Image copyrightREUTERSImage captionThe 12-year-old earlier claimed she did not tell her family, fearing retaliation
A 12-year-old African-American girl who accused three white peers of cutting off her dreadlocks has recanted the allegations, according to the school.
Amari Allen last week claimed the boys had insulted her hair and attacked her.
Her family apologised "for the pain and anxiety these allegations have caused" in a statement given to the BBC by the Immanuel Christian School.
The case made national headlines as the US vice-president's wife teaches part time at the school.
A spokeswoman for the Immanuel Christian School in Springfield, Virginia, told the BBC that officials met the Allen family on Monday morning.
Ms Allen acknowledged the claims she made were false.
"To those young boys and their parents, we sincerely apologise for the pain and anxiety these allegations have caused," the Allen family said in a statement.
They apologised to the school for "the damage this incident has done to trust within the school family and the undue scorn it has brought".
Media captionSpeaking last week, the girl said she was held down as her dreadlocks were cut
"To the broader community, who rallied in such passionate support for our daughter, we apologise for betraying your trust. We understand there will be consequences, and we're prepared to take responsibility for them.
"We know that it will take time to heal, and we hope and pray that the boys, their families, the school and the broader community will be able to forgive us in time."
The head of the school, Stephen Danish, said in a statement that Immanuel Christian was "relieved to hear the truth" but felt "tremendous pain for the victims and the hurt on both sides of this conflict".
"This ordeal has revealed that we as a school family are not immune from the effects of deep racial wounds in our society," Mr Danish said.
"We view this incident as an opportunity to be part of a learning and healing process, and we will continue to support the students and families involved."
Image copyright
When Ms Allen's grandmother noticed her hair was suddenly at different lengths on 25 September, Ms Allen claimed that three white classmates attacked her during break time.
She alleged the boys "ambushed" her, covered her mouth, held her down on a slide and began cutting off her hair.
Ms Allen also told local media that the boys called her hair "ugly" and stole her lunch, telling her to "just starve".
Local police and the school were investigating the incident.
The school, which costs Ms Allen's family almost $12,000 (£9,800) a year, is where Karen Pence - the wife of Vice-President Mike Pence - teaches art part-time.
Mrs Pence resumed working at the school earlier this year, amid scrutiny over its anti-LGBT requirements of staff.
Media captionFootage of the shooting shows the police officer fire one bullet at close range
An activist involved in anti-government protests in Hong Kong has been shot in the chest by police during a clash.
The incident came as thousands of people demonstrated in defiance of a protest ban for the 70th anniversary of Communist rule in China.
Though people have been shot by rubber bullets in previous protests, this is the first injury from a live round.
In the Chinese capital Beijing, 15,000 soldiers paraded with advanced military hardware to mark the anniversary.
Nearly four months of protests in Hong Kong have challenged Chinese President Xi Jinping's vision of national unity.
Earlier, the Chinese flag was raised at a special ceremony in the territory. Security was tight and the 12,000 invited guests watched the event on a live video feed from inside a conference centre.
On what is being described by protesters as a "day of grief", people took to the streets in central Hong Kong and at least six other districts, blocking roads in some areas.
Image copyrightAFPImage captionPolice detain demonstrators in Hong Kong's Sha Tin district
At least 51 people were injured during the clashes, as police fired tear gas and protesters threw petrol bombs. Two people are in a critical condition and two others are seriously injured, the Hospital Authority said.
According to local media, at least 96 people were arrested.
What happened to the man who was shot?
The man, who is aged 18, has been undergoing surgery and his condition is not known. Police say he was hit "near his left shoulder".
A video shows the injured man lying on the ground, saying: "Send me to hospital. My chest is hurting, I need to go to hospital."
Video of the shooting shows a police officer opening fire with a revolver at a man swinging what appears to be a metal pole, as a group of protesters confront riot police.
Image copyrightAFPImage captionA protester wearing a Guy Fawkes mask was seen holding a Chinese national flag that had been coloured black
Police say their officers were attacked and the policeman who opened fire did so as he "felt his life was under serious threat".
"He fired a round at the assailant to save his own life and his colleagues' lives," Senior Supt Yolanda Yu Hoi-kwan was quoted as saying by the South China Morning Post.
"The police force really did not want to see anyone being injured, so we feel very sad about this," she added.
Police tried to disperse the protesters - some of whom were armed with petrol bombs, projectiles or poles - with tear gas, rubber bullets and water cannon, the latter spraying blue dye to make it easier to identify them later.
A visual guide to how one peaceful protest turned violent
At least 15 metro stations and numerous shopping centres in the city were closed, and some 6,000 officers have been deployed in the territory.
A day of mayhem
Tessa Wong, BBC News, Hong Kong
Protests in Hong Kong by now follow a familiar rhythm - peaceful mass marches which then slide into violence. What stood out today however was the swiftness with which the violence exploded.
Across many parts of Hong Kong, protesters fought pitched battles with police, with one clash culminating in a young man getting shot by a live round. Authorities appeared to show a new determination to put down the protests as quickly as possible on China's National Day, perhaps to save face, but it inevitably only angered the protesters even more.
On the streets of Hong Kong Island's dense downtown, from Admiralty to Causeway Bay, protesters played cat and mouse with police, setting a trail of fires in their wake. Police relentlessly attempted to box them in by sending water cannon and elite forces down main roads, while smaller teams charged up side streets.
But at times the tactics on both sides made little sense and threatened to spin out of control. Petrol and smoke bombs alike were sometimes hurled inexpertly, seriously endangering reporters and passers-by.
In Admiralty, I saw protesters armed just with umbrellas repeatedly attempting to charge up an escalator to an overhead bridge where riot police were stationed. After shouting warnings, police let loose a barrage of tear gas directly at the protesters, shot at extremely close range. Then in Wan Chai, protesters decided to set a bonfire right in front of a petrol station that was sandwiched between residential blocks - it was quickly put out.
Dusk settled; the mayhem continued. I'd started the day with a peaceful, festive mass march that began in Causeway Bay, and hours later found myself right back where I'd started, this time dodging fires and running from police, following that Hong Kong protest rhythm to its inevitable bitter end.
What is the background to this?
Hong Kong has been a part of China since 1997 but has its own system of law and government - known as One Country Two Systems.
In recent years, there has been increasing opposition to what has been seen as the growing influence of Beijing on Hong Kong's society and politics.
Media captionHow Hong Kong got trapped in a cycle of violence
Hong Kong always sees anti-Beijing protests on 1 October, and this year they were expected to be larger than ever, because of the months of unrest triggered by proposed changes to the extradition law.
The changes would have made it possible for China to extradite people to the mainland from Hong Kong, something opponents felt put Hong Kongers at risk of persecution in unfair trials.
The proposed changes to the extradition law have officially been scrapped but that has failed to quell the unrest, which has now evolved into an existential battle over Hong Kong's future.
Quick guide to HK protests
The Hong Kong protesters are mostly young people worried about their future. Their demands have now expanded to include genuine universal suffrage and an inquiry into police behaviour.
The protests started peacefully, but as the weeks have worn on, the violence has escalated. The alleged involvement of gangs and undercover police officers has led to fear, mistrust and suspicion.
Hong Kong is part of China, but enjoys "special freedoms". Those are set to expire in 2047, and many in Hong Kong do not want it to become "another Chinese city".
How was the anniversary celebrated in mainland China?
Speaking from a podium in Beijing's Tiananmen Square, President Xi said no force could stop China's onward march.
Media captionChina at 70: Tanks parade on Tiananmen Square
Dressed in a Mao suit, he watched the parade which featured some military hardware on show for the first time.
The highly choreographed event presented an image of national unity and gave no hint at the difficulties faced by China's Communist leaders during the last seven decades, or their current problems.
Globetrotting Prince Harry has renewed his global call for increased climate consciousness in Africa, appealing on Monday for more efforts to be made to protect the environment against human “greed, apathy and selfishness.”
Says the man who lives in a mansion, does not fly commercial, has a wedding that created more CO2 than a small city...Et tu Al Gore?
Keep these in mind as you contemplate the direction of the American government over the past 50 years and especially since the Obama election.
The Goals of Communism
(as read into the congressional record January 10, 1963, from "The Naked Communist" by Cleon Skousen)
1. U.S. acceptance of coexistence as the only alternative to atomic war.
2. U.S. willingness to capitulate in preference to engaging in atomic war.
3. Develop the illusion that total disarmament of the United States would be a demonstration of moral strength.
4. Permit free trade between all nations regardless of Communist affiliation and regardless of whether or not items could be used for war.
5. Extension of long-term loans to Russia and Soviet satellites.
6. Provide American aid to all nations regardless of Communist domination.
7. Grant recognition of Red China. Admission of Red China to the U.N.
8. Set up East and West Germany as separate states in spite of Khrushchev's promise in 1955 to settle the German question by free elections under supervision of the U.N.
9. Prolong the conferences to ban atomic tests because the United States has agreed to suspend tests as long as negotiations are in progress.
10. Allow all Soviet satellites individual representation in the U.N.
11. Promote the U.N. as the only hope for mankind. If its charter is rewritten, demand that it be set up as a one-world government with its own independent armed forces. (Some Communist leaders believe the world can be taken over as easily by the U.N. as by Moscow. Sometimes these two centers compete with each other as they are now doing in the Congo.)
12. Resist any attempt to outlaw the Communist Party.
13. Do away with all loyalty oaths.
14. Continue giving Russia access to the U.S. Patent Office.
15. Capture one or both of the political parties in the United States.
16. Use technical decisions of the courts to weaken basic American institutions by claiming their activities violate civil rights.
17. Get control of the schools. Use them as transmission belts for socialism and current Communist propaganda. Soften the curriculum. Get control of teachers' associations. Put the party line in textbooks.
18. Gain control of all student newspapers.
19. Use student riots to foment public protests against programs or organizations which are under Communist attack.
20. Infiltrate the press. Get control of book-review assignments, editorial writing, policymaking positions.
21. Gain control of key positions in radio, TV, and motion pictures.
22. Continue discrediting American culture by degrading all forms of artistic expression. An American Communist cell was told to "eliminate all good sculpture from parks and buildings, substitute shapeless, awkward and meaningless forms."
23. Control art critics and directors of art museums. "Our plan is to promote ugliness, repulsive, meaningless art."
24. Eliminate all laws governing obscenity by calling them "censorship" and a violation of free speech and free press.
25. Break down cultural standards of morality by promoting pornography and obscenity in books, magazines, motion pictures, radio, and TV.
26. Present homosexuality, degeneracy and promiscuity as "normal, natural, healthy."
27. Infiltrate the churches and replace revealed religion with "social" religion. Discredit the Bible and emphasize the need for intellectual maturity which does not need a "religious crutch."
28. Eliminate prayer or any phase of religious expression in the schools on the ground that it violates the principle of "separation of church and state."
29. Discredit the American Constitution by calling it inadequate, old-fashioned, out of step with modern needs, a hindrance to cooperation between nations on a worldwide basis.
30. Discredit the American Founding Fathers. Present them as selfish aristocrats who had no concern for the "common man."
31. Belittle all forms of American culture and discourage the teaching of American history on the ground that it was only a minor part of the "big picture." Give more emphasis to Russian history since the Communists took over.
32. Support any socialist movement to give centralized control over any part of the culture--education, social agencies, welfare programs, mental health clinics, etc.
33. Eliminate all laws or procedures which interfere with the operation of the Communist apparatus.
34. Eliminate the House Committee on Un-American Activities.
35. Discredit and eventually dismantle the FBI.
36. Infiltrate and gain control of more unions.
37. Infiltrate and gain control of big business.
38. Transfer some of the powers of arrest from the police to social agencies. Treat all behavioral problems as psychiatric disorders which no one but psychiatrists can understand.
39. Dominate the psychiatric profession and use mental health laws as a means of gaining coercive control over those who oppose Communist goals.
40. Discredit the family as an institution. Encourage promiscuity and easy divorce.
41. Emphasize the need to raise children away from the negative influence of parents. Attribute prejudices, mental blocks and retarding of children to suppressive influence of parents.
42. Create the impression that violence and insurrection are legitimate aspects of the American tradition; that students and special-interest groups should rise up and use united force to solve economic, political or social problems.
43. Overthrow all colonial governments before native populations are ready for self-government.
44. Internationalize the Panama Canal.
45. Repeal the Connally reservation so the United States cannot prevent the World Court from seizing jurisdiction over nations and individuals alike.
REUTERS
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