the sidelines are included.

There is not one unique way of dealing with these challenges, but experts have warned that it's essential to understand and communicate that it's an imprecise activity.

One specialist, Stephan Poppe, from the University of Leipzig, recommended not using exact figures but instead providing a range. He told broadcaster Bayerischer Rundfunk that this would be "more honest" and "important for the political discourse."

More protests against far-right extremism are planned in Germany shortly, so this debate about how and when to count participants will likely continue.

Silja Thoms contributed to this report.