That riddle has become the latest flashpoint of feuding over Ukraine — with the nation's fledgling government and the Kremlin giving starkly different interpretations of events that could either undermine or bolster the legitimacy of the new rulers.
Ukrainian authorities are investigating the Feb. 18-20 bloodbath, and they have shifted their focus from ousted President Viktor Yanukovych's government to Vladimir Putin's Russia — pursuing the theory that the Kremlin was intent on sowing mayhem as a pretext for military incursion.
Russia suggests that the snipers were organized by opposition leaders trying to whip up local and international outrage against the government.
The government's new health minister — a doctor who helped oversee medical treatment for casualties during the protests — told The Associated Press that the similarity of bullet wounds suffered by opposition victims and police indicates the shooters were trying to stoke tensions on both sides and spark even greater violence, with the goal of toppling Yanukovych.
"I think it wasn't just a part of the old regime that (plotted the provocation), but it was also the work of Russian special forces who served and maintained the ideology of the (old) regime," Health Minister Oleh Musiy said.
Putin has pushed the idea that the sniper shootings were ordered by opposition leaders, while Kremlin officials have pointed to a recording of a leaked phone call between Estonia's foreign minister and the European Union's foreign policy chief as evidence to back up that version.
This much is known: Snipers firing powerful rifles from rooftops and windows shot scores of people in the heart of Kiev.
Some victims were opposition protesters, but many were civilian bystanders clearly not involved in the clashes.
Among the dead were medics, as well as police officers.
A majority of the more than 100 people who died in the violence were shot by snipers; hundreds were also injured by the gunfire and other street fighting.
On Tuesday, Interior Minister Arsen Avakov signaled that investigators may be turning their attention away from Ukrainian responsibility.
"I can say only one thing: the key factor in this uprising, that spilled blood in Kiev and that turned the country upside down and shocked it, was a third force," Avakov was quoted as saying by Interfax.
"And this force was not Ukrainian."
The next day, Prosecutor General Oleh Makhntisky said officials have found sniper bullet casings on the National Bank building a few hundred yards up the hill from Maidan, the square that became the center and the symbol of the anti-government protests.
He said investigators have confirmed snipers also fired from the Hotel Ukraine, directly on the square, and the House of Chimeras, an official residence next to the presidential administration building.
Deputy Interior Minister Mykola Velichkovych told AP that commanders of sniper units overseen by the Berkut police force and other Interior Ministry subdivisions have denied to investigators that they had given orders to shoot anyone.
Musiy, who spent more than two months organizing medical units on Maidan, said that on Feb. 20 roughly 40 civilians and protesters were brought with fatal bullet wounds to the makeshift hospital set up near the square.
But he said medics also treated three police officers whose wounds were identical.
Forensic evidence, in particular the similarity of the bullet wounds, led him and others to conclude that snipers were targeting both sides of the standoff at Maidan — and that the shootings were intended to generate a wave of revulsion so strong that it would topple Yanukovych and also justify a Russian invasion.
Russia has used the uncertainty surrounding the bloodshed to discredit Ukraine's current government.
During a news conference Tuesday, Putin addressed the issue in response to a reporter's question, suggesting that the snipers in fact "may have been provocateurs from opposition parties."
That theory gained currency a day later when a recording of a Feb. 26 private phone call between Estonian Foreign Minister Urmas Paet and European Union foreign policy chief Catherine Ashton was leaked and broadcast by the Russian government-controlled TV network, Russia Today.
In the call, Paet said he had heard from protesters during a visit to Kiev that opponents of Yanukovych were behind the sniper attacks.
Paet said another physician who treated victims, Dr. Olha Bohomolets, told him that both police and protesters were killed by the same bullets — and "there is now stronger and stronger understanding that behind snipers it was not Yanukovych, but it was somebody from the new (government) coalition."
On Wednesday Paet confirmed the recording was authentic, and told reporters in Tallinn that he was merely repeating what Bohomolets had told him.
He said he had no way of verifying the claims, though he called Bohomolets "clearly a person with authority."
Bohomolets couldn't be immediately reached by the AP for comment.
She did not answer repeated calls to her cellphone or respond to text messages.
In an interview earlier this week with a correspondent from British newspaper The Telegraph, Bohomolets said she didn't know if police and protesters were killed by the same bullets, and called for a thorough investigation.
"No one who just sees the wounds when treating the victims can make a determination about the type of weapons," she was quoted as saying.
"I hope international experts and Ukrainian investigators will make a determination of what type of weapons, who was involved in the killings and how it was done. I have no data to prove anything."
On Thursday, Russia's U.N. envoy said he discussed the leaked phone call during a closed-door meeting of the U.N. Security Council.
If the call represents the truth, Ambassador Vitaly Churkin told reporters, "it is hard to imagine how such a parliament ... can be regarded as a legitimate parliament that can pass legitimate decisions on the future of Ukraine."
A former top security official with Ukraine's main security agency, the SBU, waded into the confusion, in an interview published Thursday with the respected newspaper Dzerkalo Tizhnya.
Hennady Moskal, who was deputy head of the agency, told the newspaper that snipers from the Interior Ministry and SBU were responsible for the shootings, not foreign agents.
"In addition to this, snipers received orders to shoot not only protesters, but also police forces. This was all done in order to escalate the conflict, in order to justify the police operation to clear Maidan," he was quoted as saying.
One of the victims of the snipers was Alexander Tonskikh, 57.
He told AP that at around 10 a.m. on Feb. 20, he and dozens of opposition fighters moved south out of the main battleground on Maidan.
Riot police withdrew suddenly, he said, and an instant later snipers began firing from at least two different directions, from what seemed to be the rooftops of government buildings, between 200 and 300 yards away.
He said dozens of people were "mown down like grass" as he and others crouched behind a waist-high stone wall, holding wooden clubs and metal riot shields.
At least 10 people, he said, were killed instantly, and many others wounded.
The bodies piled up on top of each other like fallen tree branches.
Shooting then began from a third direction, he said.
As he crouched with his back to a tree, he was hit by a bullet that entered his right arm, went through his right side, punctured his lung and lodged just below his heart.
He then lost consciousness.